Mathematics education
An inventory
Ben Wilbrink
See also the special page on word problems, the prototype question for everything that can go wrong in designing achievement test items.
My intention in creating these 'education pages' is to assemble materials from several disciplines to investigate how they are handling common sense ideas, folk ideas, naive ideas, whatever they might get called, that are inconsistent with the scientific ideas in that particular discipline. The prime example is the folk physics of pupils that is frustrating their learning the classical mechanics of Newton, while most programs or teachers do not explicitly handle this problem, or even are aware of it. While this kind of problem evidently is frustrating the efficiency of education, it also touches on what is valid assessment of knowledge of physics. Designing physics tests should touch on this issue.
There is a flipside to this kind of issue: there are also intuitions etcetera that are consistent or might be regarded as consistent with scientific ideas. They could be of great significance in education, because they might make it possible to introduce complex ideas much earlier, much simpler. Among others Andrea DiSessa is running some projects along this line, in matheducation. For a more general approach to research on intuitions see for example the work of Gerd Gigerenzer (site).
One reason to have separate pages for clusters of disciplines is the following. In primary education the two most important subjects are language and arithmetics. The difference between these disciplines is that the first one comes hardwired in the brain, while the second one has to be built by the student almost from scratch by exercise. It is really amazing how enormous this difference is, even without my slight exaggeration, and yet schools and teachers spend almost the same amount of time on language as on arithmetics. Apart from that, it seems evident that designing test items must be rather different in these two domains. I do not yet know if such necessarily is the case.

mother language Comes with mother, prewired in the brain, already well exercised at the time of school entry. special techniques needed: reading, writing, grammar, sytactics. The relation between language development as such, and proficiency in the techniques mentioned, might be rather strained in daily school practice

second language Learning a second language should be possible in a natural way, to begin with. Exercise must be important here. Some special techniques will be needed, for instance spelling. How should obne learn grammatical and syntactical rules: in a natural way (how do you learn to ride a bike?) or in an artificial way (first learn the rules by heart, then practice them?)

arithmetics, mathematics This is an artificial discipline, in the sense that it has to be learned in school mainly, while our brain is not particularly adapted to it in the way it is to the learning of language.

physics This is a marriage between our natural environment as we experience it, scientific method, and mathematics. This must be a difficult discipline to master in an appropriate way, and so it has proved to be

history What kind of discipline is this? Listening to stories, reproducing those stories? Or is it to actively construct an understanding of the world as it is related to past events? Partly natural (finding one's way in society), partly cultural (discovering andassimilating knowledge of past events)

humanities Kind of a combination of scientific method and the characteristics of history? Some mathematics, of course. Our social and cultural environment as we experience it?

life sciences Kind of a combination of the characteristics of physics and history?
Alan Schoenfeld articulates exactly what kinds of problems are manifest in assessing mathematical proficiency. Take a few minutes off, to skim the chapter pdf. Thanks. Knowing the kinds of problems involved, you are better prepared to absorb the information in this webpage.
Alan H. Schoenfeld (2007) Issues and Tensions in the Assessment of Mathematical Proficiency. In Alan H. Schoenfeld:. Assessing mathematical proficiency (pp 321). Cambridge University Press. pdf
Never take anything for granted
2 + 2 = 4 /
2 apples + 2 apples = 4 apples /
2 apples + 2 persons = ??? /
2 cm + 2 cm = 4 cm? /
2°C + 2°C = .. °C?
The clumsy example is my own. The sum in cm is contextual, the sum in °C highly so. This example was triggered by Catherine Sophian's (2007) emphasis on the units referred to by particular numbers (see her book, mentioned below), in a way turning arithmetics into 'physical arithmetics'. The sum in °C, of course, is a variant on the physical experiment of mixing water of 0°C and 100°C, definitely an adding operation, yet the result is (better: should theoretically be) 50°C, not 100°C (on inventing temperature see the book by Hasok Chang (2008) under that title here. It is different, though, because one can't possibly tell what the answer of 2°C + 2°C = .. °C? should be, without knowing the context or intention the questioner had in mind. Research into word problems (Lieven Verschaffel, a.o.) does not seem to touch on the kind of problem in this example. My sum in °C is definitely not in the same class as that of the 'What is the age of the captain' problem.
There is a deep problem involved in the case in the box above.
Can this be true? Wat is mathematical truth anyway?
2/3 + 3/4 = 5/7
Most teachers would recognize this as a familiar kind of mistake. Is it necessarily a mistake? Can you think of an adequate justification given by the student?
By the way, always grant your students the opportunity to justify their answers, especially on MC items and short answer items. Better still: ask them to always do so in nontrivial cases.
Morris Kline, 1980, p. 94. For the answer, see here.
Arithmetics as taught in our schools, as a cultural artefact, is not quite in touch with the world at large. Might not this be somewhat confusing to young minds trying to grasp what their teachers are after? What is the particular slice of reality their teachers are talking about? Think of Euclid's world of points and lines; where does it make contact with the world as we experience it in daily life (does Robin Hartshorne, 2000, comment on this issue? (here). Morris Kline does, see his 1980, p. 95)? At the level of mathematical analysis, Rafael Núñez sees inconsistencies between mathematician's ways of talking about or explaining mathematics as movements, and the thorougly static character of rigorous mathematics itself: nothing moves at all (here). Strange things happening everywhere. Think of it: naive arithmetics and school arithmetics have in common that they are predicated on the bying and selling of goods in the market place, as shown by arithmetics books in the seventeenth century, such as the Dutch one by Bartjens (see pictures cyfferinge). Important as that is, there are lots of other things and events that do not behave in the 2 + 2 = 4 counting fashion. Economics is an example. Taking stock of happiness is not the same as taking stock of financial assets. Welfare economics is an example. Comparing quantities is more basic than counting them, why then is it that in eduation we disregard the comparison of quantities? By the way, Catherine Sophian (2007) doesn'tbelow.
Suppes and Zinnes do not (take anything . . .. )
". . . as elementary science students we are constantly warned that it 'does not make sense' (a phrase often used when no other argument is apparent) to add numbers representing distinct properties, say, height and weight. Yet as more advanced physics students we are taught, with some effort no doubt, to multiply numbers representing such things as velocity and time or to divide distance numbers by time numbers. Why does multiplication make 'more sense' than addition?"
Patrick Suppes and Joseph L. Zinnes (1963). Basic measurement theory. In R. D. Luce, R. R. Bush, and E. Galanter: Handbook of mathematical psychology. Volume 1 (the quote is from its opening paragraph). Wiley. Reprinted in Bernhardt Lieberman (Ed.) (1971). Contemporary problems in statistics. A book of readings for the behavioral sciences (3974). London: Oxford University Press. pdf
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Patrick Suppes, in the source quoted in the box, is not in the business of teaching arithmetics to primary school kids (elsewhere he is, though, in the Stanford projects on developing courseware for arithmetics). Yet children find themselves confronted with some of the problems that Suppes and Zinnes discuss at a very fundamental level. Small kids having very fundamental problems, is that possible? Yes, it is. Conceptual learning is rather difficult, notwithstanding that it is a natural thing to do. It is especially difficult in formal territories such as arithmetics and mathematics in general. We had better be aware of these difficulties. Catherine Sophian (2007), and a host of other researchers mentioned in this web page, are aware of at least some of these problems, and are developing adequate instructional strategies to handle them.
Suppes treats the subject of representing parts of the world in numbers and structures of numbers comprehensively in his 2002.
Patrick Suppes (2002). Representation and invariance of scientific structures. MIT Press. Yes, it is online, it is a very large pdf: pdf 8 Mb
Oliver Byrne (1847/2010). The First Six Books of the Elemensts of Euclid with coloured diagrams and symbols. London: William Pickering. Taschen facsimile reprint. isbn 9783836517751, boxed with Werner Oechslin’s Essay Byrne’s edition page by page html
 The figure of his page ix is on the website referred to. I do not know whether his claim about the efficiency of his teaching method is true (see the middle of the page, if you can’t read it, click on it). Page 134 is a scan on page http://www.fulltable.com/vts/b/byrne/101.jpg
 For another colourful way to present the books of Euclid, see David Joyce, below, and his website)
Stephan Körner (1960/1968/1986). The Philosophy of Mathematics. Dover.
 some older views
 m. as logic
 m. as the science of formal systems
 m. as the activity of intuitive constructions
 the nature of pure and applied mathematics
The last chapter is of special interest. The first section is on ‘exact and inexact conceptsrsquo;. Of course, this is philosophy of mathematics, not psychology of doing mathematics. It might be useful, nevertheless, to be clear about some basic issues regarding what pure mathamatics is as differentiated from applied mathematics. Or is there no such demarcation?
Morris Kline (1980). Mathematics. The loss of certainty. Oxford University Press.
 p. 94. 2/3 and 3/4 are batting averages in two games. The batting average over two games then is 5/7, not 1 and 5/12. Also, a batting average of 4/6 is not the same as an average of 2/3. Some fractions might be averages.
 p. 95: "Thus the sad conclusion which mathematicians were obliged to draw is that there is no truth in mathematics, that is, truth in the sense of laws about the real world. The axioms of the basic structure of arithmetic and geometry are suggested by experience, and the structures as a consequence have a limited applicability. Just where they are applicable can be determined only by experience. The Greeks' attempt to guarantee the truth of mathematics by starting with selfevident truths and by using only deductive proof proved futile."
 p. 92: See also Hermann von Helmholtz (1887/1930). Zahlen und Messen, in Philosophische Aufsätze. / Counting and measuring. Translated by Charlotte Lowe Bryan, with an Introduction and Notes by Harold T. Davis. New York, Van Nostrand, 1930. [not yet available on the www, as of nov 2008?]
Mathematics—I want to say—teaches you, not just the answer to a question, but a whole languagegame with questions and answers.
May 2008 I bought a nice antiquarian copy, opened it, and the first thing I read was, believe it or not, the passage cited here. The book, of course, is:
Ludwig Wittgenstein (1956/1964). Bemerkungen über die Grundlagen der mathematik. Remarks on the foundations of mathematics. Oxford: Basil Blackwell. p. 173e.
The inventory will contain studies, web pages etc. that in one way or another might touch on the topic of designing mathematics/arithmetic test items. The literature on mathematics teaching is quite extensive, I will use the principle of serendipity in regard of the literature in general: items in my library or that I have seen otherwise.
September 2007 I do need a kind of basic position or outlook in tackling mathematics education and its methods of assessment. First and foremost: I will not consider as such the graduate study of mathematics. For my item design purposes mathematics will be the discipline that is used by other disciplines in one way or another, or used by normal people in daily life in one way or another. One important way mathematics is used is in teaching mathematics, and therefore also in learning mathematics. Other sciences than mathematics proper will have significant things to say about the uses made of mathematics, and some mathematicians do not like that idea.
This position in no way detracts from the essence or the importance of mathematics. As you might have seen already in the introduction to my physicseducation.htm page, mathematics might be the only instrument available to describe the phenomena that a particular discipline studies. For example in the case of free fall in physics: the only way to 'understand' free fall is through the mathematical desciption that is adequate to its phenomena, simply because there is not a mechanistic 'cause' of free fall that might explain free fall. Does this sound a bit involved? That does not matter, I will in this page as well as in 'Designing test items' return to this kind of issue again and again.
See also my web page Wiskunde in de overgang van vwo naar wo.
Peter Lehrer mp3
Aha
[December 2007]
Having searched the math education literature for over a year now, I am stuck with a lot of interesting material, without the crucial insight into the nature of (the relation between) mathematics, education, and learning that I was looking for. And then there is this chapter by Rafael Núñez (2007 pdf), containing the exact ingredients of the analysis that is pertinent to the place of mathematics in education. The ingredients being: the thoroughly metaphorical character of most of math talk, of teachers as well as of textbooks, the inconsistency between this mathematical language and math's modern formal character, the metaphors used being thoroughly human including a rich gestural 'vocabulary' that itself is displayed without the actor (the math teacher) being even aware of it.
A prototype example of this kind of metaphor is that of time: TIME PASSING IS MOTION OF AN OBJECT, and TIME PASSING IS MOTION OVER A LANDSCAPE. The time line concept. Language expressions like: Christmas lies ahead. Etcetera. Of course, time is not something that 'moves', nor do we ourselves 'move' in time. Moving is a spatial concept. Its use in connection with time is metaphorical. What about mathematics? Mathematics is about static concepts, yet math talk is thoroughly dynamic. Núñez presents limits and its modern epsilondelta εδ method as an example. Nothing is moving here, yet it typically is 'explained' using motion methaphors. Indeed, the older concept of limits was based on the natural concept of continuity, in the nineteenth century to be replaced by the CauchyWeierstraß εδ method. The deeper problem involved here is that the older concept, allowing natural talk about movement, is quite another concept than the static one using the εδ method. The big mistake in math education is to regard the older concept as merely an imprecise version of the later one. It is a big mistake, because the older concept and its dynamic language does possess quite a different inferential structure from that of the modern εδ concept of limits. In that way students are in a constant state of bewilderment. Fantastic. Of course, limits is not the only subject suffering from a big divide between formal and informal language, and corresponding concepts.
In passing, Núñez also explains why I am not able to understand the work of Hans Freudenthal (1973, see below) on the didactics of mathematics education. Freudenthal, like many mathematicians, fails to recognize that the older concept of limits is quite another concept than that using the εδ method; if only we explain the latter better, students will see how their old ideas about continuity etcetera are imprecise, and had better be replaced with the εδ concept. The Freudenthal idea is: here is my formal mathematics, I will explain it to you in as clear a way as possible, just learn it. But that is not how the human mind works! [Yes, indeed, Freudenthal despised psychology, much to his detriment, and that of his many followers].
Look for publications from the research line established by Lakoff and Núñez—his publications web page—that are available for free download or download from questia.com.
Rafael Núñez (2007). The cognitive science of mathematics: Why is it relevant for mathematics education? pdf In Richard Lesh, Eric Hamilton and James J. Kaput, Foundations for the future of mathematics education (pp. 127154). Erlbaum contents
Stanislas Dehaene (1997). The Number Sense. How the Mind Creates Mathematics. Oxford University Press. isbn 0195110048
info newer revised edition
characteristics of the discipline
Euclid's elements of geometry. The Greek text of J. L. Heiberg (18831885) from
Euclidis Elementa, edidit et Latine interpretatus est I. L. Heiberg, in aedibus B. G. Teubneri, 18831885. Edited, and provided with a modern English translation, by Richard Fitzpatrick. isbn 9780615179841 (printing on demand, see website Fitzpatrick
site), online for free 8 Mb on that site
pdf or (Lulu)
pdf
Mathematics is a special discipline, even a highly idiosyncratic one. Mathematics as a science went its own way in the 19th century, for England at the end of the 19th century. Until then, at least in England, at least in Cambridge, it was almost identical with math education at Cambridge. The split was one between math in education, and math as a science itself, and has been beautifully described by Joan S. Richards (1988) (quotations and annotations: http://www.benwilbrink.nl/literature/richards.1988.htm).
Another split is that between physics as a science, and mathematics as a science. Mathematical physics is the territory of attraction and repulsion.
The point I am trying to make is the following. Mathematics might be an extreme example of a great divide between the science itself, and what is called arithmetics and mathematics in education, possibly even in university curricula, or mathematics curricula themselves. In spite of the great divide, mathematicians continue to influence the mathematics as taught in secondary schools and tertiary institutions, even the arithmetics as taught in primary schools, through scores of special commissions manned almost exclusively by mathematicians, through their professional organizations, and labor market mechanisms favoring the professional mathematician. Nothing wrong with all those institutional forces in itself, of course. Yet this state of affairs might be a scenario for disaster as far as the educational curriculum is concerned, and disastrous signs should be visible in much of serious educational research on math education, math educators, the education of math educators, math education's results and failures, and the connectedness of math education with other disciplines in the curriculum, or the lack of connectedness. And math education's techniques and methods of assessment of students, of course. Most actors in the field will be aware of the existence of grave educational problems, without necessarily being able to pinpoint exactly what they are and what the mechanisms behind them might be. It might take a few outsiders, such as Thorndike in the beginning of the last century, to articulate the issues and point to promising ways to address them.
Mathematics as a discipline is not special in having this kind of problem in the relations between its scientific progress, and its implementation in educational curricula, but it surely is the one having them in a very pronounced way. That makes it the choice discipline to go looking for the kind of educational derailments that in the very long run might follow from mathematical powerplay. In the very long run: developments extending over many decennia tend to be somewhat invisible to the actors involved, as well as to society at large, because of a natural human tendency to accept as normal what one has known to be the case for as long as one's own educational career. Again, the history of the Mathematical Tripos in the nineteenth century, and ultimately the demise of the ranking of the students in 1907 is a prime example (Richards, 1988). Or take the phenomenon so rightly criticized by Hans Freudenthal: to construct math curricula by rather straightforwardly projecting academic mathematics into it, without any serious didactictal or psychological reflection, let alone empirical evidence of the appropriateness of the resulting courses.
The broader issue then is not only that of a folk mathematics unconnected to the mathematics that properly might figure in the educational curriculum, but also the failure the other way around: the absence of a proper mapping of scientific mathematics on the needs of the educational curriculum, be it primary, secondary or even tertiary education.
Not being a mathematician myself [I have had a sturdy math program in secondary education, and the beginnings of a course in econometrics, though] I will have a hard time to come to grips with the issues indicated above. If you think you can give me a useful hint, please do. If you find some of my material useful in one way or another, please let me know. If it is your conviction that some of this or even all of this is bulshit, please let me hear your reasons why. [Until may 2008: no response. For reasons I do not understand, these pages in English, coded as English in the HTML lingo, receive no hits at all from the AngloAmerican world. And maybe my Dutch landgenoten do not like to read English?]
Paolo Mancosu, Klaus Frovin Jørgensen and Stig Andur Pedersen (Eds) (2005). Visualization, Explanation and Reasoning Styles in Mathematics. Synthese Library vol. 327. Springer contents — books.google example pages {I have not yet studied this books. For the amazingly relevant chapters see the contents. Relevant: to my quest that is. b.w.]
Paolo Mancosu (Ed) (2008). The Philosophy of Mathematical Practice. Oxford University Press.
 An expensive book. I have not yet had the opportunity to read in it.
 contents
Daniel L. Schwartz, Taylor Martin and Jay Pfaffman (2005). How mathematics propels the development of physical knowledge. Journal of Cognition and Development, 6, 6588. pdf
 "Three studies demonstrated the value of number and mathematics for 9 to 11yearold children’s development of physical understanding of the balance scale."
direct hits
See the Aha paragraph above.
The search for 'folk math' concepts will probably not be as easy as that for 'folk physics' concepts. In the field of statistics, a special branch of mathematics, a lot of relevant research is known, see below. In mathematics 'proper,' such is not evidently the case. But what, then, is 'proper'? Mathematics used to be oneofakind with physics, in the good old days of Galilei, Huygens, Leibniz, and Newton. It has artificially been detached from its realistic domains. I am beginning to suspect that the search for naive conceptions of mathematics might most profitably be on exactly its aloofness from the world (Lave, 1988, and his Adult Math Project, is one such search). My search has only just begun, there still is a lot of hope I will find what I am looking for. There are a number of specialised journals in math education, not only the English ones, that I have not yet seen, or do not even know the existence of. I will skim the contents of a number of them. What about Google, does 'folk math' result in any hits?
Mathematics being insulated from the real world makes it rather special, though. The exception is everything probable ( = mathematical statistics), of course. Arithmetic is another exception, let's say whatever mathematics that is being taught in primary education. Most of mathematics in secondary and higher education is unconnected to real life experiences of students. That might be the reason that Talia BenZeev and Jon Star (2001) find occasion to speak of intuitive or folk mathematics only in the sense of intuitions formed in education itself. What is interesting about this notion is that it might open up possibilities to study folk concepts  in particular intuitive mathematics  in a kind of natural laboratory: the school context. It's a pity not much relevant research seems to be available, quite in contrast to research on intuitive arithmetics.
C. Lebiere (1999). The dynamics of cognition: An ACTR model of cognitive arithmetic. Kognitionswissenschaft, 8, 519 pdf
 The Ph.D. dissertation itself is available also at http://actr.psy.cmu.edu/?post_type=publications&p=13870
 abstract "Cognitive arithmetic studies the mental representation of numbers and arithmetic facts and the processes that create, access, and manipulate them. The contradiction between the apparent straightforwardness of its exact formal structure and the difficulties that every child faces in mastering it provides an important window into human cognition. An ACTR model is proposed which accounts for the central results of the field through a single simulation of a lifetime of arithmetic learning. The use of the architecture’s Bayesian learning mechanisms explains how these effects arise from the statistics of the task. Because of the precise predictions of the simulation, a number of lessons are derived concerning the teaching of arithmetic and the ACTR architecture itself. A formal analysis establishes that the simulation can be viewed as a dynamical system whose ultimate learning outcome is fundamentally dependent upon some architectural parameters. Finally, an empirical study of the sensitivity of the simulation to its parameters determines that the values that yield the best fit to the data also provide optimal performance. The implications of these findings for the fundamental adaptivity of human cognition are discussed."
 Well, this is more or less the cognitive theory of arithmetics as learned and practiced in schools (in life). The ACTR model has been developed over the last 40 years by John Anderson and his colleagues and students.
 For a recent description of the ACTR model see John R. Anderson (2007). How can the human mind occur in the physical universe? Oxford University Press.
 The dedicated website http://actr.psy.cmu.edu/ has numerous papers and publications, many of which are available for download.
Kinga Morsanyi and Denes Szucs (2014). Intuition in Mathematical and Probabilistic Reasoning. abstract [pdf downloaded via https://qub.academia.edu/KingaMorsanyi February 19 2015]
Although the concept of intuition might be associated with something puzzling, uncontrollable, and unconscious, and students might be unaware of the sources of their intuitive preferences, this does not mean that intuition has to be a mystery for researchers and educators. In fact, intuitive tendencies are based on students’ personal experiences and their (explicit and implicit) learning history. Intuitive tendencies also tend to be similar across individuals. Whereas teaching students about the normative rules of mathematics and probability is essential, educational and training efforts should also aim to develop the right intuitions, together with an awareness of when and how these intuitions can be employed successfully.
from the Summary and Conclusion
Rafael Núñez (2007). The cognitive science of mathematics: Why is it relevant for mathematics education? pdf In Richard Lesh, Eric Hamilton and James J. Kaput, Foundations for the future of mathematics education (pp. 127154). Erlbaum contents
 This chapter is an introduction to a line of research that goes right to the heart of the question why it is that mathematics is such a troublesome subject in education. I must not exaggerate this, of course, the troublesome didactics of mathematics is not a onecauseissue. Some items from this line of research:
 George Lakoff and Rafael Núñez (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
 Introduction and first 4 chapters available as pdf on the Núñez site
 George Lakoff and Rafael Núñez: Reply to Bonnie Gold's review of "Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being" html MAA Online. There is a difference between mathematics proper, and the cognitive science of mathematical ideas. A difference that is not always easy to understand, especially so for mathematicians. This particular kind of misunderstanding is also frustrating the field of didactics of mathematics, of course: this didactics is not itself part of mathematics, it is educational psychology etcetera. It does not help at all that some mathematicians do not understand very well what makes the empirical sciences different from that of mathematics.

Rafael Núñez (2004). Do real numbers really move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. scan In Fumiya Iida, Rolf Pfeifer, Luc Steels and Yasuo Kuniyoshi (Eds). Embodied artificial intelligence. Springer.
 George Lakoff and Rafael Núñez (1997). The Metaphorical Structure of Mathematics: Sketching Out Cognitive Foundations for a MindBased Mathematics. In L. English, Mathematical Reasoning: Analogies, Metaphors, and Images (pp. 2189). Erlbaum. questia

Rafael Núñez, Laurie D. Edwards and João Filipe matos (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, 39, 4565. pdf

Rafael Núñez and George Lakoff (1998). What did Weierstrass really define? The cognitive structure of natural and εδ continuity. Mathematical Cognition, 4, 85101. pdf

Rafael Núñez(1994). Cognitive development and infinity in the small: Paradoxes and consensus. In A. Ram and K. Eiselt, Proceedings of the 16th Annual Conference of the Cognitive Science Society (pp. 670674). Erlbaum. pdf
 How do 12 till 14 year old children respond to one of Zeno's paradoxes?

Jim Kaput (1979). Mathematics and learning: Roots of epistemological status. In J. Lockhead and J. Clement, Cognitive process instruction (pp. 289303). Philadelphia: Franklin Institute Press. questia
 Jim Kaput cv and publications
 Illustrations used by Jim Kaput in this chapter: " The symbol system of mathematics denies the reality or importance of the knowerlearner. Our use of mathematical equality systematically denies the process/product distinction, a distinction that is fundamental and real in the universe of human knowing. It also denies the various and distinct heuristic/ linguistic functions of equality.
The processual meaning of mathematical operations is achieved through an essential, yet covert and unacknowledged, act of anthropomorphism, a projection from our internal cognitive experience onto the timeless, abstractstructural mathematical operations. (This includes, for example, operations in arithmetic, algebra and calculus.) In some respects this anthropomorphism acts as a metaphorical "structure preserving mapping," a morphism.
The basic, irreducible and essential metaphoric nature of human thinking has only an accidental, unacknowledged, and denigrated role in mathematics. As it is in any circumstance, the metaphorizing process in mathematics is our primary means for creating and, especially, transferring meaning from one universe to the other. However in mathematics this process is forced into a smuggling and bootlegging role, and never acknowledged for its crucial function. For example, virtually all of basic calculus (the study of change) achieves its primary meaning through an absolutely essential collection of motion metaphors. These metaphors control the notation. Hence we write limit statements using arrows and use imageladen words such as 'diverge,' 'converge,' 'increasing,' "'constant,' and 'transform.' However, the formal mathematical definitions associated with these notations, being atemporal, are not connected to motion."
Absolutely a direct hit. This analysis is what inspired Núñez.
Wim van Dooren, Dirk de Bock, Dirk Janssens and Lieven Verschaffel (2008). The linear imperative: An inventory and conceptual analysis of students' overuse of linearity. Journal of Research in Mathematics Education, 39, 311342.
 abstract "(. . .) This article provides an overview and a conceptual analysis of students' tendency to use linear methods beyond their applicability range."
James Stigler (1997). Lessons in perspective: How culture shapes math instruction in Japan, Germany and the United States. pdf Zie vooral http://www.timssvideo.com
 “The new study that I want to talk with you about today is the video component of the Third International Math and Science Study (TIMSS). This study was conducted in 41 different countries, examining student achievement in fourth, eighth and twelfth grades. As part of this study, the U.S. decided to do something that had never been done before: they took a national probability sample of teachers in three countries—the U.S., Germany and Japan—and videotaped them teaching in their classrooms.”
 Hilary Hollingsworth (2004). From research to practice: Using the TIMSS Video Study findings to enhance Australian mathematics teaching.Paper at the Mathematics Association of Western Australia (MAWA) 2004 Conference. pdf “The broad purpose of the TIMSS 1999 Video Study was to investigate and describe Year 8 mathematics and science teaching practices in a variety of countries, including several with varying cultural traditions and with relatively high achievement on TIMSS assessments. Countries participating in the mathematics component of the TIMSS 1999 Video Study were Australia, the Czech Republic, Hong Kong SAR, Japan, the Netherlands, Switzerland, and the United States.” “One of the core findings of the TIMSS 1999 Video Study is that there is no single method for teaching mathematics across countries; different methods of teaching can lead to high achievement. ”
Mark Levi (2009). The mathematical mechanic. Using physical reasoning to solve problems. .
 sample chapter 1
 Pythagorean Theorem (fragment from chapter 2) pdf
 About the artificial demarcation of problems mathematical and problems physical. The idea is not a new one, Levi refers to Polya’s Mathematics and plausible reasoning, Volume 1. Levi’s book is a gem.
Amy B. Ellis (2007). The influence of reasoning with emergent quantities on students' generalizations. Cognition and Instruction, 25, 439478.
 abstract
 Subject: what it means to reason algebraically.
Stellan Ohlsson & Ernest Rees (1991). The function of conceptual understanding in the learning of arithmetic procedures. Cognition and Instrction, 8, 103179.
Robyn Arianrhod (2005). Einstein's Heroes: Imagining the World Through the Language of Mathematics. Oxford University Press.
 This is an unexpected 'direct hit.' Somewhat out of character, this is a popular book. Never mind, it is delightful. What makes it a direct hit is its exposition of the interplay between mathematics and physics, using the lives and work of Newton, Farady and Maxwell (and some others, such as Galilei and Einstein).
 As far as the mathematics is concerned, a number of points are of specific interest:
 The first point is Galileo trying experimentwise to improve upon the gravitation theory of Aristotle, mathematising his results.
 The second point is the power of the Newtonian laws of motion, adequate to the empirical data on the orbits of the planets, to predict the existence and location of a new planet Neptune. Almost two centuries later, but I will not start nitpicking the point made here. Is this the power of mathematics?
 The theory of Newton can still be understood in physical terms, if one is prepared to look away from the disturbing fact that the working of gravitational force is a complete unknown. Mathematical terms seem to map into physical concepts, and vice versa.
 The third point: Faraday has powerful intuitions about electromagnetism, he does not know anything of mathematics however (except some geometry).
 The fourth point: along comes Maxwell, recognizing the power of Farady's 'field' construct. Maxwell invests heavily in mathematizing the electromagnetic 'field.' To get thing going he uses the imagery of mechanical devices, a lot of them: the 'ether' (not his own invention). He intentionally uses only emprical physical facts to base his mathematics on, succeeds in doing so, and never again mentions any 'ether.'
 The intriguing thing now is that it seems not to be possible to call Maxwell's mathematics a 'mathematical model' of something called 'electromagnetic fields.' The 'field' concept was a useful analogue, suggested by the filings pattern in strong magnetic fields, but there is nothing mechanical one might call a 'field.' This mysterious 'field' is what the mathematics says, nothing more. What baffles me, as a psychologist, is that Maxwell’s mathematical theory of electromagnetism is not something that can be 'understood' by mapping the mathematics on the real world. Of course, there is a lot left that one can try to understand about electromagnetic phenomena, including the fact that Maxwell’s theory has been able to predict new phenomena (radio waves, for example), just like the existence of Neptune was predicted from the mathematics of Newton's theory.
 Can you see what this implies? How is it possible to teach children the basic concepts of physics and mathematics, without inducing hosts of misunderstandings? Robyn Arianrhod does not go into this question, of course. It would take her at least one other book. Would be lovely, though, to have her view on this problem.
Catherine Sophian (2007). The origins of mathematical knowledge in childhood. Lawrence Erlbaum. [for an annotation see here
In physics the didactical problem as located by Slotta and Chi (2006 pdf) (see also physicseducation.htm) centers on the specific character of its theoretical concepts being emergent processes, not material substances. BenZeev and Star (2001) (see below) refer to this line of research, but they do not see any direct implications for the didactics of mathematics as well. Regrettably, there does not seem to be research exploring this possible connection. My intuition about the possible link is rather straightforward: historically the attempts to describe and research these emergent processes (light, force, speed, mass, to mention some physics concepts) were the occasion to invent or develop the mathematics (for example: the calculus) enabling one to do so. Therefore the mathematics concerned should somehow or other be taught in its proper context, isn't it? That context being: emergent processes. A lot of other disciplines, such as psychology, economics, sociology, know these emergent processes as well. A particularly interesting one is the way experience is being grafted into neural networks, allowing later to 'remember' it.' In many cases one or another branch of statistics is used to describe or research these emergent processes.
Therefore, the work of Slotta and Chi must tell us also something about the possibilities to develop a 'true didactics' for the mathematical topics involved. Do not assume this applies only to the calculus, why should not the number concept itself be studied as an 'emergent' concept? Maybe it is not, then why should that be so, and what can we learn from that result about the way kids might learn it (research by, among others, Susan Carey, see her site)?
What import does this quest for the lost grail have? Lots of kids, pupils and students loose their interest in mathematics while in school, high school or college. This loss of interest results from a number of different causes, is itself therefore an 'emergent process,' one of the causes being the perennial problem of the very high levels of abstractness of mathematics course content, another cause undoubtedly is mathematics' hidden curriculum (ab)use to sort kids, pupils or students into different classes of intellectual abilities. The losses to individuals as well as to society at large, resulting from these stultifying causes of mathematical 'drop out,' are enormous. If bad didactics enables these kinds of abuse of the mathematics curriculum, we should try to change the didactics. An adequate didactical theory would be very helpful in developing an instructional design theory that is adequate to the task of empowering almost all students with adequate mathematical intuitions (in BenZeev and Star's (2001, see below) terms). Such a theory being available, the design theory for achievement test items will follow suit. What is more: designing test items according to such a design theory will invite instruction and instructors to practice these better didactics. I am sure many experimental courses nowadays are using some of these didactical insights I am looking for in this webpage.
A significant part of the literature on teaching mathematics and researching teaching mathematics has been published by Lawrence Erlbaum, and is available in the data base questia.com for online reading. If you are not a member, it is always possible to read contents, and first pages of chapters and articles. If need be, for reading from cover to cover, a free online period of seven days is available.
topics and key publications or key researchers/projects
core knowledge: Feigenson, Dehaene & Spelke (2004) pdf
(cardinal) number: Carey (2004) pdf; Corre, Brannon, Van de Walle & Carey (2006) look for a copy on this site
rational number: Mack (1993) here; Carpenter, Fennema & Romberg (1993) here
proportion and ratio: Singer, Kohn & Resnick (1997) here; Empson (1999) here
adding and multiplying: Lebière and Anderson (1998) here; Lebière (1998) here; Riviera, Reiss, Eckert and Menon (2005). pdf
word problems: Verschaffel, Greer and De Corte (2000) here
procedural and conceptual knowledge (algorithms versus understanding): Zamarian, LópezRolón and Delazer (2007) here
understanding: Reif & Allen (1992) here
math text book 1st college: Daepp and Gorkin (2003). Kevin Houston (2009)
algebra: Brizuela & Schliemann (2004) here
geometry: Robin Hartshorne (1997/2000) Geometry: Euclid and beyond. Hafner/Dover. isbn 0486605094
calculus: Boyer (1949/1959) The history of the calculus and its conceptual development.
statistics: Garfield (2002) pdf
applications = mathematical models of phenomena in the world: Suppes (2002) Representation and invariance of scientific structures. pdf 8 Mb
axiomatics, formalism: Patrick Suppes (2002). Representation and invariance of scientific structures. pdf 8 Mb
intuitive mathematics: Fischbein (1987) The intuitive sources of probabilistic thinking in children.; BenZeev & Star (2001) pdf
intuition: Gerd Gigerenzer (2007). Gut feelings; Robin M. Hogarth (1993). Educating intuition
philosophy: Kitcher (1984) The Nature of Mathematical Knowledge; Mancosu (2008)? The Philosophy of Mathematical Practice.
psychology of knowing/doing/explaining mathematics
psychology: Sternberg & BenZeev (Eds) (1996) The nature of mathematical thinking questia
didactics: Bransford, Brown & Cocking (1999) How People Learn: Brain, Mind, Experience, and School html; constructivisme: zie constructivisme.htm; Enzensberger (1997) Der Zahlenteufel. Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben. [The number devil].
epistemological beliefs
assessment: Kulm (1990)? Assessing Higher Order Thinking in Mathematics questia; Schoenfeld (2007) Assessing mathematical proficiency pdf
history: John Fauvel and Jan van Maanen (2000) — Fibonacci (1202) Liber abaci translated by Sigler (2002). — Ellerton & Clements (2013). Rewriting the History of School Mathematics in North America 16071861
Cambridge Mathematical Tripos: Richards (1988) Mathematical visions. The pursuit of geometry in Victorian England.
situatedness: [ ideology; pseudoscience ] Lave, 1988; Anderson, Reder & Simon (1996, 2000) pdf & html; Watson and Winbourne (2008); Anna Sfard (2008) here; Kelso (1995)
isolatedness: Doorman (2005) Modelling motion: from trace graphs to instantaneous change access to chapter pdf's
'math wars': Klein (2007) The state of the state math standards 2005
math disabilities: Geary (1993) pdf; Berch & Mazzocco (2007) Why is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities contents
TIMSS home international
Small children do have a lot of intuitive understanding of numbers, proportions, etcetera (protoquantitative conceptions). It might just be the case that instructional methods emphasize too early and too much the working with numbers, instead of with the intuitions (for example: Singer, Kohn and Resnick, 1997). What makes me think so? Research on word problems (Verschaffel et al. 2000) shows that school arithmetics is quite different from the mathematics people like you en me need in daily life, or in professional life for that matter. Something has gone sour in education, probably already very long ago. Take a look at an arithmetics book from the 15th or 16th century, and you will begin to suspect that the emphasis on algorithmically working with numbers, as contrasted with understanding what one is doing while performing the algorithm, was already formly established, and never since has that been changed in any fundamental way.
The table above of topics and key publications is only for starters. I will add topics, and replace preliminary choices of key publications with better ones. Some keys function better than others, no offense intended. Remark that the point of departure defnitely is not a mathematical, but an empirical psychological one, make it neuropsychological if you like that better. After all, there is very little 'mathematical' in the arithmetics in primary education. The same goes for the algebra, calculus and geometry of secundary education. Quite another thing is that teachers need some grounding in mathematics proper, to prevent them from doing some crazy things with the children we entrust them. I am somewhat preoccupied by the mathematics as taught in primary and secundary education. My hunch is that especially here a lot of things can and do go wrong in severe ways. Mathematics at tertiary and university levels might have some specific problems also, but I have not seen much research yet that touches on the contrast between naive conceptions and the scientific ones.
It might be the case that intuition in arithmetics does not pose the kind of problems intuition in physics does, meaning that it might be possible to build arithmetics education on those very intuitive concepts. The case reminds one of that in the field of decisionmaking: formal decisonmaking is quite different from intuitive decisionmaking, formal methods are rather complex while intuitive ones are rather simple, and yet intuitive methods might give results that are nearly as good as the formal ones (or veen better, recognizing the cost of formal methods?). See for example the work of Gigerenzer, or early publications by Herbert Simon on bounded rationality.

Gerd Gigerenzer (2007). Gut feelings. Penguin.

Robin M. Hogarth (2001). Educating intuition. The University of Chicago Press. isbn 0226348601
Hugh Burkhardt (2007). Mathematical Proficiency: What Is Important? How Can It Be Measured? In Alan H. Schoenfeld: Assessing mathematical proficiency (7797). Cambridge University Press. pdf .
 Examples of nontrivial assessment of mathematics.
 Illustrates work from institutions that otherwise is rather inaccessible.
 Burkhardt describes what is possible. For empirical support he does not mention sources in the research literature.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sensemaking in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334370). New York: MacMillan. pdf
Anne Watson (2006). Raising achievement in secondary mathematics. Open University Press. website [I have not seen the book yet, there is an eBook as well as a softpver edition. Research articles on the subject may be available for download on Anne Watson's home page]
Candia Morgan & Anne Watson (2002). The interpretive nature of teachers' assessment of students' mathematics: issues for equity. Journal for Research in Mathematics Education, 78107. first page only
Anne Watson (2004) Red herrings: post14 ‘best'’ mathematics teaching and curricula. British Journal of Educational Studies pdf
Anne Watson & Els De Geest (2005) Principled Teaching for Deep Progress: Improving mathematical learning beyond methods and materials. Educational Studies in Mathematics 58, 209234 pdf
 “In this paper we report on a project to improve attainment in mathematics among low attaining secondary students. (. . .) The main aim of the paper is to report the practices of teachers involved in the project and the complexities of identifying what was common about the teaching which led to improved engagement and learning.”
Alan H. Schoenfeld (Ed.) (2007). Assessing mathematical proficiency. Cambridge University Press.
 Chapters are available for download here
 contents
 Issues and Tensions in the Assessment of Mathematical Proficiency, by Alan H. Schoenfeldpdf
 What Is Mathematical Proficiency? by R. James Milgram, 3158 pdf
 5. What Is Mathematical Proficiency and How Can It Be Assessed? by Alan H. Schoenfeld, 5973 pdf
 6. Mathematical Proficiency: What Is Important? How Can It Be Measured? by Hugh Burkhardt, 7797 pdf
 7. Aspects of the Art of Assessment Design, by Jan de Lange, 99111 pdf
 8. Mathematical Proficiency for Citizenship, by Bernard L. Madison, 111124 pdf
 9. Learning from Assessment, by Richard Askey, 125136 pdf
 10. When Assessment Guides Instruction: Silicon Valley's Mathematics Assessment Collaborative, by David Foster, Pendred Noyce, and Sara Spiegel, 137154 pdf
 11. Assessing the Strands of Student Proficiency in Elementary Algebra, by William G. McCallum, 157162 pdf
 12. Making Meaning in Algebra: Examining Students' Understandings and Misconceptions, by David Foster, 163176 pdf
 13. Task Context and Assessment, by Ann Shannon, 177191 pdf
 14. Learning About Fractions from Assessment, by Linda Fisher, 195211 pdf
 15. Assessing a Student's Mathematical Knowledge by Way of Interview, by Deborah Loewenberg Ball with Brandon Peoples, 213267 pdf video
 16. Reflections on an Assessment Interview: What a Close Look at Student Understanding Can Reveal, by Alan H. Schoenfeld, 269277 pdf
 17. Assessment in France, by Michèle Artigue, 283309 pdf
 18. Assessment to Improve Learning in Mathematics: The BEAR Assessment System, by Mark Wilson and Claus Carstensen, 311332 pdf
 19. English Learners and Mathematics Learning: Language Issues to Consider, by Lily Wong Fillmore, 333344 pdf
 20. Beyond Words to Mathematical Content: Assessing English Learners in the Mathematics Classroom, by Judit Moschkovich, 345352 pdf
 21. Assessment in the Real World: The Case of New York Ci ty, by Elizabeth Taleporos, 345355 pdf
 22. Perspectives on State Assessments in California: What You Release Is What Teachers Get, by Elizabeth K. Stage, 357363 pdf
J. Singer, A. Kohn and L. B. Resnick (1997). Knowing about proportions in different contexts. In P. Bryant and T. Nunes: Learning and teaching mathematics: An international perspective (pp. 115132). Hove, England: Psychology Press.
 "This chapter proposes that children's knowledge of proportions is composed of three distinct components. At the direct level, children have an immediate, nonanalyzed understanding of proportions via a perceptual or analogous process. At the covariational level, children know something about variables and how they may covary, either directly or inversely. At the formal level, children know how to manipulate numbers and variables to describe proportional relationships between entities. This conception of proportional knowledge helps explain why children sometimes behave appropriately in proportional reasoning tasks and sometimes do not; that is, different tasks tap different kinds of knowledge." p. 116: The issue in this chapter is whether "children can first reason protoquatitatively about situations involving proportions or ratios and later carry this knowledge into quatified, mathematically exact forms of representation and reasoning. (. . .) We present evidence of intuitive schemas for reasoning about densities and rates—both of which count as protoquatitative forms of reasoning about ratios and proportions—but also suggest that quantification of these schemas does not proceed smoothly or directly." The authors then use a distinction (Schwartz) between extensive and intensive quantities, the first being of the kind that add (apples, lengths), the second of the kind that mixes (temperatures, densities, and especially rates), but I am way too sloppy now.
 protoquantitative schemas is one of the theoretical ideas. See also Resnick & Singer (1993). Protoquantitative origins of ratio reasoning. In T. Romberg: Rational numbers: An integration of research (pp. 107—130). Hillsdale, Erlbaum. questia
 For a recent paper in this line of research see Olof Bjorg Steinthorsdottir (2005). Girls journey towards proportional reasoning. pdf
James W. Stigler and Ruth Baranes (1988). Culture and mathematics learning. In Ernst Z. Rothkopf: Review of research in education volume 15 198889. (253306). Washington, D.C.: American Educational Research Association.
Lyle V. Jones (1988). School achievement trends in mathematics and science, and what can be done to improve them. In Ernst Z. Rothkopf: Review of research in education volume 15—198889. (307341). Washington, D.C.: American Educational Research Association. [Jstor]
Jean Lave (1988). Cognition in practice. Mind, mathematics and culture in everyday life. Cambridge University Press.
 Kind of anthropological approach to research the mathematics people use in everyday situations, such as cooking, shopping. The empirical material probably is fascinating, the theorizing less so.
 catch phrase: situated learning; the thousand ways word problems tend to go wrong in instruction (Verschaffel a.o. 2000) result from situated learning aspects of instruction instructors typically do not notice.
 This book is a much cited one, not quite deservedly so. It ridicules a lot of good scientific thinking and research, declaring situatedness the one and only circumstance that is important in learning, or cognition, for that matter. The book is a position statement, not a stern report on a particular line of research. The theme surely is fascinating, and there is a lot of truth in it as far as school learning goes. It is the kind of research as reported by Verschaffel Greer and De Corte (2000) that makes clear to what extent the solving of typical word problems in arithmetics is situated in the loose sense of Lave. That extent is staggering.
 There is one topic that intrigues me, and that I would like to know more of: the way price comparisons in the school/textbook world differ from those made in the shopping malls of this real world where to most people it suffices to evaluate marginal costs of the extra number or weight of items in the larger package (p. 119: "I will get two ounces more for six cents. Is it worth it?"). The general point seems to be that there are potentially many different mathematical possibilities to model a given situation, some of them being preferred by educationalists, others by real people in real places making real choices. So much the worse for the educational tradition. The research refered to here is: Noel Capon and Deanna Kuhn (1979). Logical reasoning in the supermarket: Adult females' use of a proportional reasoning strategy in an everyday context. Developmental Psychology, 15, 450452 (ERIC abstract: "Results showed that only 32 percent of adult female shoppers in a supermarket were able to use a proportional reasoning strategy to determine which of two sizes of a common item (size ratio 2:3) was the better buy. Performance declined when the ratio was more complex. (JMB)". And M. Murtough (1985). The practice of arithmetic by American grocery shoppers. Anthropology and Education Quarterly, 16, 186192 (his dissertation, 1985: A hierarchical decision process model of American grocery shopping.)
 T. Nunes, A. Schliemann and D. Carraher (1993) Street Mathematics and School Mathematics. Cambridge University Press. [I have not yet seen this one]
 David Kirshner and James A. Whitson (Eds) (1997). Situated cognition. Social, semiotic, and psychological perspectives. Erlbaum. questia a.o. Philip E. Agre: Living math: Lave and Walkerdine on the meaning of everyday arithmetic 7183. Carl Bereiter: Situated cognition and how to overcome it 281300
 Watson and Winbourne, 2008, see below
Theodore M. Porter (1995). Trust in numbers. The pursuit of objectivity in science. Princeton University Press. UP questia
 p. viii: " My approach here is to regard numbers, graphs, and formulas first of all as strategies of communication. They are intimately bound up with forms of community, and hence also with the social identity of the researchers. To argue this way does not imply that they have no validity in relation to the objects they describe, or that science could do just as well without them. The first assertion is plainly wrong, while the latter is absurd or meaningless. Yet only a very small proportion of the numbers and quantitative expressions loose in the world today make any pretense of embodying laws of nature, or even of providing complete and accu rate descriptions of the external world. They are printed to convey results in a familiar, standardized form, or to explain how a piece of work was done in a way that can be understood far away. They conveniently summarize a multitude of complex events and transactions. Vernacular languages are also available for communication. What is special about the language of quantity?
My summary answer to this crucial question is that quantification is a technology of distance. The language of mathematics is highly structured and rulebound. It exacts a severe discipline from its users, a discipline that is very nearly uniform over most of the globe. That discipline did not come automatically, and to some degree it is the aspiration to a severe discipline, especially in education, that has given shape to modern mathematics [Joan Richards (1988). Mathematical Visions. The Pursuit of Geometry in Victorian England. Academic Press]. Also, the rigor and uniformity of quantitative technique often nearly disappear in relatively private or informal settings.
Terezinha Nunes and Peter Bryant (1996). Children doing mathematics. Blackwell. [I have not yet seen this book, it has been referred to by Brizuela (2004). reviewed by Derek Haylock.
Lucas Michiel Doorman (2005). Modelling motion: from trace graphs to instantaneous change. CDβ Press, Center for Science and Mathematics Education. Dissertation Utecht University. access to chapter pdf's or integral text 4 Mb pdf
 "Students in secondary education experience mathematics and physics as strictly separate disciplines. They do not realise for instance that the mathematics used to describe change (calculus) is used in the topic kinematics in physics. The goal of this research was to examine whether it is possible to develop understanding of both subjects and of their mutual relationship. Furthermore, it has been examined what role computer tools could play in learning mathematics and physics."
 I have yet to study this research. Its promise is that it purports to bridge the gap that has grown in the last one or two centuries between (school) mathematics and the real world it is supposed to model. Has Doorman succeeded in developing an effective didactics that is true to the process of doing mathematics in a physics context, or doing physics is a supportive mathematics context? The historical model here might be the German research laboratory: for example Kathryn M. Olesko (1991). Physics as a calling. Discipline and practice in the Königsberg Seminar for Physics. Ithaca: Cornell University Press.
Benedikt Löwe and Thomas Müller (2005). Mathematical knowledge is context dependent. Prepublication Institute for Logic, Language & Computation, University of Amsterdam. pdf
Annie Selden and John Selden (1993). Collegiate Mathematics Education Research: What Would That Be Like? The College Mathematics Journal, 24, 431445. pdf JStor
John R. Anderson, Lynne M. Reder, and Herbert A. Simon (1996). Situated learning and education. Educational Researcher, 25(4), 511. pdf
 from the abstract We review the four central claims of situated learning with respect to education: (1) action is grounded in the concrete situation in which it occurs; (2) knowledge does not transfer between tasks; (3) training by abstraction is of little use; and (4) instruction must be done in complex, social environments. In each case, we cite empirical literature to show that the claims are overstated and that some of the educational implications that have been taken from these claims are misguided.
 p. 5: "In this paper, we want to concentrate on empirical evidence and its implications for mathematics education [my emphasis, b.w.]
 Situated learning (e.g. Lave and Wenger 1991): "emphasizes the idea that much of what is learned is specific to the situation in which it is learned."
 Be aware that the authors themselves have done a lot of empirical research and theory building that is relevant to the issues discussed here.
 Their 1995 unpublished article has been published in 2000, see below. It is still available online, however the URL now is http://actr.psy.cmu.edu/papers/misapplied.html
Edward Silver (2010). Examining what teachers do when they display their best practice: Teaching mathematics for understanding. Journal of Mathematics Education at Teachers College, 1, 16. gratis beschikbaar pdf
 Edward A. Silver, Vilma M. Mesa, Katherine A. Morris, Jon R. Star & Babette M. Benken (2009). Teaching mathematics for understanding: An analysis of lessons submitted by teachers seeking NBPTS certification. American Educational Research Journal, 46, 501531.
John R. Anderson, Lynne M. Reder, and Herbert A. Simon (2000, Summer). Applications and Misapplications of Cognitive Psychology to Mathematics Education. Texas Educational Review, Summer. html
 abstract There is a frequent misperception that the move from behaviorism to cognitivism implied an abandonment of the possibilities of decomposing knowledge into its elements for purposes of study and decontextualizing these elements for purposes of instruction. We show that cognitivism does not imply outright rejection of decomposition and decontextualization. We critically analyze two movements which are based in part on this rejectionsituated learning and constructivism. Situated learning commonly advocates practices that lead to overly specific learning outcomes while constructivism advocates very inefficient learning and assessment procedures. The modern informationprocessing approach in cognitive psychology would recommend careful analysis of the goals of instruction and thorough empirical study of the efficacy of instructional approaches.
 Anderson mentioned the article in 1996 as being submitted for publication. By 2000 Herbert Simon was deceased. The other authors: Department of Psychology, Carnegie Mellon University. Be warned: Simon is a Nobel Prize winner, Anderson is especially known by the ACTR theory of cognition he and his colleagues developed.
Christian Lebiere and John R. Anderson (1998). Cognitive arithmetic. In John R. Anderson, Christian Lebiere, and others: The atomic components of thought (297342). Lawrence Erlbaum. questia
 Christian Lebiere (1998). The Dynamics of Cognition: An ACTR Model of Cognitive Arithmetic. Dissertation Carnegie Mellon University pfd. From the abstract: "Cognitive arithmetic, the study of the mental representation of numbers and arithmetic facts and the processes that create, access and manipulate them, offers a unique window into human cognition. Unlike traditional Artificial Intelligence (AI) tasks, cognitive arithmetic is trivial for computers but requires years of formal training for humans to master. Understanding the basic assumptions of the human cognitive system which make such a simple and wellunderstood task so challenging might in turn help us understand how humans perform other, more complex tasks and engineer systems to emulate them. The wealth of psychological data on every aspect of human performance of arithmetic makes precise computational modeling of the detailed error and latency patterns of cognitive arithmetic the best way to achieve that goal."
Susan B. Empson (1999). Equal Sharing and Shared Meaning: the Development of Fraction Concepts in a FirstGrade Classroom. Cognition and Instruction, 17, 283342. questia
 from the abstract The study provides an account of children's learning that examines the relation between classroom talk and children's evolving fraction concepts, with a focus on the analysis of several key classroom interactions that resulted in cognitive change. Pretests and posttests indicated that children's understanding of fractions changed in important ways. The results suggest that how children think about fractions is influenced not only by how their own knowledge is structured but, perhaps more profoundly, by how the context for thinking about and discussing fractions is structured.
 p. 284: " most research on children's fraction thinking is founded on models of cognition that highlight universal structures of understanding ( Behr, Harel, Post, & Lesh, 1992; Hiebert & Carpenter, 1992). These models focus on the products of understanding, downplaying the processes involved in understanding. (. . .) A burgeoning body of research posits that participation in communities of practice is fundamental to mathematical understanding ( Greeno & Goldman, 1998; Greeno & MiddleSchool Math through Applications [MMAP], 1997; Lave & Wenger, 1991). From this perspective, understanding is relative to participation in a community, and to understand cognitive change, we need to consider the socially organized processes that motivate activity and shape the products of thinking. "
 p. 334 (from the conclusions): How children think about fractions is influenced not only by how their own knowledge is structured but, more importantly, by how the context for thinking about and discussing fractions is structured.
Leone Burton (2004). Mathematicians as Enquirers: Learning about Learning Mathematics. Kluwer.
 "This volume reports on an empirical study with 70 research mathematicians, 35 females and 35 males. The purpose of the study was to explore how these mathematicians came to know mathematics and to match their descriptions against a theoretical model of coming to know mathematics derived from the literature of the history, philosophy and sociology of science and mathematics. The assumption underlying the research was that, when researching, mathematicians are learning and, consequently, their experiences are valid for less sophisticated learners in classrooms. The study provided major surprises particularly with respect to the mathematical thinking of the mathematicians and to the ways in which they organised their practices. It also contradicted longstanding stereotypes.
This book applies the learning from the study to learning and teaching mathematics. It offers a rationale, based on the practices of research mathematicians, to support and encourage recent schoolbased developments in the learning of mathematics through enquiry."  I have not yet seen the book. I am really curious.
 Leone Burton (2001). Research Mathematicians as Learnersand what mathematics education can learn from them. British educational research journal, 27, 589600
 Leone Burton (1998). Advice to Prospective Authors  The Practices of Mathematicians: What Do They Tell Us About Coming to Know Mathematics? Educational studies in mathematics; 37, 121144
Anna Sierpinska (1992). On understanding the notion of function, in Guershon Harel and Ed. Dubinsky: The Concept of Function: Aspects of Epistemology and Pedagogy MAA (Math. Ass of Am.) Notes (Vol. 251, 1992, pp. 2558). [I have not yet located this one. It was referred to by Kieran, 1997, p. 133 (in Nunes and Bryant)]
Talia BenZeev and Jon Star (2001). Intuitive mathematics: theoretical and educational implications. In Robert J. Sternberg and Bruce Torff: Understanding and teaching the intuitive mind: student and teacher learning. Erlbaum. pdf of concept or questia
 p. 29: " . . . can we identify a set of naive beliefs that are applied to solving abstract mathematics problems? If so, how do these intuitions hinder or facilitate problem solving? The answers to these questions have implications for both psychology and education. By examining the nature of intuitive mathematics we could help (a) improve our understanding of people's formal and informal reasoning skills, and (b) create more effective instructional materials."
 p. 30: "In this chapter, we examine the nature and origin of what we term as symbolic intuition, or the intuitive understanding of mathematical symbols that develops as a result of experience with formal and abstract schoolbased procedures."
 The article is a sloppy review of the field, as perceived by the authors. It is useful as an introduction to some of the literature touching on deeper questions of didactics in mathematics. It is rather superficial in its treatment of, for example, the 'what is the age of the captain' problem, a question that in fact does not allow any numerical answer, and yet many pupils obligingly will produce one. This kind of pupil behavior is not necessarily as simple as these authors tell you it is. The hidden curriculum these pupils have experienced is: all questions always have definite answers. Now here is this new question about the age of the captain, containing some numbers, none of them useful to find his age. 'What kind of game are they playing with me?' the pupil might think. Or simply: well, obviously some mistake has been made, why not use the numbers given to show my skill in adding, subtracting or multiplying them? If Jon Star is puzzled by the controls on some new appliance coming in his way, he will probably just try one or two to see what happens. Is that very much different from what pupils are contriving as the age of this captain?
 What might be useful in this article, is its characterization of two kinds of 'intuition,' and inferences from there to the learning of mathematics.
Meg Schleppenbach, Michelle Perry, Kevin F. Miller, Linda Sims and Ge Fang (2007). The Answer Is Only the Beginning: Extended Discourse in Chinese and US Mathematics Classrooms. Journal of Educational Psychology, 99, 380396. abstract Also via researchgate.net.
This is a key publication on the topic of justifying your answer on partical achievement test items. Be warned, however: this might be a constructivist approach.
Efraim Fischbein (1975). The intuitive sources of probabilistic thinking in children. Dordrecht: Reidel.
 Jane M. Watson and Jonathan B. Moritz (2002). School students' reasoning about conjunction and conditional events. International Journal of Mathematical Education in Science and Technology
 Jenni Way (2003). The development of young children's notions of probability. European Research in Mathematics Education III Proceedings of the Third Conference of the European Society for Research in Mathematics Education 28 February  3 March 2003pdf
Efraim Fischbein (1987) Intuition in science and mathematics. An education approach. Reidel.
 I will borrow it from the KB
 The first 73 pages are available from Google Book Search
Kathleen E. Metz (1998). Emergent Understanding and Attribution of Randomness: Comparative Analysis of the Reasoning of Primary Grade Children and Undergraduates. Cognition and Instruction, 16, 285365. questia
 " In this study, I examined primary grade children's emergent understanding and attribution of randomness, as reflected in the classicist and frequentist forms of objectivist probability. Participants included kindergartners, 3rd graders, and, in order to address the complex issue of developmental versus nondevelopmental difficulties, university undergraduates (n = 36)."
 "Comparison of the undergraduate data with the children's indicates that many of the deficiencies in the children's performance cannot be attributed to developmental shortcomings. Many aspects of interpreting these random phenomena constituted a nontrivial challenge for kindergartners, 3rd graders, and these relatively welleducated adults alike, including
 assessing the bounds of the agent's control,
 assessing the extent to which different devices or apparatuses would support confidence of predictions,
 assessing the information given in a data set, and, more generally,
 transcending an overly deterministic interpretation, as well as
 integrating a conceptualization of the uncertainty of the situation with the patterns one could expect across many repetitions of the event (corresponding with the conceptual integration underlying the randomness construct)."
R. Duncan Luce and Patrick Suppes (1968). Mathematics. pdf
 A 24 column introduction to the major fields of mathematics, by the masters of measurement theory  and that is what mathematics is about, isn't it: measurement.
Bárbara M. Brizuela (2004). Mathematical development in young children. Exploring notations. New York: Teachers College.
 A beautiful book/study. Emphasizes how important it is not to exclude notational usage from the mathematical concepts themselves.
 p. 58 cites Alfred Noirth Whitehead (1911), from Cajori, 1929, p. 332: "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems." Imagine having to do your arithmetics in the old Roman notation, instead of algebraic numbers! This idea generalizes to most mathmatical notation.
 Richard Lehrer in the foreword: "Mathematical doing and conceiving are mediated by powerful and often complicated systems of writing, so mathematics is also a particular kind of written discourse."
 p. 102: "The two main points explored throughout this book  that there is a constant interaction between mathematical notations and conceptual understandings and that there is a similar interplay between invented and conventional mathematical notations  pervade each of the other issues put forth. Similarly, connections established with the history of mathematical notations pervade each of the chapters  the similarities between the events in the history of mathematical notations and children's development of mathematical notations highlight the types of difficulties encountered by both matematicians of antquity and contemporary children.
 For the history of notations see especially the generally known work of Ifrah, or studies on the early arithmetics printed books.
 One of the chapters is about fractions. See also: Bárbara M. Brizuela (2006). Young children's notations for fractions. Educational Studies in Mathematics, 62, 281305. pdf
 M. V. Martinez and B. M. Brizuela (2006). A third grader's way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285298. pdf
 D. W. Carraher, A. D. Schliemann, B. M. Brizuela and D. Earnest (2006). Arithmetic and Algebra in Early Mathematics Education. Journal for Research in Mathematics Education 37(2), 87115. pdf
Bárbara M. Brizuela and Analúcia Schliemann (2004). Tenyearold students solving linear equations. pdf
 p. 33: "The crux of the argument of this article is that if we can present evidence of younger, elementary school children engaging with algebra, and using and understanding the syntactic rules of algebra, we have to ask ourselves why so many adolescents face difficulties with algebra. Perhaps it is not that the students are not prepared or ready for learning algebra, but that the teaching or curriculum to which the students have been exposed has been preventing them from developing mathematical ideas and representations they would otherwise be capable of developing."
 Other publications (in pdf) in the Early Algebra Project, see the publications page of its site
Paul Ernest (1992). The Philosophy of Mathematics Education. Falmer Press. questia
 I have yet to browse this one. It does not abundantly cross refer to Kitcher, 1984.
 p. 102 cites Freudenthal (1973) as criticizing Piaget on mathematical grounds (NB: Pieter van Hiele, reviewed in 1999, says Freudenthal is mistaken in his assessment of the work of Piaget)
Pierre M. van Hiele (1986). Structure and Insight: A Theory of Mathematics Education. Academic Press.
 [nog niet ingezien, aanwezig UB Leiden] [uitvoerig review in Educational Studies in Mathematics 1991, 95103, evenmin ingezien, niet in UB Leiden]
 Ik ben hier heel terughoudend. Het werk van Van Hiele heeft enorme invloed gehad op het denken van Hans Freudenthal, en op het Nederlandse reken en wiskundeonderwijs, neem ik aan. Maar vooralsnog zie ik het als een slecht met de literatuur verbonden persoonlijke theoretische constructie, niet goed geschraagd in empirisch onderzoek.
K. Gravemeijer, R. Lehrer, B. van Oers and L. Verschaffel (Eds) (2002). Symbolizing, modeling, and tool use in mathematics education. Kluwer. preview preview [Ik heb dit boek niet gezien, het is aanwezig in de KB maar niet ook als eBook. Ik verwacht niets nieuws]
 Lieven Verschaffel, Brian Greer, Erik de Corte: Everyday Knowledge and Mathematical Modeling of School Word Problems. 257276. abstract. Covers the ame ground as their 2000 book on word problems?
Koeno Gravemeijer (1997). Mediating between concrete and abstract. In P. Bryant and T. Nunes: Learning and teaching mathematics: An international perspective. Hove, England: Psychology Press.
 Explains Realistic Mathematics Education in a rather clumsy way.
 The problem, again, is sloppy theory and the lack of empirical research.
 However sympathetic the idea of guided discovery of mathematizing abstractions, it is still a very long shot from a full grown curriculum theory with proven applicability.
 One of the crucial things missing is research on implementation processes and how they affect the effectiveness of RME.
 The intentions of the protagonists place RME in my direct hits category, only to immediately get disqualified. Be warned that RME is a world wide hype in math education, however.
 The chapter bristles with naive notions about problem solving, expertness, what it is that mathematicians do when they mathematize, etcetera, etcetera. It is a pity to see some good ideas drowned in irrelevance. Koeno Gravemeijer is not the one to blame, the RME movement itself seems to be the source of the problems mentioned.
Eugene Maier (1977). Folk math. Instructor, 86 Feb, 8492.
 abstract Folk music has been defined as music that folks sing. Folk math is defined as math that folks do. It's logical, useful, sensible, discusses the ways you and others really use mathematics in the outside world, and it uses language that folks use. (ERIC Editor/RK)
 Maier's folk math does not seem to be exactly what I am looking for. There is another article by Eugene, available online, that will be interesting.
 Eugene Maier (1987). Paper and paper skills 'impede' math progress. Education Week, June 10, html

Eugene Maier (1998). Number sense/number sens. http://www.mlc.pdx.edu/articles/resources/gene/article6037.htm [dead link?]. The title is a pun on the word num.ber/numb.er. The article reviews a book by Dehaene:

Stanislas Dehaene (1997). The Number Sense: How the Mind Creates Mathematics. Oxford University Press. questia
 [Maier:] Further, experimental evidence shows that the human brain has not evolved "for the purpose of formal calculations." Remembering multiplication facts and carrying out algorithmic procedures are not our brains' forte. To do this successfully we turn to verbatim memory  that is, memorization without meaning  at the expense of intuition and understanding. The danger is that we become "little calculating machines that compute but do not think."
Thus, the author suggests that we deemphasize memorizing arithmetic tables and mastering paperandpencil algorithms. Instead, we should take advantage of our strength, which is our associative memory. This is what enables us to connect disparate data, use analogies to advantage, and apply knowledge in novel settings  all things that calculators don't do well. And above all, whatever we do in school, we should honor and nurture the vast amount of intuitive knowledge about numbers children bring to the educational process.
Cornoldi D.L.C. (1997). Mathematics and Metacognition: What Is the Nature of the Relationship? Mathematical Cognition, 3, 121139.
Bourne P.P.L.E.; Birbaumer J. N. (1998). Extensive Practice in Mental Arithmetic and Practice Transfer Over a Tenmonth Retention Interval. Mathematical Cognition, 4, 2146.
Mathematical Cognition, 1998, volume 4, number 2. Interesting articles:
 Editorial Mathematical Reasoning: Nature, Form, and Development , pp. 8183(3),
 What Did Weierstrass Really Define? The Cognitive Structure of Natural and _Continuity , pp. 85101(17) , Authors: Nunez R. E.; Lakoff G.,
 The Relationship between Young Children's Analogical Reasoning and Mathematical Learning , pp. 103123(21) , Authors: White C.S.; Alexander P. A.; Daugherty M.,
 Reasoning by Analogy in Solving Comparison Problems , pp. 125146(22) , Author: English L. D.,
 Pedagogical, Mathematical, and RealWorld ConceptualSupport Nets: A Model for Building Children's Multidigit Domain Knowledge , pp. 147186(40) , Author: Fuson K. C.
Hoard M. K.; Geary D. C.; Hamson C. O. (1999). Numerical and Arithmetical Cognition: Performance of Low and AverageIQ Children. Mathematical Cognition, 5, 6591.
 Abstract:
Neuropsychological and developmental models of number, counting, and arithmetical skills, as well as the supporting working memory and speed of articulation systems, were used as the theoretical framework for comparing groups of low and averageIQ children. The lowIQ children, in relation to their averageIQ peers, showed an array of deficits, including difficulties in retaining information in working memory while counting, more problem solving errors, shorter memory spans, and slower articulation speeds. At the same time, the lowIQ children's conceptual understanding of counting did not differ from that of their higherIQ peers. Implications for the relation between IQ and mathematics achievement are discussed.
Spinillo A. G.; Bryant P. E. (1999). Proportional Reasoning in Young Children: PartPart Comparisons about Continuous and Discontinuous Quantity. Mathematical Cognition, 5, 181197.
Dave Pratt and Richard Noss (2002). The MicroEvolution of Mathematical Knowledge: The Case of Randomness. Journal of the Learning Sciences 11, 453488
 p. 455: "In this article, we explore the growth of mathematical knowledge and in particular, seek to clarify the relation between abstraction and context. Our method is to gain a deeper appreciation of the process by which mathematical abstraction is achieved and the nature of abstraction itself, by connecting our analysis at the level of observation with a corresponding theoretical analysis at an appropriate grain size. In this article, we build on previous work to take a further step toward constructing a viable model of the microevolution of mathematical knowledge in context." (. . .) "Our explanation will employ the notion of situated abstraction as an explanatory device that attempts to synthesize existing micro and macrolevel descriptions of knowledge construction. One implication will be that the apparent dichotomy between mathematical knowledge as decontextualized or highly situated can be usefully resolved as affording different perspectives on a broadening of contextual neighborhood over which a network of knowledge elements applies."
Nancy K. Mack (1993). Learning rational numbers with understanding: The case of informal knowledge. In Thomas P. Carpenter, Elizabeth Fennema and Thomas A. Romberg: Rational numbers. An integration of research (p. 85132). Erlbaum. questia
 abstract Students come to instruction with a rich store of informal knowledge related to rational number concepts and procedures. Initially this informal knowledge is limited in three ways: (a) Students' informal strategies treat rational number problems as whole number partitioning problems, (b) students' informal conception of rational number influences their ability to reconceptualize the unit, and (c) students' informal knowledge initially is disconnected from their knowledge of formal symbols and procedures associated with rational numbers. However, appropriate instruction can extend students' informal knowledge so that these limitations are redressed and the informal knowledge provides a base for developing an understanding of formal symbols and procedures.
Stanislas Dehaene (2004). Evolution of human cortical circuits for reading and arithmetic: The 'neuronal recycling' hypothesis. In S. Dehaene, J. R. Duhamel, M. Hauser and G. Rizzolatti: From monkey brain to human brain. Cambridge, Massachusetts: MIT Press. pdf
of concept.
 This is a few steps removed from what is happening in classrooms. What intrigues me is that it is possible at all, for researchers like Dehaene, to connect highly cultural skills like reading and mathematics, to specific brain events.
 More work by Dehaene: see his site
Hilary Barth, Kristen La Mont, Jennifer Lipton, Stanislas Dehaene, Nancy Kanwisher, Elizabeth Spelke (2005). Nonsymbolic arithmetic in adults and young children. pdf of concept
 abstract Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on nonsymbolic numerosities. Previous research has demonstrated that human adults,
human infants, and nonhuman animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these nonsymbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on nonnumerical continuous quantities,
modalityspecific quantity information, the adoption of alternative nonarithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.
 [from the general discussion:] Advances in understanding of nonsymbolic
numerical abilities may allow educators to harness this primitive number sense to enhance early mathematics instruction.
Elizabeth S. Spelke (2000). Core knowledge. American Psychologist, 55, 1233—1243. (award address, Award for Distinguished Scientific Contributions) pdf
 What are core knowledge systems? Studies of human infants suggest that they are mechanisms for representing and reasoning about particular kinds of ecologically important entities and eventsincluding inanimate, manipulable objects and their motions, persons and their actions, places in the continuous spatial layout and their Euclidean geometric relations, and numerosities and numerical relationships.
 "My story begins with two core knowledge systems found in human infants and in nonhuman primates: a system for representing objects and their persistence through time and a system for representing approximate numerosities. Then I ask how young children may build on these two systems to learn verbal counting and to construct the first natural number concepts. Finally, I consider how the same systems may contribute to mathematical thinking in adults."
 . . . a large body of work, beautifully reviewed by Stanislas Dehaene (1997) in his book, The Number Sense, provides evidence that the large, approximate numerosity system plays an important role in our mature capacities to compare numbers and perform mental arithmetic.
 David Dobbs (2005). Big answers from little people. In infants, Elizabeth Spelke finds fundamental insights into how men and women think. Scientific American October issue

Pinker vs Spelke. The science of gender and science. A debate. Edge The Third Culture html [In a way, Lawrence Summers started this with his comment on sex deifferences in januari 2005. Summers, then president of Harvard, was 'een beetje dom' [meaning: he should not have said what he said]
Stanislas Dehaene (2001). Précis of the number sense. Mind \& Language, 16, 1636. pdf
 from the abstract: I postulate that higherlevel cultural developments in arithmetic emerge through the establishment of linkages between this core analogical representation (the ' number line ') and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution.
Lisa Feigenson, Stanislas Dehaene and Elizabeth Spelke (2004). Core systems of number. TRENDS in Cognitive Sciences, 8 July pdf
 Conclusion: Why is number so easy and yet so hard?
Read the answers in the article! Otherwise I would have to cite way too many words. Why did Newton as well as Leibniz invent a calculus "stretching their systems of numerical and mechanical knowledge so as to reconcile them"?
 I will stop my search here, for the moment, having returned to my initial observation about mathematics being artifically disconnected from the real world. Remember?

In research on conceptual change by Susan Carey, the work of DeHaene and others is used. Rmember: conceptual change is what is involved in changing from folk physics to Newtonian physics insights. More on the meno.htm page.
Well, where are we now? Dehaene (1997) puts my thesis upside down. It is not the case that intuitive notions about mathematics hamper understanding the 'real thing,' as it is in physics, but intuitive understanding is the strong point of the human mind that education should build on. I am surprised, I must have been naive in my search for a folk math on collision course with its scientific counterpart. I could have been warned by what has happened in the field of rational decision making, read the research by people like Gigerenzer: operating on the basis of intuitive notions in a number of real world cases gives better results than following the academic routines using expected utilities and all the rest of it. The concept of 'bounded rationality' probably is better known, first proposed by Herbert Simon. For a short review of a Gigerenzer book see here.
While most instructors would like their students to develop statistical reasoning, research shows that it is not enough to instruct students about the correct rules and concepts in order for them to develop an integrated understanding to guide their reasoning. It may be tempting to conclude that if students have been well taught and have performed well on exams, that they are able to reason correctly about statistical information. However, unless their reasoning is carefully examined, especially in applied contexts, these students may only be at the early stages of reasoning and not have an integrated understanding needed to make correct judgments and interpretations. [Garfield, 2002, from the summary]
Harold Jeffreys and Bertha Swirles Jeffreys (1946). Methods of mathematical physics. Cambridge at the University Press.
 This is pretty advanced stuff. The authors place many remarks on the appropriateness of presentation, as well as on typical difficulties experienced by physicists in using mathematics.
 p. 49: "Any physical measurement is the assignment of a single magnitude. Such magnitudes are called scalars. Physics may be defined as the study of the relations between scalars, so that from one set of measurements other sets, given the conditions of observation, can be predicted."
Kelly S. Mix, Janellen Huttenlocher, Susan Cohen Levine (2003). Quantitative Development in Infancy and Early Childhood. Oxford University Press.
Nora S. Newcombe & Janellen Huttenlocher (2000). Making space. The development of spatial representation and reasoning. MIT Press info [no, Susan Carey is not an author of this book]
Michael J. Jacobson and Robert B. Kozma: Innovations in Science and Mathematics Education. Advanced Designs for Technologies of Learning (p. 1146). Erlbaum. questia
Anna Sfard (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 136. pdf 3Mb
Situationist ideology.
Anna Sfard and Irit Lavie (2005). Why cannot children see as the same what grownups cannot see as different? — early numerical thinking revisited. Cognition and Instruction, 23, 237309. pdf
Situationist ideology.
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press info.
I have read portions of the book. Pseudoscience, in my opinion.
BenYehuda, M., Lavy, I., Linchevski, L., Sfard, A. (2005), Doing wrong with words or What bars students' access to arithmetical discourses. The Journal for Research in Mathematics Education, 36, 176—247. doc. Constructivistsituationist ideology.
David C. Geary (2006). Development of mathematical understanding. In D. Kuhl and R. S. Siegler: Cognition, perception, and language, Vol 2 (pp. 777810). W. Damon (Gen. Ed.), Handbook of child psychology (6th Ed.). New York: John Wiley & Sons. concept pdf
 See his site for more of his publications on math education and dyscalculia
Jill L. Quilici and Richard E. Mayer (1996). Role of Examples in How Students Learn to Categorize Statistics Word Problems. Journal of Educational Psychology, 88, 144. questia
 p. 144: "This study is concerned with using examples to help students recognize which problems require which tests, such as tests of means and correlations, rather than using examples to help students learn how to compute statistical tests."
Bharath Sriraman & Lyn English (Eds) (2010). Theories of mathematics education. Seeking new frontiers. Springer.
 a.o.: Bharath Sriraman & Lyn English: Surveying theories and philosophies of mathematics education (useful references from the literature, o.a. Brousseau) 732)  Richard Lesh and Bharath Sriraman: Reconceptualizing Mathematics Education as a Design Science  Gerald A. Goldin: Problem Solving Heuristics, Affect, and Discrete Mathematics: A Representational Discussion  Lyn English and Bharath Sriraman: Problem Solving for the 21st Century  Günter Törner, Katrin Rolka, Bettina Rösken, and Bharath Sriraman: Understanding a Teacher’s Actions in the Classroom by Applying Schoenfeld’s Theory TeachingInContext : Reflecting on Goals and Beliefs .
Reuben Hersh (2006). 18 Unconventional Essays on the Nature of Mathematics. Springer. [nog niet gevonden/gezien, http://www.springer.com/mathematics/book/9780387257174 voor smaple hoofdstuk, contents (o.a. Nunez 'Do real numbers move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. This seems to be another collection of superfluous materials (nothing really new presented here? The combination is the interesting thing, according to Hersh), quite interesting materials though. Not especially directed to questions of education/didactics. The preface is quite informative: pdf)
Deborah L. Bandalos, Sara J. Finney & Jenneke A. Geske (2003). A Model of Statistics Performance Based on Achievement Goal Theory. Journal of Educational Psychology, 95, 604616.
Dirk T. Tempelaar, Wim H. Gijselaers and Sybrand Schim van der Loeff (2006). Puzzels in statistical reasoning. Journal of Statistics Education, 14. html
 From the discussion: One of Garfield's (2002) conclusions is that the quality of teaching, and the performance of students on their exams, does not tell that much about students' reasoning skills and their level of integrated understanding. This study adds to that that also specific aspects of the quality of learning, such as approaching learning tasks in a committed but reproduction directed way, do not guarantee proper reasoning skills. Chance (2002) describes several instructional tools that allow 'thinking beyond the textbook'. The outcomes of this study emphasize the importance of using those types of activities and other tools discussed by Chance; neither traditional lecturing, nor textbookbased independent learning, can assure success. The study at the same time indicates what those tools should do beyond teaching some specific skills or knowledge: strengthen e.g. critical processing, and create a better balance in learning orientations and mental models of learning, since these are important in achieving statistical reasoning skills.
 The abstract and discussion of this article are somewhat mysterious. I will have to study the article itself, as well as some of its references, to get the whole picture. The article's discussion uses big words, the bulk of the article however does not seem to touch directly on the issues mentioned in the discussion. The Garfield and Chance references are available online
Arthur Bakker (2004). Design research in statistics education : on symbolizing and computer tools. Dissertation Utrecht University. 4 Mb pdf
 Research in the tradition of Realistic Mathematics Education RME.
 'Design research' is research in classroom situations, using didactical methods under the control of the researcher.
 From the abstract: "Diagrammatic reasoning consists of three steps: making a diagram, experimenting with it, and reflecting on the results. The research shows the importance of letting students make their own diagrams and discussing these. The computer tools seemed most useful during the experimentation phase. Remarkably, the best diagrammatic reasoning occurred only during class discussions without computers around. One recommendation is: only invest in using computer tools if all educational factors such as teaching, end goals, instructional activities, tools, and assessment are tuned to each other."
Beth L. Chance (2002). Components of Statistical Thinking and Implications for Instruction and Assessment. Journal of Statistics Education, 10 html
 This is an excellent article on the concept of statistical thinking. Nevertheless, there is no mention of naive statistical thinking. The reason for this omission might be that involving students in a course where they learn to think statistically in the way statisticians intend them to do, might quickly suppress any common sense tendencies to interpret the world's chance aspects. I do not for one second believe such to be the case, however. One reason to be a nonbeliever is the scientific research of Tversky, Kahneman, Gigerenzer, etcetera. Incidentally, Gigerenzer's research suggests that there are limits to the reasonableness of approaching all chance events in daily life in the statistician's way.
Joan Garfield (2002). The Challenge of Developing Statistical Reasoning. Journal of Statistics Education, 10 pdf
 "Research by this author on assessing statistical reasoning (see Garfield 1998a, 1998b), revealed that students can often do well in a statistics course, earning good grades on homework, exams, and projects, yet still perform poorly on a measure of statistical reasoning such as the Statistical Reasoning Assessment (Garfield 1998b). These results suggest that statistics instructors do not specifically teach students how to use and apply types of reasoning. Instead, most instructors tend to teach concepts and procedures, provide students opportunities to work with data and software, and hope that reasoning will develop as a result. However, it appears that reasoning does not actually develop in this way. Current research (see delMas, Garfield and Chance 1999) is focused on exploring and describing the development (and assessment) of statistical reasoning skill, particularly in the area of statistical inference."

"There is an abundance of research on incorrect statistical reasoning, indicating that statistical ideas are often misunderstood and misused by students and professionals alike. Psychologists (such as Kahneman, Slovic, and Tversky 1982) and educators (such as Garfield and Ahlgren 1988) have collected convincing information that shows how people often fail to use the methods learned in statistics courses when interpreting or making decisions involving statistical information. This body of research indicates that inappropriate reasoning about statistical ideas is widespread and persistent, similar at all age levels (even among some experienced researchers), and quite difficult to change." Garfield then discusses typical examples of these errors and misconceptions.

Garfield then goes into the question how best to teach statistics, and test for understanding statistics
 Garfield also reports on an interview study on how students who had completed an introductory course go about handling statistical problems. This has been reported on more fully in Chance, delMas and Garfield, in a book by BenZvi and Garfield (Kluwer), not available to me.
John B. Garfield (1998). The Statistical Reasoning Assessment: Development and Validation of a Research Tool. In Proceedings of the Fifth International Conference on Teaching Statistics, ed. L. PereiraMendoza, Voorburg, The Netherlands: International Statistical Institute, 781786. pdf
 abstract This paper describes the development and validation of the Statistical Reasoning Assessment (SRA), an instrument consisting of 20 multiplechoice items involving probability and statistics concepts. Each item offers several choices of responses, both correct and incorrect, which include statements of reasoning explaining the rationale for a particular choice. Students are instructed to select the response that best matches their own thinking about each problem. The SRA provides 16 scores which indicate the level of students' correct reasoning in eight different areas and the extent of their incorrect reasoning in eight related areas. Although the 16 scales represent only a small subset of reasoning skills and strategies, they provide useful information regarding the thinking and reasoning of students when solving statistical problems.
P. Sedlmeier (1999). Improving Statistical Reasoning: Theoretical Models and Practical Implication. Erlbaum. (Mentioned in Garfield, 2002) [I have to borrow this one: UB Leiden F.S.W. M&T 70.03/6427] questia
E. R. Michener (1978). Understanding understanding mathematics. Cognitive Science, 2, 361383.
Frederick Reif (1987). Interpretation of scientific or mathematical concepts: Cognitive issues and instructional implications. Cognitive Science: A Multidisciplinary Journal, 11:4, 395416.
 abstract Scientific and mathematical concepts are significantly different from everyday concepts and are notoriously difficult to learn. It is shown that particular instances of such concepts can be identified or generated by different possible modes of concept interpretation. Some of these modes use formally explicit knowledge and thought processes; others rely on less formal casebased knowledge and more automatic recognition processes. The various modes differ in attainable precision, likely errors, and ease of use. A combination of such modes can be used to formulate an "ideal" model for interpreting scientific concepts both reliably and efficiently. Comparisons are made with the actual concept interpretations of expert scientists and novice students. The discussion elucidates some cognitive and metacognitive reasons why the learning of scientific or mathematical concepts is difficult. It also suggests instructional guidelines for teaching such concepts more effectively.
Frederick Reif and Sue Allen (1992). Cognition for Interpreting Scientific Concepts: A Study of Acceleration. Cognition and Instruction, 9, 144.
 abstract Interpreting a scientific concept, that is, identifying or generating it properly in any particular instance, is a complex cognitive task. We analyze the underlying knowledge required to achieve such concept interpretation accurately and efficiently. This analysis is used to examine detailed observations of expert scientists and novice students interpreting the physics concept of acceleration. Most experts interpret the concept well in expected ways; however, even some experienced scientists exhibit marked deficiencies in concept interpretation. Novice students, even after using a scientific concept for some months, interpret it incorrectly in many cases. Their poor performance can be traced to concept knowledge that is largely incoherent, consisting of disconnected knowledge elements leading to frequent paradoxes. These knowledge elements are often flawed because of deficient applicability conditions or lack of discriminations. Furthermore, students' definitional or other general knowledge often cannot be properly applied, even if correctly stated. By directly addressing such deficiencies, instruction can substantially improve students' ability to interpret a scientific concept.
Nisbett, R. (1993), Rules for Reasoning, Mahwah, NJ: Lawrence Erlbaum. (Mentioned in Garfield, 2002) [I have to borrow this one: UB Leiden PSYCHO C6.2.38] questia
John D. Bransford, Ann L. Brown, and Rodney R. Cocking (Eds) (1999). How People Learn: Brain, Mind, Experience, and School. National Research Council. html.
 The book is online available
 from the executive summary "Science now offers new conceptions of the learning process and the development of competent performance. Recent research provides a deep understanding of complex reasoning and performance on problemsolving tasks and how skill and understanding in key subjects are acquired. This book presents a contemporary account of principles of learning, and this summary provides an overview of the new science of learning."
 See especially Chapter 7, Effective Teaching: Examples in History, Mathematics, and Science html: Multiplication with meaning, Understanding negative numbers, Guided discussion, Modelbased reasoning
 " Increasingly, approaches to early mathematics teaching incorporate the premises that all learning involves extending understanding to new situations, that young children come to school with many ideas about mathematics, that knowledge relevant to a new setting is not always accessed spontaneously, and that learning can be enhanced by respecting and encouraging children to try out the ideas and strategies that they bring to schoolbased learning in classrooms. Rather than beginning mathematics instruction by focusing solely on computational algorithms, such as addition and subtraction, students are encouraged to invent their own strategies for solving problems and to discuss why those strategies work. Teachers may also explicitly prompt students to think about aspects of their everyday life that are potentially relevant for further learning. "
Pearla Nesher (1986). Learning mathematics. A cognitive perspective. American Psychologist, 41, 1141122. Reprinted in Open University Press, Readings in the Psychology of Education, and in C. Hedley, J. Houtz and A. Baratta (Eds) (1990): Cognition, Curriculum, and Literacy. Norwood, NJ: Ablex.
 p. 1121: "It has become evident that there are conceptual guiding principles that underlie the execution of procedures end that systematic errors can make the underlying incorrect principles apparent. These errors evolve over a long process of learning and appear consistently; they have their roots in the learner's prior meaning systems."
 This article is dated, of course. It contains however an short description of a kind of 'mental model' research by the author, on procedures involving decimal numbers. I have used one example in ch. 5 of Toetsvragen ontwerpen.
Mitchell Rabinowitz and Kenneth E. Woolley (1995). Much Ado About Nothing: the Relation Among Computational Skill, Arithmetic Word Problem Comprehension, and Limited Attentional Resources. Cognition and Instruction, 13. questia
 from the abstract These results suggest that the cognitive processes involved in understanding an arithmetic word problem and in performing the required computations are best explained by a serial processing model. The absence of an interaction between problem comprehension and computational processes questions the notion that automatized retrieval facilitates problem solving and assertions suggesting that increasing computational requirements can interfere with problemsolving performance.
M. Le Corre, E. M. Brannon, G. van de Walle & S. Carey (2006). Revisiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52, pp. 130169.
 Look at Carey's site for a copy.
 about the concept of number, and how do toddlers do on it? It is the beginning of mathematics, isn't it?
Susan Carey (2001). Bridging the gap between cognitive development and developmental neuroscience: A case study of the representation of number. In C. A. Nelson & M. Luciana (Eds.) The Handbook of Developmental Cognitive Neuroscience. Cambridge, MA: MIT Press, 415432. pdf
Jerry Uhl and William Davis (1999). Is the mathematics we do the mathematics we teach? Contemporary Issues in Mathematics Education. MSRI Publications, vol. 36 pdf
 Calculus&Mathematica at the University of Illinois UrbanaChampaign site
Carol L. Smith, Gregg E. A. Solomon and Susan Carey (2005). Never getting to zero: Elementary school students' understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51, 101140. pdf
Susan Carey (2004). Bootstrapping and the origins of concepts. Daedalus, 5968. pdf
 p. 68: "We cannot just teach our children to count and expect that they will then know what 'two' or 'five' means. Learning such words, even without fully understanding them, creates a new structure, a structure that can then be filled in by mapping relations between these novel words and other, familiar concepts. And so eventually our children do know what 'five' means: through the medium of language and the bootstrapping process sketched here they have acquired a new concept."
Susan Carey (1998). Knowledge of number: Its evolution and ontogenesis. Science, 242, 641642.
 Look at Carey's site for a copy.
Sal Restivo and Deborah Sloan (2007). The Sturm und Drang of Mathematics: Casualties, Consequences, and Contingencies in the Math Wars. Philosophy of Mathematics Education Journal No. 20 (June 2007) doc
 abstract What is behind and what is at stake in the 'math wars?" In this chapter, we take a sociological step backward to consider the antagonists in this "war" and the sociocultural and historical contexts of their enmities. We explain what it means to claim that mathematics, particularly as taught in our schools, is a social construction, a social institution, and dependent upon social relations. This explanation is crucial to understanding the emergence of multicultural mathematics, ethnomathematics, alternative math, and radical math as valid alternatives to the study of traditional mathematics. It also gives a context for understanding the reactions these different perspectives have provoked within various factions of science education and mathematics education. We will demonstrate that this conflict has battlegrounds running all the way from the classroom to the Oval Office, and contradicts the goals of higher learning in our diverse society. In our conclusion we will explore the cultural significance of the math wars and pathways to resolution.
 The Math Wars interest me here only insofar as they touch on diactical matters. See also Klein (2007) [below]
David Klein and others (2005). The state of the state math standards 2005. Washington, D. C., Thomas B. Fordham Foundation. pdf
 Among the 'others': Bastiaan Braams.
 p. 6 "Indeed, as the reader will see in the following pages, the essential finding of this study is that the overwhelming majority of states today have sorely inadequate math standards. Their average grade is a 'high D' and just six earn 'honors' grades of A or B, three of each. Fifteen states receive Cs, 18 receive Ds and 11 receive Fs." "Tucked away in these bleak findings is a ray of hope. Three states — California, Indiana, and Massachusetts  have firstrate math standards, worthy of emulation. If they successfully align their other key policies (e.g., assessments, accountability, teacher preparation, textbooks, graduation requirements) with those fine standards, and if their schools and teachers succeed in instructing pupils in the skills and content specified in those standards, they can look forward to a topnotch K12 math program and likely success in achieving the lofty goals of NCLB."
David Klein (2007). A quarter century of US 'math wars' and political partisanship. Journal of the British Society for the History of Mathematics, 22. 2233. html preprint
 "This article traces the history of the US 'math wars' from 1980, and discusses the political polarizations that fuelled and resulted from the disagreements."
Gert Schubring (2012). 'Experimental pedagogy' in Germany, elaborated for mathematics  a case study in searching the roots of PME, Research in Mathematics Education, 14:3, 221235, DOI: 10.1080/14794802.2012.734968 abstract
epistemological beliefs
Not exactly an epistemological theme, but I nevertheless place it here: regard the design of achievement test items in mathematics as analogous to the training and/or practices of the mathematics teacher. The welldesigned test item 'teaches' as the mathematics teacher ideally would do; badly designed items are the ones showing the flaws mathematics teachers might exhibit also. A recent review is Da Ponte and Chapman (2006), directed however primarily to papers from the PME proceedings, PME being Psychology of Mathematics Education.
João Pedro da Ponte and Olive Chapman (2006). Mathematics teachers' knowledge and practices. In Angel Gutiérrez and Paolo Boero: Handbook of research on the psychology of mathematics education (p. 461494). Sense Publishers.
Bharath Sriraman & Lyn English (Eds). Theories of Mathematics Education. Seeking New Frontiers. Springer.
 Richard Lesh & Bharath Sriraram: Reconceptualizing mathematics education as a design science. (123146)
 Stephen R. Campbell: Embodied minds and dancing brains: New opportunities for research in mathematics education. (309332)
 Guershon Harel: DNRbased instruction in mathematics as a conceptual framework. (343367) [DNR: Duality, Necessity, Repeated Reasoning] “This section facuses mainly on two central concepts of DNR: way of understanding and way of thinking. As was explained in Harel (2008), these are fundamental concepts in DNR, in that they define the mathematics that should be tought in school. Judging from contemporary textbooks and years of classroom observations, teachers at all grade levels, including college instructors, tend to view mathematics in terms of ‘subject matter,’ such as definitions, theorems, proofs, problems and their solutions, and so on, not in terms of ‘conceptual tools’ that are necessary to construct such mathematical objects. (p. 355) ”
 Andy Hurford: Complexity theories and theories of learning: Literature reviews and syntheses. (567567591) [dynamical systems theory  general systems theory  radical embodied coginition  situated cogniiton]
 Bharath Sriraman, Matt Roscoe & Lyn English: Politicizings mathematics education: Has politics gone too far? Or not far enough? (621638)
E. J. Dijksterhuis (1925). Beschouwingen over de universitaire opleiding tot leeraar in wis en natuurkunde. (Commissie, ook: I. van Andel, H. J. E. Beth, P. Cramer) Bijvoegsel op het Nieuw Tijdschrift voor Wiskunde II, 8195. pdf
 "Daarnaast echter kan historische ontwikkeling ook voor het onderwijs in de wis en natuurkunde zelve zegenrijk werken. Er bestaat onder docenten in deze vakken niet zelden een vrij vergaand onvermogen, om zich te kunnen indenken in de soms bijna onoverkomelijke moeilijkheden, die de leerlingen kunnen ondervinden bij onderwerpen, waarmede hun eigen wetenschappelijk geoefende denken zoo volkomen vertrouwd is geraakt, dat ze de noodzakelijkheid van een nadere uitlegging heelemaal niet inzien, en een eerlijk gemeende, van alle inzicht verstoken verbazing over het telkens weer voorkomen van telkens dezelfde, toch zoo vaak waarschuwend aangewezen fouten. Wanneer echter een docent, die deze methodische fout begaat (want het is een fout, niet te kunnen begrijpen, dat men niet begrepen wordt) eens enkele eeuwen teruggaat in de geschiedenis der wetenschap, welker tegenwoordige rijkdommen hij bezit, dan zal hij menigmaal òf de denkers zelve, aan wier werk hij die rijkdommen dankt òf hun onmiddellijke voorloopers bevangen vinden in dezelfde fouten, die hem bij zijn leerlingen ergeren of worstelend met dezelfde moeilijkheden, waarin hij hen met ongeduld verstrikt ziet.
Er bestaan bezwaren tegen, om het geheel algemeen uit te spreken, maar voor tal van vakken kan men de stelling volhouden, dat de normale (d.w.z. telkens weer bij normaal begaafde leerlingen voorkomende) denkfouten van de tegenwoordige jeugd bij het aanleeren van een wetenschap menigmaal de denkfouten uit de jeugd dier wetenschap zelve zijn, waaruit onmiddellijk de conclusie volgt, dat hij, die de jeugd geestelijk te leiden heeft, vertrouwd moet zijn met den groei der wetenschappen, in welker beginselen hij hen inwijdt.
Er is wellicht geen sprekender voorbeeld voor de juistheid der uitgesproken stelling dan het onderwijs in mechanica, waarbij kennis van de geschiedenis dezer wetenschap ieder oogenblik weer in de foutieve denkbeelden en redeneeringen der leerlingen de beroemde historische dwalingen doet herkennen en waarbij zij dus inplaats van ergernis over wat domheid schijnt, begrip van moeilijkheid en inzicht in den weg tot verheldering doet ontstaan.
De historische beschouwingswijze schenkt echter nog meer dan deze verruiming en verzachting van oordeel; zij verleent in menig geval eerst de juiste waardeering voor de intellectueele waarde van de eerste beginselen van een vak en het juiste inzicht in hare beteekenis. Zij wekt veel tot nieuw leven, wat den, uitsluitend op het heden ingestelden, beoefenaar der wetenschap, misschien slechts een stoffige curiositeit lijkt; zij voert terug tot de tijden, toen de bekoring van het nieuwe, onvermoede, gedurfde al dat nu schijnbaar vanzelfsprekende omgaf. Zou zij dan niet de ware spheer scheppen, om al deze oude wetenschap weer levend te maken voor jonge menschen, waarvoor zij nog niet vanzelfsprekend is?
Historische ontwikkeling zou men daarom den docent in wis en natuurkundige vakken willen toewenschen, historische ontwikkeling echter, verkregen op wetenschappelijke wijze en behoed voor het dilettantisme, dat men menigmaal bij beoefening der wetenschapsgeschiedenis toelaatbaar acht. "
Beliefs of participants, of course, will impact on the effectiveness of mathematics education. I suspect the book edited by Leder and others (see below) is about this kind of thing. These beliefs seem to be beliefs about mathematics etcetera, not mathematical beliefs.
Leder, G.C.; Pehkonen, Erkki; Törner, Günter (Eds.) (2003). Beliefs: A Hidden Variable in Mathematics Education? Springer. Series: Mathematics Education Library, Vol. 31.
 I have not seen this one. For its contents, see pdf Promising chapters might be:
 Gilah C. Leder and Helen J. Forgasz: Measuring mathematical beliefs and their impact on the learning of mathematics: A new approach  Norma Presmeg: Beliefs about the nature of mathematics in the bridging of everyday and school mathematical practices
Op 't Eynde, P., De Corte, E. and Verschaffel, L. (2006). Epistemic dimensions of students' mathematicsrelated belief systems. International journal of educational research, 45, 5770 [KB electronisch, alleen in publieksruimte]
J. A. Scott Kelso (1995). Dynamic patterns. The selforganization of brain and behavior. Cambridge, Massachusetts: The MIT Press.
 Kelso does not specifically address mathematics or mathematics education. However, his treatment of situated cognition is directly relevant to the field, especially in contrast to epistemic beliefs of a distincly platonic character regarding what it is to do math.
Joe Redish and David Hammer (project: 20052009). Learning the Language of Science:
Advanced Math for Concrete Thinkers. University of Maryland Physics Education Research Group.
 "A project to study and model student difficulties with applying advanced mathematics in physics. A critical issue is the integration of modeling, interpretation, and evaluation skills with the more commonly stressed math processing skills."
 Proposal pdf =
 "Physics faculty have known for years that many of the students in their physics classes have trouble with math — both at the introductory and at the advanced level. Sometimes, they blame the math classes, calling for more math prerequisites. Sometimes, they blame the students, writing off large fractions of their class as 'just unable to do physics.' In our detailed study of an algebrabased physics class, neither lack of preparation in math nor lack of ability turned out to be the students' biggest problem. [23] In this project we videotaped nearly 1000 hours of student behavior in lab, tutorial, and group problem solving, took surveys, and collected thousands of pages of homework and exam data. We found that deciding what to do with the math was a bigger problem than how to do the math. In order to understand this result we need to consider how professionals use math and say something about how students think."
 The publications resulting from the project will be made available here. The first one is:
 E. F. Redish (2005). Problem solving and the use of math in physics courses. To be published in Proceedings of the conference World View on Physics Education 2005, Delhi India. pdf
 "From this analysis of the use of math in physics (and in science in general), we have learned a number of important results that have implications for our teaching. There's more to problem solving than learning 'the facts' and 'the rules.' What expert physicists do in even simple problems is quite a bit more complex than it may appear to them and is not 'just' what is learned (or not learned) in a math class. Helping students to learn to recognize what tools (games) are appropriate in what circumstances is critical."
Ian Stewart (2006). Letters to a young mathematician. The art of mentoring. Basic Books.
 On what it is to be a mathematician, to do mathematics, to learn mathematics
 Ian Stewart is directior of the Mathematics Awareness Center, University of Warwick. Amazing.
 The book "tells readers what Stewart wishes he had known when he was a student." It is, therefore, about the motivation of mathematics: why mathematics? Why me?
Ulrich Daepp and Pamela Gorkin (2003). Reading, writing, and proving. A closer look at mathematics. Springer.
 "Students will follow Pólya's fourstep process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them."
 "Historical connections are made throughout the text (. . ..)"
 I like the approach chosen by the authors of this course. The book is especially strong in motivating steps, proofs, etcetera. Quite extraordinary for a course in mathematics. The authors make it a point that the student should always be able to motivate her answers: explain why the answer is a good or a correct one. It can be overdone, of course: see issues in Common Core teaching in the US.
math disabilities
Math disabilities is not particularly the subject of this page. Yet a lot of children, think in the order of 1 in 20, might suffer in one way or another from one or more dosabilities touching on the capability to count, perform algorithmic tasks, etcetera. Standard curricula surely might harm these children, while it might be possible to teach them mathematics using another choice from the see of possible topics and themes. Therefore, a few recent articles on the subject.
Daniel B. Berch and Miché M. M. Mazzocco (Eds) (2007). Why is math so hard for some children? The nature and origins of mathematical learning difficulties. Paul H. Brookes Publishing.
 contents
 Laura Zamarian, Alex LópezRolón and Margarete Delazer: Neuropsychological Case Studies on Arithmetic Processing 245263.
 The discussion paragraph summarizes exactly what is important to know for instructional designers, I give a very extensive quote:
 p.259: "Neuropsychological case studies provide clear evidence that in adults, dufferent types of knowledge contribute to arithmetic processing. These types of knowledge are selectively vulnerable to brain degenration of acquired bain lesions and are functionally independent. Double dissociations have been described between fact knowledge and procedural knowledge, between fact knowledge and conceptual knowledge, and between procedural knowledge and conceptual knowledge. There is also evidence of a double dissociation between exact and approximate number knowledge. Though evidence from case studies suggests that these different components are separately implemented in the human brain, they benefit from their linking. Fact knowledge is only meaningful when supported by conceptual knowledge and is very often compensated for by procedural backup strategies. Procedures are less error prone when effectively supported by conceptual knowledge. Conceptual knowledge is more advantageous when more memorybased facts and procedures can be used. Exact fact knowledge is more efficiently processed when estimation abilities and approximate knowledge of number are available. Approximation is also essential in checking the plausibility of a result obtained by exact calculation. Although the cognitive architecture of number processing seems to be modularly organized, the cooperation of different types of knowledge leads to meaningful and efficient processing.
Regarding the numerical abilities of children, one should be cautious not to draw toosimple parallels between adults' and children's cognitive architecture."  'double dissociation' is a technical term: 'contrasting patterns of of impairment in two people.'
 'arithmetic facts': onedigit problems and their answers that are directly available from memory.
 'procedural knowledge' is knowledge of the sequence of steps that will solve a particular kind of problem.
 'conceptual knowledge': "the understanding of arithmetic operations and principles." This definition is not particularly helpful, basically the difference is that procedures might just be memorized without understanding. In fact, most or all arithmetics books from the middel ages well into the 18th century relied on tobememorized procedures only.
 For exact and approximate calculation, see Dehaene's publications.
Nancy C. Jordan, Laurie B. Hanich, and David Kaplan (2003). Arithmetic fact mastery in young children: A longitudinal investigation. Journal of Experimental Child Psychology 85, 103119.http://www.udel.edu/dkaplan/jordan_arithmetic.pdf [broken link? 122008]
Alfonso Caramazza and Alex Martin (Eds). The Organisation of Conceptual Knowledge in the Brain: Neuropsychological and Neuroimaging Perspectives. Psychology Press. questia
S. M. Riviera, A. L. Reiss, M. A. Eckert and V. Menon (2005). Developmental changes in mental arithmetic: evidence for increased functional specialization in the left inferior parietal cortex. Cerebral Cortex, 15, 17791790. pdf
Lee Swanson and Olga Jerman (2006). Math disabilities: A selective metaanaysis of the literature. Review of Educational Research, 76, 249274.
 p. 249: Several studies (. . .) estimate that approximately 6% to 7% of the schoolage population has mathematical disabilities."
 p. 270: "A primary problem for students with MD is their difficulty in peforming WM [Woring Memory] tasks."
 p. 249: "Although not a quantitative analysis, one of the most comprehensive syntheses of the cognitive literature on MD was provided by Geary (1993; see also Geary, 2003, for a review)."
 Personally, I get the impression from this article that MD is a container concept, that because of a possible multitude of causes it is effectively a continuous condition, not an allornothing phenomenon, and that it is rather difficult to discriminate between MD, Reading Disabilities, and somewhat lower intellectual capabilities.
D. C. Geary (1993). Mathematical disabilities: Cognitive, neuropsychological and genetic components. Psychological Bulletin, 114, 345362. pdf
 abstract Cognitive, neuropsychological, and genetic correlates of mathematical achievement and mathematical disability (MD) are reviewed in an attempt to identify the core deficits underlying MD. Three types of distinct cognitive, neuropsychological, or cognitive and neuropsychological deficits associated with MD are identified. The first deficit is manifested by difficulties in the representation or retrieval of arithmetic facts from semantic memory. The second type of deficit is manifested by problems in the execution of arithmetical procedures. The third type involves problems in the visuospatial representation of numerical information. Potential cognitive, neuropsychological, and genetic factors contributing to these deficits, and the relationship between MD and reading disabilities, are discussed
 For recent publications see his website
J. I. D. Campbell: Handbook of mathematical cognition Psychology Press.

About numerical representations : insights from neuropsychological, experimental, and developmental studies 3 abstract
 Number recognition in different formats 23 abstract
 Spatial representation of numbers 43 abstract
 Automaticity in processing ordinal information 55 abstract
 Computational modeling of numerical cognition 67abstract
 What animals know about numbers 85 abstract
 Rafael Nunuz and George Lakoff: The cognitive foundations of mathematics : the role of conceptual metaphor 109 abstract
 The young numerical mind : when does it count? 127 abstract
 Development of arithmetic skills and knowledge in preschool children 143 abstract
 Learning mathematics in China and the United States : crosscultural insights into the nature and course of preschool mathematical development 163 abstract
 Magnitude representation in children : its development and dysfunction 179 abstract
 Robert S. Siegler and Julie L. Booth: Development of numerical estimation : a review 197 abstract
 Karen C. Fuson and Dor Abrahamson: Understanding ratio and proportion as an example of the apprehending zone and conceptualphase problemsolving models 213 abstract
 Stereotypes and math performance 235 abstract
 David C. Geary and Mary K. Hoard: Learning disabilities in arithmetic and mathematics : theoretical and empirical perspectives 253 abstract
 Math performance in girls with Turner or fragile x syndrome 269abstract
 Number processing in neurodevelopmental disorders : spina bifida myelomeningocele 299 abstract
 Mark H. Ashcraft and Kelly S. Ridley: Math anxiety and its cognitive consequences : a tutorial review 315 abstract
 What everyone finds : the problemsize effect 331 abstract
 Campbell, Jamie I. D.; Epp, Lynette J.: Architectures for arithmetic. abstract
 J.A. LeFevre, D. DeStefano, B. Coleman & T. Shanahan (2005). Mathematical cognition and working memory. (361378). abstract No pdf available online; if you do not have access to the book, see instead Ashcraft & Krause 2007
 Mathematical problem solving : the roles of exemplar, schema, and relational representations 379 abstract
 Aging and mental arithmetic 397 abstract
 Calculation abilities in expert calculators 413 abstract
 Stanislas Dehaene, Manuela Piazza, Philippe Pinel and Laurent Cohen: Three parietal circuits for number processing 433 abstract
 Developmental dyscalculia 455 abstract
 Rehabilitation of acquired calculation and numberprocessing disorders 469
David C. Geary and Mary K. Hoard (2005). Learning disabilities in arithmetic and mathematics: Theoretical and empirical perspectives. In J. I. D. Campbell: Handbook of mathematical cognition (pp. 253267). New York: Psychology Press. concept pdf
David C. Geary, Mar K. Hoard, Lara Nugent, and Jennifer ByrdCraven (2007). Strategy use, longterm memory, and working memory capacity. In D. B. Berch and M. M. M. Mazzocco: Why is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities. New York. Brookes Publishing.
Miura (1987). Mathematics achievement as a function of language. Journal of Educational Psychology, 79, 7982. Google's html version of notforfree pdf
 p. 8182: "Japanese speakers in this study were more likely than English speakers to use a canonical base 10 construction for representing numbers concretely. If this is assumed to be an accurate behavioral representation of the child's mental image of number, the evidence suggests that for speakers of Asian languages, numbers are organized as structures of tens and ones; place value seems to be an integral part of that cognitive representation. Because this in turn affects mathematics performance, the results also help explain why it may be unnecessary to include placevalue representation with manipulative materials as a separate exercise in the Japanese mathematics curriculum."
 p. 82 "Further inquiry into the contribution of the placevalue concept to mathematics computation is needed. This also raises the question of how American children come to understand place value because it is not inherent in their numerical language."
 Wow.
Sharon Griffin (2007). Early intervention for children at risk of developing mathematical learning difficulties. In D. B. Berch and M. M. M. Mazzocco: Why is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities (p. 373395). New York. Brookes Publishing.
Steven Alan Hecht, Kevin J. Vagi, and Joseph K. Torgesen (2007). fraction skills and proportional reasoning. In D. B. Berch and M. M. M. Mazzocco: Why is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities (p. 121132). New York. Brookes Publishing.
Steven A, Hecht, (1998). Toward an information processing account of individual differences in fraction skills. Journal of Educational Psychology, 90, 545559. questia
 This is also the title of his dissertation
Anna J. Wilson (www). Dyscalculia Primer and Resource Guide. OECD html
 "The purpose of this primer is to explain the cognitive neuroscience approach to dyscalculia (including the state of research in this area), to answer frequently asked questions, and to point the reader towards further resources on the subject."
A. J. J. M. Ruijssenaars (1992). Rekenproblemen. Theorie, diagnostiek, behandeling. Lemniscaat. isbn 9060698576, 294 blz., tweede druk 1997, ingenaaid,
regulars
regulars French
Stella Baruk (1973). Échec et maths. Editions du seuil.
 François Boule (2002). Les difficultés rencontrées par les enfants en mathématiques. pdf
Renaud d'Enfert (2003). Inventer une géométrie pour l'école primaire au XIXe siècle. Tréma #22, 4149. html (better for the pictures); pdf. There is no English abstract.
The following works may be consulted/downloaded on http://gallica.bnf.fr/:
BUISSON Ferdinand (dir.), Dictionnaire de pédagogie et d'instruction primaire , Paris, Hachette, 1887.
DALSÈME Jules, Enseignement de l'arithmétique et de la géométrie, Mémoires et documents scolaires publiés par le Musée pédagogique, 2 e série, fascicule n° 32,Paris, Impr. nationale, 1889.
F. P. B., Abrégé de géométrie pratique appliquée au dessin linéaire, au toisé et au lever des plans, suivi des Principes de l'architecture et de la perspective, Tours, Mame ; Paris, Vve PoussielgueRusand, 1851 (21 e éd.).
LAMOTTE Louis, Cours méthodique de dessin linéaire et de géométrie usuelle applicable à tous les modes d'enseignement. Deuxième partie  Cours supérieur , Paris, Hachette, 1843.
SARDAN, Dessin linéaire géométrique, ou Géométrie pratique à l'usage des écoles primaires, Paris, L. Colas, 1876 (5 e éd.).
regulars German
Helge Lenné (1969). Analyse der Mathematikdidaktik in Deutschland. Nach dem Nachlass hrsg. von Walter Jung. Stuttgart: Ernst Klett Verlag.
 gründlich! & historisch!
 Makes use of Friedrich Paulsen (1885/191921/1960). Geschichte des gelehrten Unterrichts auf den deutschen Schulen und Universitäten vom Ausgang des Mittelalters bis zur Gegenwart. Mit besonderer Rücksicht auf den klassischen Unterricht. Berlin: de Gruyter (Unveränderter photomechanischer Nachdruck 1960).
 A rich source on the genesis of the current mathematics curriculum in Germany, and therefore in the Netherlands as well.
 For a recent update, a bird's eye's view in English, Günter Törner & Bharath Sriraman (2005). Issues and tendencies in German mathematics didactics: An epochal perspctive. pdf (the last contribution; other papers are interesting as well).
 Aufgabendidaktik p. 34:"Die in der Traditionellen Mathematik sichtbar werdende Stofforganisation läßt sich also folgendermaßen charakterisieren: Jedes Teilgebiet ist durch einen Aufgabentypus bestimmt, der systemaisch von einfachen zu komplexen Formen hin abgehandelt wird. Komplexe Aufgaben lassen sich dabei als Kombinationen einfacher Aufgaben auffassen. Die einzelnen Gebiete zeigen so in sich eine strenge Systematik. Sie sind jedoch untereinander wenig verknüpft, sondern werden jeweils relativ isoliert behandelt. 'Anwendungsaufgaben' werden jedem Gebiet gesondert zugeteilt und nur die Reihenfolge der Gebiete wird so festgelegt, daß ein Gebiet mögligst die notwendigen Voraussetzungen für die nächtsfolgenden liefert. Gebiete, die einmal behandelt worden sind, gelten insoweit als erledigt; der betreffende Stoff wird als bekannt vorausgesetzt; Querverbindungen anhand Uuml;bergreifender Ideen oder Strukturen werden — jedenfalls systematisch — kaum grundsätzlich herausgearbeitet. Es gilt stets 'das haben wir gehabt' oder 'das haben wir nicht gehabt'. Die Mathematik im ganzen tritt daher dem Schüler weniger als innere ideelle Einheit, sondern vielmehr als eine Sammlung von Aufgabentypen entgegevn. Dieses Prinzip der Stofforganisation in der Traditionellen Mathematik soll als 'Aufgabendidaktik' bezeichnet werde,
Die Benühungen von Felix Klein um 1900, einige den ganzen Unterricht verknüpfende Leitideen (Funktion und Abbildung) in die Gymnasialmathematik einzuführen, können nunmehr als erster nachhaltiger Versuch interpretiert werden, die Aufgabendidaktik zu durchbrechen und eine 'Fusion' der Einzelstoffe zu bewirken."
Götz Krummheuer (2007). Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions. Journal of Mathematical Behavior 26, 60—82. [May 2007: the first issue of 2007 is a sample issue, see the site]
 The articl's research is (geographically) located in the German situation, and theoretically in, among others, Toulmin's theory of argumentation. Video taped classroom sessions.
Rainer Kaenders (2006). Zahlbegriff, zwischen dem Teufel und der tiefen See.
Der Mathematikunterricht, Jahrgang 52 pdf
 "In diesem Artikel beschreiben wir zunächst einige Hintergründe des niederländischen Mathematikunterrichts soweit sie für eine Unterrichtsreihe zur Zahlentheorie von Bedeutung sind und gehen dann auf die Lernprozesse der Schüler ein. Ausgehend von überlegungen zur Kreativität stellen wir dar, inwiefern das Buch Der Zahlenteufel zur Lösung der gestellten didaktischen Probleme bei 17jährigen beitragen kann und erläutern dies anhand von Schülerarbeiten."
 Hans Magnus Enzensberger (1997). Der Zahlenteufel. Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben.
H. M. Enzensberger (1997/1998). The number devil. New York: Metroplitan Books, Henri Holt. isbn 0805057706
Erich Ch. Wittmann (2005). Eine Leitlinie für die Unterrichtsentwicklung vom Fach aus:
(Elementar) Mathematik als Wissenschaft von Mustern. pdf
Rainer Kaenders (2006). Zahlbegriff, zwischen dem Teufel und der tiefen See. pdf
Wolfram Meyerhöfer (2005). Was misst TIMSS? Einige Überlegungen zum Problem der Interpretierbarkeit der erhobenen Daten. pdf
 abstract The design and interpretation of aptitude tests in mathematics provoke questions as to what each of the set tasks actually measures. With structural or objective hermeneutics, this article introduces a methodology capable of discerning the various dimensions of skills required for a particular task. Not only does this approach allow for the recognition of the technical requirements of the task, its offputting factors and the image of the subject conveyed. The methodology is also able to locate the elements addressing the kind of skill that can more accurately be classified as 'test ability'. Focusing on an example selected from a TIMSS aptitude test, the discussion seeks to demonstrate that the theoretical construction employed in setting the test is hardly suited to define with any sense of permanence what is measured by each task.
Der Mathematikunterricht
regulars Dutch
Because of it's length as well as it's language, this chapter has been moved to a special webpage matheducation.dutch.htm.
regulars English
Thomas A. Romberg (1983). A common curriculum for mathematics. In Gary D. Fenstermacher & John I. Goodlad (Eds.) (1983). Individual Differences and the Common Curriculum (121159). NSSE, University of Chicago Press