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 hard-wired 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.


Physics education
Mathematics education
Life sciences education
Humanitieses education
Language education
conceptual change (paradigm shift)

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 3-21). 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 non-trivial 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 (39-74). London: Oxford University Press. pdf

vv 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

Stephan Körner (1960/1968/1986). The Philosophy of Mathematics. Dover.

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.

Mathematics—I want to say—teaches you, not just the answer to a question, but a whole language-game 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


[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 epsilon-delta ε-δ 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 Cauchy-Weierstraß ε-δ 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 ε-δ 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 ε-δ 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 ε-δ 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 ε-δ 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

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. 127-154). 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 (1883-1885) from Euclidis Elementa, edidit et Latine interpretatus est I. L. Heiberg, in aedibus B. G. Teubneri, 1883-1885. Edited, and provided with a modern English translation, by Richard Fitzpatrick. isbn 978-0-6151-7984-1 (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:

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 Anglo-American 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 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.

Daniel L. Schwartz, Taylor Martin and Jay Pfaffman (2005). How mathematics propels the development of physical knowledge. Journal of Cognition and Development, 6, 65-88. pdf

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 one-of-a-kind 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 Ben-Zeev 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 ACT-R model of cognitive arithmetic. Kognitionswissenschaft, 8, 5-19 pdf

Kinga Morsanyi and Denes Szucs (2014). Intuition in Mathematical and Probabilistic Reasoning. abstract [pdf downloaded via February 19 2015]

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. 127-154). Erlbaum contents

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, 311-342.

James Stigler (1997). Lessons in perspective: How culture shapes math instruction in Japan, Germany and the United States. pdf Zie vooral

Mark Levi (2009). The mathematical mechanic. Using physical reasoning to solve problems. .

Amy B. Ellis (2007). The influence of reasoning with emergent quantities on students' generalizations. Cognition and Instruction, 25, 439-478.

Stellan Ohlsson & Ernest Rees (1991). The function of conceptual understanding in the learning of arithmetic procedures. Cognition and Instrction, 8, 103-179.

Robyn Arianrhod (2005). Einstein's Heroes: Imagining the World Through the Language of Mathematics. Oxford University Press.

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. Ben-Zeev 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 Ben-Zeev 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 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, López-Rolón 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.; Ben-Zeev & 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 & Ben-Zeev (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 1607-1861
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 neuro-psychological 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 decision-making: formal decison-making is quite different from intuitive decision-making, 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.

Hugh Burkhardt (2007). Mathematical Proficiency: What Is Important? How Can It Be Measured? In Alan H. Schoenfeld: Assessing mathematical proficiency (77-97). Cambridge University Press. pdf .

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). 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, 78-107. first page only Anne Watson (2004) Red herrings: post-14 ‘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, 209-234 pdf

Alan H. Schoenfeld (Ed.) (2007). Assessing mathematical proficiency. Cambridge University Press.

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. 115-132). Hove, England: Psychology Press.

James W. Stigler and Ruth Baranes (1988). Culture and mathematics learning. In Ernst Z. Rothkopf: Review of research in education volume 15 1988-89. (253-306). 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—1988-89. (307-341). Washington, D.C.: American Educational Research Association. [Jstor]

Jean Lave (1988). Cognition in practice. Mind, mathematics and culture in everyday life. Cambridge University Press.

Theodore M. Porter (1995). Trust in numbers. The pursuit of objectivity in science. Princeton University Press. UP questia

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. CD-β Press, Center for Science and Mathematics Education. Dissertation Utecht University. access to chapter pdf's or integral text 4 Mb pdf

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, 431-445. pdf JStor

John R. Anderson, Lynne M. Reder, and Herbert A. Simon (1996). Situated learning and education. Educational Researcher, 25(4), 5-11. pdf

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, 1-6. gratis beschikbaar pdf

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

Christian Lebiere and John R. Anderson (1998). Cognitive arithmetic. In John R. Anderson, Christian Lebiere, and others: The atomic components of thought (297-342). Lawrence Erlbaum. questia

Susan B. Empson (1999). Equal Sharing and Shared Meaning: the Development of Fraction Concepts in a First-Grade Classroom. Cognition and Instruction, 17, 283-342. questia

Leone Burton (2004). Mathematicians as Enquirers: Learning about Learning Mathematics. Kluwer.

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. 25-58). [I have not yet located this one. It was referred to by Kieran, 1997, p. 133 (in Nunes and Bryant)]

Talia Ben-Zeev 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

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, 380-396. abstract Also via

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.

Efraim Fischbein (1987) Intuition in science and mathematics. An education approach. Reidel.

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, 285-365. questia

R. Duncan Luce and Patrick Suppes (1968). Mathematics. pdf

Bárbara M. Brizuela (2004). Mathematical development in young children. Exploring notations. New York: Teachers College.

Bárbara M. Brizuela and Analúcia Schliemann (2004). Ten-year-old students solving linear equations. pdf

Paul Ernest (1992). The Philosophy of Mathematics Education. Falmer Press. questia

Pierre M. van Hiele (1986). Structure and Insight: A Theory of Mathematics Education. Academic Press.

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]

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.

Eugene Maier (1977). Folk math. Instructor, 86 Feb, 84-92.

Cornoldi D.L.C. (1997). Mathematics and Metacognition: What Is the Nature of the Relationship? Mathematical Cognition, 3, 121-139.

Bourne P.P.L.E.; Birbaumer J. N. (1998). Extensive Practice in Mental Arithmetic and Practice Transfer Over a Ten-month Retention Interval. Mathematical Cognition, 4, 21-46.

Mathematical Cognition, 1998, volume 4, number 2. Interesting articles:

Hoard M. K.; Geary D. C.; Hamson C. O. (1999). Numerical and Arithmetical Cognition: Performance of Low- and Average-IQ Children. Mathematical Cognition, 5, 65-91.

Spinillo A. G.; Bryant P. E. (1999). Proportional Reasoning in Young Children: Part-Part Comparisons about Continuous and Discontinuous Quantity. Mathematical Cognition, 5, 181-197.

Dave Pratt and Richard Noss (2002). The Micro-Evolution of Mathematical Knowledge: The Case of Randomness. Journal of the Learning Sciences 11, 453-488

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. 85-132). Erlbaum. questia

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.

Hilary Barth, Kristen La Mont, Jennifer Lipton, Stanislas Dehaene, Nancy Kanwisher, Elizabeth Spelke (2005). Non-symbolic arithmetic in adults and young children. pdf of concept

Elizabeth S. Spelke (2000). Core knowledge. American Psychologist, 55, 1233—1243. (award address, Award for Distinguished Scientific Contributions) pdf

Stanislas Dehaene (2001). Précis of the number sense. Mind \& Language, 16, 16-36. pdf

Lisa Feigenson, Stanislas Dehaene and Elizabeth Spelke (2004). Core systems of number. TRENDS in Cognitive Sciences, 8 July pdf

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.

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. 11-46). 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, 1-36. 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, 237-309. 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.

Ben-Yehuda, 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. Constructivist-situationist ideology.

David C. Geary (2006). Development of mathematical understanding. In D. Kuhl and R. S. Siegler: Cognition, perception, and language, Vol 2 (pp. 777-810). W. Damon (Gen. Ed.), Handbook of child psychology (6th Ed.). New York: John Wiley & Sons. concept pdf

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

Bharath Sriraman & Lyn English (Eds) (2010). Theories of mathematics education. Seeking new frontiers. Springer.

Reuben Hersh (2006). 18 Unconventional Essays on the Nature of Mathematics. Springer. [nog niet gevonden/gezien, 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, 604-616.

Dirk T. Tempelaar, Wim H. Gijselaers and Sybrand Schim van der Loeff (2006). Puzzels in statistical reasoning. Journal of Statistics Education, 14. html

Arthur Bakker (2004). Design research in statistics education : on symbolizing and computer tools. Dissertation Utrecht University. 4 Mb pdf

Beth L. Chance (2002). Components of Statistical Thinking and Implications for Instruction and Assessment. Journal of Statistics Education, 10 html

Joan Garfield (2002). The Challenge of Developing Statistical Reasoning. Journal of Statistics Education, 10 pdf

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. Pereira-Mendoza, Voorburg, The Netherlands: International Statistical Institute, 781-786. pdf

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, 361-383.

Frederick Reif (1987). Interpretation of scientific or mathematical concepts: Cognitive issues and instructional implications. Cognitive Science: A Multidisciplinary Journal, 11:4, 395-416.

Frederick Reif and Sue Allen (1992). Cognition for Interpreting Scientific Concepts: A Study of Acceleration. Cognition and Instruction, 9, 1-44.

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.

Pearla Nesher (1986). Learning mathematics. A cognitive perspective. American Psychologist, 41, 114-1122. 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.

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

M. Le Corre, E. M. Brannon, G. van de Walle & S. Carey (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52, pp. 130-169.

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, 415-432. 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

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, 101-140. pdf

Susan Carey (2004). Bootstrapping and the origins of concepts. Daedalus, 59-68. pdf

Susan Carey (1998). Knowledge of number: Its evolution and ontogenesis. Science, 242, 641-642.

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

David Klein and others (2005). The state of the state math standards 2005. Washington, D. C., Thomas B. Fordham Foundation. pdf

David Klein (2007). A quarter century of US 'math wars' and political partisanship. Journal of the British Society for the History of Mathematics, 22. 22-33. html preprint

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, 221-235, 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 well-designed 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. 461-494). Sense Publishers.

Bharath Sriraman & Lyn English (Eds). Theories of Mathematics Education. Seeking New Frontiers. Springer.

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, 81-95. pdf

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.

Op 't Eynde, P., De Corte, E. and Verschaffel, L. (2006). Epistemic dimensions of students' mathematics-related belief systems. International journal of educational research, 45, 57-70 [KB electronisch, alleen in publieksruimte]

J. A. Scott Kelso (1995). Dynamic patterns. The self-organization of brain and behavior. Cambridge, Massachusetts: The MIT Press.

Joe Redish and David Hammer (project: 2005-2009). Learning the Language of Science: Advanced Math for Concrete Thinkers. University of Maryland Physics Education Research Group.

Ian Stewart (2006). Letters to a young mathematician. The art of mentoring. Basic Books.

Ulrich Daepp and Pamela Gorkin (2003). Reading, writing, and proving. A closer look at mathematics. Springer.

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.

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, 103-119. [broken link? 12-2008]

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, 1779-1790. pdf

Lee Swanson and Olga Jerman (2006). Math disabilities: A selective meta-anaysis of the literature. Review of Educational Research, 76, 249-274.

D. C. Geary (1993). Mathematical disabilities: Cognitive, neuropsychological and genetic components. Psychological Bulletin, 114, 345-362. pdf

J. I. D. Campbell: Handbook of mathematical cognition Psychology Press.

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. 253-267). New York: Psychology Press. concept pdf

David C. Geary, Mar K. Hoard, Lara Nugent, and Jennifer Byrd-Craven (2007). Strategy use, long-term 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, 79-82. Google's html version of not-for-free pdf

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. 373-395). 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. 121-132). New York. Brookes Publishing.

Steven A, Hecht, (1998). Toward an information processing account of individual differences in fraction skills. Journal of Educational Psychology, 90, 545-559. questia

Anna J. Wilson (www). Dyscalculia Primer and Resource Guide. OECD html

A. J. J. M. Ruijssenaars (1992). Rekenproblemen. Theorie, diagnostiek, behandeling. Lemniscaat. isbn 9060698576, 294 blz., tweede druk 1997, ingenaaid,


regulars French

Stella Baruk (1973). Échec et maths. Editions du seuil.

Renaud d'Enfert (2003). Inventer une géométrie pour l'école primaire au XIXe siècle. Tréma #22, 41-49. html (better for the pictures); pdf. There is no English abstract.

The following works may be consulted/downloaded on

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 Poussielgue-Rusand, 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.

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]

Rainer Kaenders (2006). Zahlbegriff, zwischen dem Teufel und der tiefen See. Der Mathematikunterricht, Jahrgang 52 pdf

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

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 (121-159). NSSE, University of Chicago Press

Romberg is sincere, I believe. He is not a declared constructivist. Romberg might offer a good sample of the thinking about education in de community of mathematicians.

B. Christiansen, A. G. Howson & M. Otte (Eds.) (1986). Perspectives on Mathematics Education. Reidel.

Sorry, I wrote my comment in Dutch. De bijdragen in dit boek zijn ongelijk van kwaliteit, maar mijn indruk is dat de kwaliteit hoog is. Het gaat niet om protagonisme van constructivistische ideeën. Er ontbreken twee belangrijke onderwerpen in het boek. Het is is een stevige basis in cognitieve psychologie, zoals dat ook begin tachtiger jaren goed mogelijk was geweest. Nu staan er in meerdere bijdragen in dit boek heel zinvolle gedachten over zaken die cognitief-psychologisch van aard zijn, terwijl de lezer eigenlijk niet meer in handen krijgt dan ervaringswijsheid van de betreffende auteurs. Het tweede onderwerp dat mist: de staat van het onderwijs in het rekenen, en dat vooral toegespitst op de ernstig tekortschietende kwalificaties van de leerkrachten in landen als de VS, UK, Nederland ook (maar misschien was het in Nederland tot de tachtiger jaren relatief nog relatief goed in orde). Zie voor de UK ene kort commentaar van Shad Moarif op de linkedin discussiegroep The Math Connection, 19 augustus 2012.

Lerner, Marcia (1994). Math Smart. Essential math for these numeric times. The Princeton Review. New York: Villard Books.

Sharon L. Senk, Denisse R. Thompson & Gwendolyn Johnson (????). Reasoning and proof in high school textbooks from the USA. pdf

Heather C. Hill (2007). Mathematical knowledge of middle school teachers: Implications for the No Child Left Behind policy initiative. Educational Evaluation and Policy Analaysis, 29, 95-114. read online free

Heather C. Hill, Deborah Loewenberg Ball and Stephen G. Schilling (2008). Unpacking pedagogical content knowledge: Conceptalizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372-400. pdf

Heather C. Hill, Brian Rowan, and Deborah Loewenberg Ball (2005). Effects of teachers' Mathematical Knowledge for Teaching on student achievement. American Educational Research Journal, 42, 371-406. [pdf available for download on the LMT website]

Heather C. Hill and Deborah Loewenberg Ball (2004). Learning mathematics for teaching: Results from California's mathematics professional development institutes. Journal for Research in Mathematics Education, 35, 330-351. [pdf available for download on the LMT website]

Heather C. Hill, Merrie L. Blunk, Charalambos Y. Charalambous, Jennifer M. Lewis, Geoffrey C. Phelps, Laurie Sleep and Deborah Loewenberg Ball (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26, 430-511. download free access pdf

Heather C. Hill, Deborah Loewenberg Ball & Stephen G. Schilling (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372-400.

R. Courant (1934/1937/1970). Differential and integral calculus. Volume one - second edition. Blackie.

George Pólya (1954/68). Mathematics and plausible reasoning. Volume I: Induction and analogy in mathematics. Volume II: Patterns of plausible inference. Princeton University Press.

NCTM (****). Curriculum focal points for prekinderkarten through grade 8 mathematics. National Council of Teachers of Mathematics. downloads page (The full document is 18.9 Mb, mainly because of a fancyful front page. Use the document-by-grade instead.)

Joan Ferrini-Mundy (2000). Principles and Standards for School Mathematics: A Guide for Mathematicians. Notices of the AMS, 47, 868-876. pdf


Richard A. Lesh, Eric Hamilton and James J. Kaput (Eds) (2007). Foundations for the future in mathematics education. Erlbaum.

Michael R. Harwell, Thomas R. Post, Yukiko Maeda, Jon D. Davis, Arnold L. Cutler, Edwin Andersen and Jeremy A. Kahan (2007). Standards-based mathematics curricula and secondary students' performance on standardized achievement tests. Journal for Research in Mathematics Education, 38, 71-101. read online

Dig up the evidence on effectivenes of methods. Predictably, the results are 'no difference': “No differences on the standardized achievement subtests emerged among the Standards-based curricula studied once background variables were taken into account.”

I will have to look into the design of the empirical study, the character and philosophy of the mathematics methods involved, the training of the teachers involved, and the kind of tests used.

This study definitely is not a study on what is happening in the classroom at the micro level, therefore the results probably will not touch on anything that might be important at this level.

June 2014. This study is not an experimental one. The journal is a NCTM one, I do not trust the NCTM any more. The authors definitely are reformers, see d.e. the citation below. I have dumped my copy of the article.

Further, advances in technology have made preoccupation with algorithmic procedures unrealistic

p. 95

Thomas R. Post, Michael R. Harwell, Jon D. Davis, Yukiko Maeda, Arnold L. Cutler, Edwin Andersen, Jeremy A. Kahan & Ke Wu Norman (2008). Standards-based mathematics curricula and middle-grades students' performance on standardized achievement tests. Journal for Research in Mathematics Education, 39, 184-2121. pdf

June 2014. This study is not an experimental one. The journal is a NCTM one, I do not trust the NCTM any more. Researching the achievements of students following curricula A or B, without any experimental control, is of no use at all. This research is for promotion of NCTM standards alone, I suspect. I dumped my copy of the article.

Maria Bartolini Bussi, Lyn D. English, Graham A. Jones, Richard A. Lesh, Dina Tirosh (Eds) (2002). Handbook of International Research in Mathematics Education. Erlbaum. questia

Richard Noss and Celia Hoyles (1996). Windows on mathematical meanings. Learning cultures and computers. Springer. See for a (limited) preview of the book.

William Hook, Wayne Bishop and John Hook (2007). A quality math curriculum in support of effective teaching for elementary schools. Educational Studies in Mathematics. free pdf

I walk from home to school in 30 minutes, and my brother takes 40 minutes. My brother left 6 minutes before I did. In how many minutes will I overtake him?

from Krutetski, 1976, p. 160, as cited and elaborated on in Smith and Thompson (2006), see below

John P. Smith III and Patrick W. Thompson (2006). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher and M. Blanton: (under review) Employing children's natural powers to build algebraic reasoning in the context of elementary mathematics. concept pdf

Sue Johnston-Wilder and John Mason (Eds) (2004). Fundamental Constructs in Mathematics Education. RoutledgeFalmer. questia

[constructivisme, bewonderaars van Freudenthal] Barbara Allen and Sue Johnston-Wilder (Eds) (2004). Mathematics Education: Exploring the Culture of Learning. Routledge-Falmer. questia

D. J. Struik (1934). On the foundations of the theory of probabilities. Philosophy of Science, 1, 50-70. jstor

Hans Freudenthal (1991). Revisiting mathematics education. Dordrecht: Kluwer.

Sacha la Bastide-van Gemert (2006). “Elke positieve actie begint met critiek” Hans Freudenthal en de didactiek van de wiskunde. Proefschrift Groningen (promotoren: Van Berkel, Van Maanen) pdf

Hans Freudenthal (1973). Mathematics as an educational task. Dordrecht: Reidel.

David Hilbert (1899/1902/1950). The foundations of geometry. Authorized translation by E. J. Townsend. Reprint 1950: Open Court. The Gutenberg Project pdf eBook

Bas Braams website : Comments on the June, 2003, New York Regents Math A Exam html; Mathematics in the OECD PISA Assessment html; OECD PISA: Programme for International Student Assessment. html; Mathematics in the OECD PISA Assessment html.

Antoine Bodin (2005). What does PISA really assess? What it doesn't? A French view. Joint Finnish-French Conference Teaching mathematics: beyond the PISA survey Paris 6 - 8 octobre 2005pdf

R. Biehler, R.W. Scholz, R. Strässer and B. Winkelmann (Eds) (1994). Didactics of Mathematics as a Scientific Discipline. Mathematics Education Library. Kluwer Academic Publishers: Dordrecht.

Thomas A. Romberg (Ed.) (1992). Mathematics Assessment and Evaluation: Imperatives for Mathematics Educators.. contents

Gerald Kulm (Ed.) (1990). Assessing Higher Order Thinking in Mathematics. American Association for the Advancement of Science. questia

Denise Jarrett and Robert McIntosh (2000). Teaching mathematical problem solving: Implementing the vision. A literature review. pdf

Talia Ben-Zeev and James Ronald (not dated). Is mathematical problem solving as unstable as it seems? [dead link? 12-2008]

K. VanLehn (1986). Arithmetic procedures are induced from examples. In J. Hiebert: Conceptual and procedural knowledge: The case of mathematics (pp. 133-179). Hillsdale, NJ: Lawrence Erlbaum Associates. [I have yet to borrow this one PEDAG 47.b.72]

R. J. Sternberg and T. Ben-Zeev (Eds) (1996). The nature of mathematical thinking (pp. 55-80). Erlbaum. questia

Thomas P. Carpenter, Elizabeth Fennema and Thomas A. Romberg (Eds) (1993). Rational Numbers: An Integration of Research. questiaErlbaum.

Lauren B. Resnick and Wendy W. Ford (1981). The psychology of mathematics for instruction. Erlbaum. questia [als eBook in KB]

R. Janssens, E. De Corte, L. Verschaffel, E. Knoors and A. Colemont (2002). National assessment of new standards for mathematics in elementary education in Flanders. Educational Research and Evaluation, 8, 197-225. [I have not yet seen this one?]

Freudenthal, H. ( 1983 ). Didactical phenomenology of mathematical structures. Dordrecht, Holland: Reidel.

Marc M. Sebrechts, Mary Enright, Randy Elliot Bennett, and Kathleen Martin (1996). Using Algebra Word Problems to Assess Quantitative Ability: Attributes, Strategies, and Errors. Cognition and Instruction, 14, 285-343. jstor

S. Vinner (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall: Advanced mathematical thinking (pp. 65-81). Kluwer.

Miguel R. Wilhelmi, Juan D. Godino and Eduardo Lacasta (2007). Didactic effectiveness of mathematical definitions. The case of absolute value. International Electronic Journal of Mathematics Education, 2, numer 2. pdf

Harel, G., & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced mathematical concepts. In David Tall (Ed.), Advanced mathematical thinking, (pp 81-94), Dordrecht: Kluwer Academic Publishers.

David Tall (2005). The transition from embodied thought experiment and symbolic manipulation to formal proof. pdf

David Tall (2004). Thinking through three worlds of mathematics. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281—288. pdf

David Tall (2006). The long-term cognitive development of different types of reasoning and proof. Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Universität Duisburg-Essen, Campus Essen, November 1 — 4, 2006. pdf

Eddie Gray, Demetra Pitta, Marcia Pinto and David Tall (1999). Knowledge Construction and diverging thinking in elementary and advanced mathematics. Educational Studies in Mathematics, 38, 111-133. pdf

I. Gal and J. B. Garfield (Eds) (1997). The assessment challenge in statistics education. IOS Press. All chapters now available online at this site.

Marja van den Heuvel-Panhuizen (1996). Assessment and realistic mathematics education. dissertation University of Utrecht. html [this page gives links to pdf's of the different chapters, or the dissertation in its entirity: pdf

Marja van den Heuvel-Panhuizen and Jerry Becker (2003). Towards a didactic model for assessment design in mathematics education. In Alan J. Bishop: Second International Handbook of Mathematics Education. Springer.

But let me tell you how I feel about the teaching of calculus. I think it has completely diverged from the way in which calculus is thought about and used by professionals. What is taught under the name of calculus has become a ritual, that's all. There is a long essay on education by Alfred North Whitehead which he starts by saying that the biggest problem is how to stop teaching inert matter. Most of what we teach in calculus is inert.

Peter D. Lax, in Donald J. Albers, Gerald L. Alexanderson and Constance Reid (Eds) (1990). More mathematical people. Contemporary conversations (p. 148). New York: Harcourt Brace Jovanovich.

The solution is to sweep away the cobwebs but, as one publisher has explained to me, for economic reasons that cannot be done. All those fancy textbooks cost so much to produce that at least fifty thousand copies have to be sold to cover production costs. That means they have to include everybody's pet topic; the result is that you get monstrosities that have no point of view at all.

Peter D. Lax, in Donald J. Albers, Gerald L. Alexanderson and Constance Reid (Eds) (1990). More mathematical people. Contemporary conversations (p. 148). New York: Harcourt Brace Jovanovich.

The point made by Peter Dax in the boxed citation is true not only for calculus courses, but for the educational enterprise in general, especially also the high school level. He might have added that teachers come under public pressure to teach all the material in the textbooks, what else is it printed for? [I can't remember exactly who studied this question when, and where it was published, b.w.] It is a point to remember when making a course content inventory in preparation for the design of achievement test items on it. More is less, or if you like: Less is more. And don't be too cautious in taking these steps back. Which makes me remember that there are more phenomena of this kind resulting in more inert matter in course content. The Dutch mathematician Beth warned against it, saying that after all even the inert matter is only a tiny fraction of our mathematical knowledge, so why teach it? Which makes me wonder how much 'inert matter' I have tried to study in my own student days: we had to do our Woodworth and Schlosberg (Experimental psychology) almost from cover to cover, as all the other textbooks on the rostrum.

TIMMS Trends in International Mathematics and Science Study: International site

pdf TIMMS 2007 report 80 Mb Jan de Lange Jzn. (1987). Mathematics, insight and meaning : teaching, learning and testing of mathematics for the life and social sciences. Dissertation University of Utrecht.

Lynn Arthur Steen (Ed.) (1990). On the shoulders of giants. New approaches to numeracy. National Research Council. Washington, D.C.: National Academy Press. isbn 0309042348


Lynn Arthur Steen (Ed.) (2001). Mathematics and democracy. The case for quantitative literacy. The National Council on Education and the Disciplines. pdf

H. W. Heyman (2004). Why teach mathematics? A focus on general education. Springer. Series: Mathematics Education Library, Vol. 33.

Bernard L. Madison and Lynn Arthur Steen (Eds) (2003). Quantitative literacy. Why literacy matters for schools and colleges. Proceedings of the National Forum on Quantitative Literacy held at the National Academy of Sciences in Washington, D.C. on December 1-2, 2001. Natonal Council on Education and the Disciplines. contents available as pdf-files.

Sfard, A. & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34, 14-22. pdf. Situationist ideology.

Joy Jordan and Beth Haines (2006). The Role of Statistics Educators in the Quantitative Literacy Movement. Journal of Statistics Education, 14,

Luli Stern and Andrew Ahlgren (2002). Analysis of Students' Assessments in Middle School Curriculum Materials: Aiming Precisely at Benchmarks and Standards. Journal of research in Science Teaching, 39, 889-910. pdf

Alan H. Schoenfeld (2005). What doesn't work: The challenge and failure of the what works clearinghouse to conduct meaningful reviews of studies of mathematics curricula. pdf

Karla Ballman (1997). Greater Emphasis on Variation in an Introductory Statistics Course.Journal of Statistics Education, 5 html

Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stinulating questions, he may give them a taste for, and some means of, independent thinking. [Polya, 1971, p. v]

Barbara Allen, Sue Johnston-Wilder (Eds) (2004). Mathematics Education: Exploring the Culture of Learning. RoutledgeFalmer. questia

Lorraine Mottershead (1977). Sources of mathematical discovery. Blackwell. isbn 0631102213

A book f small projects. It does not seem to be the intnetion of Lorraine Mottershead to offer a discovery-based math course. I hope.

Allen Newell (1983). The heuristic of George Polya and its relation to artificial intelligence. In Rudolf Groner, Maria Groner & Walter F. Bischof: Methods of heuristics. Lawrence Erlbaum. pdf van een concept-versie [het hele archief van Newell staat online, gescand]
scan 21 Mb Een voorloper in de vorm van een artikel in 1970: pdf

G. Polya (1957/1971). How to solve it. A new aspect of mathematical method. Princeton, New Jersey, Princeton University Press.

G. Polya (1945/1957). How to solve it. A new aspect of mathematical method. Doubleday Anchor Books

George Polya (1954). Mathematics and plausible reasoning. Volume I: Induction and analogy in mathematics. Volume II: Patterns of plausible inference. Princeton University Press.

George Polya (1962, 1965). Mathematical discovery. On understanding, learning, and teaching problem solving. Volume I, II. Wiley.

Alan H. Schoenfeld (1985). Mathematical problem solving. London: Academic Press.

Mary C. Shafer and Thomas A. Romberg (1999). Assessment in classrooms that promote understanding. In Elizabeth Fennema and Thomas A. Romberg: Mathematics Classrooms That Promote Understanding. Erlbaum. questia

Alan H. Schoenfeld (Ed.) (1994). Mathematical thinking and problem solving. Erlbaum. questia

Carol Seefeldt (2005). How to work with standards in the early childhood classroom. New York: Teachers College, Columbia University.

Judith T. Sowder (2006). Reconceptualizing Mathematics: Courses for Prospective and Practicing Teachers. pdf

Margaret Schwan Smith, Edward A. Silver and Mary Kay Stein (2005). Improving instruction in rational numbers and proportionality. Improving instruction in algebra. Improving instruction in geometry and measurement. Using cases to transform mathematics teaching and learning Volume 1, 2, 3. New York: Teachers College Press. (Ways of Knowing in Science & Mathematics)

Lesley R. Booth (1981). Child-methods in secondary mathematics. Educational Studies in Mathematics, 12

A. Kursat Erbas (2002). Teacher knowledge of student thinking and instructional practices in algebra. pdf

Helge Lenné (1969). Analyse der Mathematikdidaktik in Deutschland. Nach dem Nachlass hrsg. von Walter Jung. Stuttgart: Ernst Klett Verlag. (ao.: I Einfürung in Problematik und gegenwärtige Hauptrichtungen der Mathematikdidaktik - II Analyse der Zielsetzungen in der Mathematikdidaktik - III Grundsätzliches zur Methodologie der Mathematikdidaktik und ihre historischen und sozialen Bedingungen)

Chris Rasmussen. Project: Differential Equations: Building a Theory of Student Reasoning and Understanding. Started in 2005.

C. R. Gallistel and Rochel Gelman (2005). Mathematical cognition. In K Holyoak & R. Morrison (Eds). The Cambridge handbook of thinking and reasoning. Cambridge University Press (pp 559-588) pdf

D. Grouws (Ed.) A handbook of research on mathematics teaching and learning. NY: MacMillan

Robert S. Siegler, Clarissa A. Thompson & Michael Schneider (2011 accepted). An integrated theory of whole number and fractions development. Cognitive Psychology. pdf

J. L. Booth & R. S. Siegler (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79(4), 1016–1031. pdf

R. S. Siegler and M. Robinson (1982). The development of numerical understandings. In H. W. Reese and L. P. Lipsett (Eds.): Advances in child development and behavior, 16, (pp. 242-312). pdf1 and pdf2

B. Rittle-Johnson and R. S. Siegler (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). East Sussex, UK: Psychology Press. pdf

B. Rittle-Johnson, R. S. Siegler and M. W. Alibali (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362. pdf

R. S. Siegler (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. G. Martin and D. E. Schifter (Eds): A research companion to principles and standards for school mathematics (pp. 119-233). Reston, VA: National Council of Teachers of Mathmatics. pdf

Annie and John Selden (1997). Should mathematicians and mathematics educators be listening to cognitive psychologists? MAA Online html

Jeremy Kilpatrick and Jane Swafford (Eds) (2002). Adding it up. Helping children learn mathematics. NAP piecewise html

Brian Butterworth (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46, 3-18. pdf

Raymond Duval (200 ). A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, 61.

Joseph Klep (2000). Arithmeticus: A DPS-based model for arithmetical competence. J. of Interactive Learning Research, 11, 465-484. pdf

Joost Klep, Jos Letschert, Annette Thijs (2004). What are we going to learn? Choosing educational content. Netherlands Institute for Curriculum Development, SLO, Enschede. pdf

Elmar Cohors-Fresenborg and Christa Kaune (2001). The metaphor 'contracts to deal with concepts' as a structuring tool in algebra. pdf

Mathematical Sciences Education Board, National Research Council (1998). High School Mathematics at Work: Essays and Examples for the Education of All Students. free pdf available



The history of mathematics is extremely important for an understanding of what is happening in classrooms and in individual learning. One of the possibilities here is to use mathematical history as an asset in math education. Quite another subject is the history of mathematics education, but it might be as revealing as the history of mathematics itself.

In the history of mathematics in education, the Mathematical Tripos examinations at Cambridge University (around 1800) must have been very influential. The examination was highly competitive, and as such has left many traces in today's examination traditions, as it has in the way we think mathematics achievement in education should be assessed. If only because legions of professors have used mathematics as a convenient vehicle to cool out a large part of the student body (in whatever curriculum you like to name, but especially, of course, economics, technology and other sciences, and mathematics itself) [If you doubt this, contact me]

Patricia Baggett & Andrzej Ehrenfeucht (not dated). Content and teaching methods in elementary school mathematics. pdf

David Baker, Hilary Knipe, John Collins, Juan Leon, Eric Cummings, Clancy Blair and David Gamson (2010). One Hundred Years of Elementary School Mathematics in the United States: A Content Analysis and Cognitive Assessment of Textbooks From 1900 to 2000. Journal for Research in Mathematics Education, 41, 383-423.

A. G. Howson (1982). A history of mathematics education in England. Cambridge University Press.

Benchara Branford (1908). A study of mathematical education; including the teaching of arithmetic. Clarendon Press.

Frank J. Swetz (2009). Culture and the development of mathematics: An historical perspective. In Brian Greer, Swapna Mukhopadhyay, Arthur B. Powell and Sharon Nelson-Barber: Culturally responsive mathematics education (11-42). Routledge. [some pages available in]

David Eugene Smith (1923). Mathematics. London: George G. Harrap. Introduction by Sir Thomas Little heath.

David E. Joyce's web page History of Mathematics page listing English works on the subject. Here also the complete text of Euclid's Elements html

direct hit historical

The idea is that the history of mathematics will show what kind of problem induced the development of a particular technique or solution, will show how difficult it was for new concepts to get broadly understood and accepted, etcetera. This kind of knowledge might make one more aware of how difficult it must be for students to master even very simple mathematical concepts (contemporary neurocognitive science will have more to say on the subject, for example: why it is that some things very simple are so difficult to master; why it is that some things very complex are relatively easy to understand). The next idea surely will be to use the history of mathematics in the classroom to motivate etcetera students.

Nerida Ellerton & Ken Clements (2013). Rewriting the History of School Mathematics in North America 1607-1861. (te leen als eBook bij de KB)

John Fauvel and Jan van Maanen (Eds) (2000). History in mathematics education: The ICMI Study. Kluwer Academic Publishers. Also distributed by Springer

Because of its ICMI ancestry I expect the contributions to be loaded on constructivism. I have not yet checked its contents on this issue.

The MacTutor History of Mathematics archive web site University of St Andrews, School of Mathematics and Statistics.

regular historical

Reinhard Laubenbacher and David Pengelley (Web site). Teaching with original historical resources in mathematics.

University of Michigan Historical Math Collection html

Ubiratan D'Ambrosio (2003). Stakes in mathematics education for the societies of today and tomorrow. Monographie de L'Enseignement Mathématique 39 (2003), p. 301—316 pdf

I. Todhunter (1949). A history of the mathematical theory of probability from the time of Pascal to that of Laplace. Chelsea Public. Available online

David Dennis (2000). The Role of Historical Studies in Mathematics and Science Educational Research. In Anthony E. Kelly and Richard A. Lesh: Handbook of Research Design in Mathematics and Science Education. Erlbaum. questia

Jens Høyrup (2002). Conceptual divergence - Canons and taboos - And critique. Reflections on explanatory categories. Paper presented to The Sixth International Conference on Ancient Mathematics Delphi, 18th to 21st July 2002. pdf

D.R. Bellhouse (2003). Probability and Statistics Ideas in the Classroom — Lessons from History. Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada N6A 5B7 ~iase/publications/13/Bellhouse.pdf [broken link? 12-2008]

Helen M. Walker (1931). Studies in the History of Statistical Method With Special Reference to Certain Educational Problems. Williams and Wilkins.

Louis M. Friedler (2004). Calculus in the US: 1940-2004. pdf

D. J. Struik (1933). Outline of a history of differential geometry. I, II. Isis, 19, 92-120, 161-191. jstor

gif/treviso.jpg Frank J. Swetz (1987). Capitalism & arithmetic. The new math of the 15th century. Including the full text of the Treviso arithmetic of 1478, translated by David Eugene Smith. La Salle, Illinois: Open Court Publishing Company.

Shinya Yamamoto (2006). The process of adapting a German pedagogy for modern mathematics teaching in Japan. Paedagogica Historica, 42, 535-545.

Cornell University Library: Historical Mathematics Monographs site

John Derbyshire (2007). Unknown quantity. A real and imaginary history of algebra. A Plume Book.

German historical

Daniel Coray, Fulvia Furinghetti, Hélène Gispert, Bernard R. Hodgson and Gert Schubring (Eds.) (2003). One Hundred Years of L’Enseignement Mathématique. Moments of Mathematics Education in the Twentieth Century. Genève. Proceedings of the EM-ICMI Symposium Geneva, 20-22 October 2000. pdf

Gert Schubring (1981). Mathematics and teacher training: Plans for a polytechnic in Berlin. Historical Studies in the Physical Sciences, 12, 161-194. fc

Gert Schubring (2006). Researching into the history of teaching and learning mathematics: The state of the art. Paedagogica Historica; International Journal of the History of Education, 42 Special Issue:   History of Teaching and Learning Mathematics. 665-677.

A. Karp and G. Schubring (eds.) (2014). Handbook on the History of Mathematics Education DOI 10.1007/978-1-4614-9155-2_2, Springer Science+Business Media New York 2014 [UB Leiden, alleen online, augustus 2016] previews

Moritz Cantor (1922-1924). Vorlesungen über Geschichte der Mathematik. 4 Bände.

Hugo Grosse (1901/1965). Historische Rechenbücher des 16. und 17. Jahrhunderts und die Entwicklung ihrer Grundgedanken bis zur Neuzeit : ein Beitrag zur Geschichte der Methodik des Rechenunterrichts. Wiesbaden: Sändig.

Lorey, Willhelm Das Studium der Mathematik an den deutschen Universitäten selt Anfang des 19. Jahrhunderts , Abhandlungen über den mathematischen Unterricht in Deutschland, veranlaßt durch die internationale mathematische Unterrichtskommission, herausgegeben von F. Klein, Bd.III, Heft 9. Leipzig und Berlin, B. G. Teubner, 1916. XII, 428 S. und 13 Abbildungen. Preis geh. M. 12.-, geb. M. 14.-.

Hermann Schubert (1898/2004). Elementare Arithmetik und Algebra. eBook #11925 Cornell University, Joshua Hutchinson, Arno Peters and the Online Distributed Proofreading Team. Gutenberg Project pdf

Dutch historical

Because of it's length as well as it's language, this chapter has been moved to a special webpage matheducation.dutch.htm.

French historical

Recherche IREM-INRP: Histoire et enseignement des mathématiques : rigueurs, erreurs, raisonnements. Journées INRP de Lyon 14 et 15 juin 2006 pdf

Michel Delord (1996). La place de la géométrie dans l'enseignement des mathématiques en France: de la réforme de 1902 à la réforme des mathématiques modernes. Les Sciences au Lycée, 1996. pdf

Alfred Binet (1899). La pédagogie scientifique. L'Enseignement Mathématique. 1, 29-38. pdf

Alexis-Claude Clairaut (1741/2006). Eléments de géométrie. Paris: David Fils. Reprint Editions Jacques Gabay site

Gallica site

English historical

IHCM International Commission on the History of Mathematics site

Jacob Klein (1936/1968). Greek mathematical thought and the origin of algebra. MIT Press. Translated by Eva Brann from Die griechische lLogistik und die Entstehung der Algebra. tweet

The historical materials show the impossibility of discovery learning in mathematics.

Walter William Rouse Ball (1919). A short account of the history of mathematics. London: Macmillan. pdf pages

David E. Joyce (www). History of mathematics: Textbooks html

David E. Joyce (www). Euclid's Elements. html

Jörn Bruhn (****). Mathematics Education and Comparative Studies: Two Examples. Mathematics Education and Comparative Studies pdf

Florian Cajori (1928,1929/1993). A history of mathematical notations. Vol. I Notations in elementary mathematics; Vol II Notations mainly in higher education. Dover re-issue.

Claude Irwin Palmer (1924). Practical calculus for home study. McGraw-Hill.

Dirk J. Struik (1948). A concise history of mathematics. Dover. questia

David Eugene Smith (1934). A history of mathematics in Amerca before 1900. Mathematical Association of America. questia

Olof Magne (2003). Literature on special educational needs in mathematics. A bibliography with some comments. (4th Ed.) (Educational and Psychological Interactions, 124). Malmö, Sweden: School of Education. pdf

E. Verhelst (1888). 1800 questions mathématiques posées aux examens d'admission avec la résolution des questions. Bruxelles: Librairie Alfred Castaigne.

Joan L. Richards (1988). Mathematical visions. The pursuit of geometry in Victorian England. Academic Press.

oral exercise Edward Lee Thorndike (1921). The New Methods in Arithmetic. Rand McNally. read online

Lee J. Cronbach and Patrick Suppes (Eds) (1969). Research for tomorrow's schools: Disciplined inquiry for education. London: Collier-Macmillan Limited. The book is a committee-report, the other members of the Committee on Educational Research of the National Academy of Education being James S. Coleman, Lawrence A. Cremin, John I. Goodlad, Calvin Gross, David M. Jackson & Israel Scheffer.

Edward L. Thorndike (1922). The psychology of arithmetic. New York: Academic Press. read online; pdf ophalen

portrait De Morgan Augustus De Morgan (1831/1910). Study and difficulties of mathematics.. Chicago: Open Court. Third reprint edition. Portrait: to the right. pdf 22 Mb or page wise here

Charles Davies (1851). The logic and utility of mathematics with the best methods of instruction explained and illustrated.. New York: Barnes. pdf 33Mb

Luis Radford (2004). From Truth to Efficiency: Comments on Some Aspects of the Development of Mathematics Education. Canadian Journal of Science, Mathematics and Technology Education, 4, 551-556. pdf

Patrick Suppes (1982). On the effectiveness of educational research. D.B.P. Kallen, G.B. Kosse, H.C. Wagenaar, J.J.J. Kloprogge, and M. Vorbeck (Eds.), Social Science Research and Public Policy-making: A Reappraisal. Windsor, Berks., England: NFER-Nelson, pp. 255-270. First published by Foundation for Educational Research in the Netherlands (SVO), 1982. pdf

Early algebra, early arithmetic site (A. D. Schliemann and others)

links history of mathematics (education)

The Math Archives Topics in Mathematics History of mathematics html - Mathematics education html



Freudenthal Instituut English site


L'Enseignement Mathématique Volume 1 (1899) - ff site


Philosophy of Mathematics Education Journal ISSN 1465-2978 (Online) html of Paul Ernest (Editor) site.

Historical Math Monographs Cornell University Library site

The MacTutor History of Mathematics Archive site

Gilbert Strang (1991). Calculus. Wellesley-Cambridge Press. Available online as MIT OpenCourseWare, including the Instructor's Manual and a Study Guide.

Recent doctoral dissertations in statistics education IASE site

International Statistical Literacy Project of the International Association for Statistical Education site

Consortium for the Advancement of Undergraduate Statistics Education site

Journal of Statistics Education site, articles are online available for free. Congratulations.

Statistics Education Research Journal archive

Journal of Mathematical Behavior site

Educational Studies in Mathematics site

Mathematical Thinking and Learning site

Mathematical Cognition site

Journal for Research in Mathematics Education site

J. of Comps in Math. & Sc. Teaching

Teaching Mathematics Applications

Journal of mathematics teacher education.

ICMI/IASE Study: Statistics Education in School Mathematics: Challenges for Teaching and Teacher Education Conference to be held june 30 - july 4 2008 in Monterrey, Mexico, ITESM. site, discussion paper pdf

The International Commission on Mathematical Instruction ICMI site

Harry C. Barber (1924). Teaching junior school mathematics. Houghton Mifflin. digitized

F. Gregory Ashby & Matthew J. Crossley (2012). Automaticity and Multiple Memory Systems Interdisciplinary Reviews; Cognitive Science (Advanced Review) dead link [try sci-hub] of abstract

Sebastian Rezat, Mathias Hattermann & Andrea Peter-Koop (Eds.) (2014). Transformation - A Fundamental Idea of Mathematics Education. Springer. [eBook in KB]

Xinrong Yang (2013). Conception and Characteristics of Expert Mathematics Teachers in China Springer. [eBook in KB] previews

Qin Dai & Ka Luen Cheung (2016). The wisdom of traditional mathematical teaching in China. pdf ch. 1

In Lianghuo Fan a.o.: How Chinese teach mathematics. Perspectives from insiders. (3-42) [I have not seen this book, not in UB Leiden, not in KB]

Ann Dowker*, Amar Sarkar and Chung Yen Looi (25 April 2016). Mathematics Anxiety: What Have We Learned in 60 Years? Front. Psychol., 25 April 2016 | webpage

Paul Ernest (2000). Why teach mathematics? This chapter is published in Why Learn Maths?, edited by John White and Steve Bramall, London: London University Institute of Education, 2000. page. Translated in Dutch: Waarom geven we wiskunde? Nieuwe Wiskrant (december 2003) 36-40. pdf .

Colleen M. Ganley and Amanda L. McGraw (2016). The Development and Validation of a Revised Version of the Math Anxiety Scale for Young Children Frontiers in Psychology open access

JSME (2000). Mathematics education in Japan. Japan Society of Mathematical Education. no isbn, not for sale.

H. B. Griffiths & A. G. Howson (1974). Mathematics: Society and curricula. Cambridge University Press. isbn 0521098920 met manuscript van bespreking door Sieb Kemme

Quite interesting.

Randall I. Charles & Edward A. Silver (Eds.) (1988). The Teaching and Assessing of Mathematical Problem Solving. Research Agenda for Mathematics Education Series. Volume 3. Erlbaum. 0873532678

Schmidt, W.H., McKnight, Curtis C., Raizen, S. (Eds.) (1997). A Splintered Vision. An Investigation of U.S. Science and Mathematics Education. Kluwer. 0792344413 info

Begle, Edward Griffith (1979). Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature. Mathematical Association of America / National Council of Teachers of Mathematics. 0883854309

Vi-Nhuan Le and others (2006). Improving Mathematics and Science Education A Longitudinal Investigation of the Relationship Between Reform-Oriented Instruction and Student Achievement. RAND Education. 9780833039644

Duytsche Mathematique and the Building of a New Society: Pursuits of Mathematics in the Seventeenth-Century Dutch Republic. Fokko Jan Dijksterhuis (2017). In Lesley B. Cormack, Steven A. Walton, and John A. Schuster. : Mathematical Practitioners and the Transformation of Natural Knowledge in Early Modern Europe. Springer. . [eBook in KB] info

Clare Tikly & Alison Wolf (Eds.) (2000). The maths we need now: Demands, deficits and remedies. Bedford Way Papers. 0854736166

I am afraid this is a bullshit book. Sorry, Alison. Analyses keep to surface features, yet that is absolutely not sufficient. Math is an extremely complicated subject, needing many, many years of sustained efforts to come to grips with it. Just the basics, mind you! Not the ‘mathematical thinking’ of professional mathematicians, of course.

Report of the Task Group on Conceptual Knowledge and Skills Francis “Skip” Fennell, Chair Larry R. Faulkner Liping Ma Wilfried Schmid Sandra Stotsky Hung-Hsi Wu pdf

Space, Time and Number in the Brain : Searching for the Foundations of Mathematical Thought Dehaene, Stanislas; Brannon, Elizabeth  Elsevier Science  2011

Development of Mathematical Cognition : Neural Substrates and Genetic Influences Berch, Daniel B.; Geary, David C.;and more  Elsevier Science  2015

Camilla Gilmore, Silke M. Göbel & Matthew Inglis (2018). An introduction to mathematical cognition. Routledge. isbn 9781138923959 info

Alyssa J. Kersey, Emily J. Braham, Kelsey D. Csumitta, Melissa E. Libertus and Jessica F. Cantlon (2018). No intrinsic gender differences in children’s earliest numerical abilities. npj Science of Learning open

Marilyn Achiron (Ed.) (2016). Equations and inequalities. Making mathematics accessible to all. OECD PISA. read

David Geary, Daniel Berch & Kathleen Mann Koepke (Eds.) (2019). Cognitive Foundations for Improving Mathematical Learning, Volume 5. Academic Press info [expected release date Feb 1, 2019] [I have no access]

Camilla Gilmore, Silke M. Göbel & Matthew Inglis (2018). An introduction to mathematical cognition. Routledge. isbn 9781138923959 info

Rhonda Douglas Brown (2018). Neuroscience of mathematical cognitive development. From infancy through emerging childhood. Springer. [UB Leiden Py\sycho M2.0.1.-70] contents and previews

Cognitive Foundations for Improving Mathematical Learning, Volume 5. Editors: David Geary Daniel Berch Kathleen Mann Koepke Imprint: Academic Press Published Date: 1st February 2019 info

Fidelity Ballmer (November 11, 2013). SPECIAL REPORT: New Common Core math standards sweep into Foothill amid criticism and doubt. The Foothill Dragon Press. article

Eric A. Nelson (n.d.). Cognitive Science and thevCommon Core Mathematics Standards. Nonpartisan Education Review article

Ann Dowker (2019 2nd). Individual differences in arithmetic. Implications for psychology, neuroscirence and education. Routledge [UB Leiden PEDAG 47.B.144] info

Informatie op een zee van onderwepen (toegangen tot recente literatuur). Heel nuttig dus.

Stephen Lerman (Editor) (2014). Encyclopedia of Mathematics Education. Springer.

James K. Bidwell (Ed.) (1970). Readings in the History of Mathematics Education. National Council of Teachers of Mathematics. lccc 74-113172

December, 2018 \contact ben at at at

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