announced June 14, 2015. I have been invited (July 14) to write a 2500 word article on psychological obscurantism in Dutch math educaion (primary and secondary education), especially so in the case of the new kind of exit exam in all branches of secondary education: the tests of functional use of arithmetics in situations of daily life called the *rekentoetsen-2F, -3F*. The article, in Dutch, is to appear in *De Psycholoog*. In preparation for it I will write the text in English, on this very web page.

“Spencer also argued for an increase in mathematics, focused on what students would need in their everyday lives.”

Kieran Egan (2002). **Getting it wrong from the beginning. Our progressivist inheritance from Herbert Spencer, John Dewey, and Jean Piaget.** New Haven: Yale University Press. (p 122)

“If more scholars were willing to expose nonsense for what it is, fewer students and teachers would waste their lives.”

Jon Elster (2015). Obscurantism and Academic Freedom. In Akeel Bilgrami & Jonathan R. Cole (Eds.) (2015). **Who's afraid of academic freedom?** (p. 82). Columbia University Press. isbn 9780231168809 info tweet

This webpage is not about the use of contexts in math education. Instead, it is about research on people using math (or failing to do so) in situations of daily life, in vocational life also, and especially about research on the idea and the practice of testing students’ mastery of the functional use of mathematics in such situations. Ask me ‘Is there a substantial body of such research?’, then my answer will be: ‘No’. There is a lot of research, however, that in more or less slanted ways bears on the problem. This webpage makes a fresh start in bringing that research together, or at least in building an inventory of research questions.

Why is this important? Because obscurantist amateur psychology is ruining math education, and because the foundations of PISA Math are constructivism/situationism/progressivism and the misconception of testing for the use of math in situations of daily life. I bet you have never encountered scientific research that attests to the soundness of the concept of testing for ‘math using in situations of daily life’. I haven’t!

- Robert L. Ebel (1984). Achievement test items: Current issues. In Barbara S. Plake :
**Social and Technical Issues in Testing. Implications for Test Construction and Usage**(Ch. 8, see pp 146-148*The merits of items based on realistic problem situations*Erlbaum. pdf

**A perfect example of what I will be writing about:**

Jan De Lange, Alla Routitsky, Kaye Stacey, Ross Turner, Margaret Wu, Andreas Schleicher, Claire Shewbridge, Pablo Zoido, and Nicola Clements. (2009). *Learning Mathematics for Life. A view perspective from PISA*.
pdf

This key publication explains the rationale of the item writing for PISA Mathematics tests. The ideology explained. Examples. Empirical data. I was not aware of the existence of this report until Joost Hulshof informed me (August 5, 2014). Of course, there is a publication in Dutch by Truus Dekker and others, with lots of adstruction and examples.

**red lines: Yet better to be formulated:**

PISA) Wait a moment—PISA Math is not a high stakes test. It definitely is not an exit exam for students! Not for students, indeed. For politicians it is, however. PISA Math is rapidly becoming the international ‘standard’ for what math education should be about, what it should ‘deliver’. The items in the Dutch *rekentoets* are in many respects identical to those in PISA Math—there is a research report by Cito on this question, it is kept a secret by the Department of Education. The problem stil not explained by Cito is: how is it possible for Dutch youth to score in the world top on PISA Math, yet do so poorly on the Dutch *rekentoets* that it has become a political problem of huge proportions?

LO) It is not only this kind of questioning itself, but also the kinds of questioning **left out** (LO) of testing: testing for mastery of algorithms of arithmetic, or for mental arithmetic. ‘Algorithm’ in the mathematical sense, not the fuzzy one like this (Everyday Math). The Dutch *rekentoets* explicitly does **not** test mastery of algorithmic arithmetic, and does not seem to take mental arithmetic seriously (no speed limit imposed, small number of items).

IC) The accusation that testing for ‘math use in situations of daily life’ really is testing for differences in intellectual capabilities (IC) is not the same as claiming that these tests will correlate highly with standardized intelligence tests. Let us say instead that, f.e., PISA Math is a funny IQ-test. One example: working memory capacity as an intellectual capability is not directly tested in most standardized IQ tests, yet it is evidently the case that differences in working memory capacity are influential in results on contextual math questions of the PISA-type.

IH) Is this a special class: the context presented may inhibit (IH) inference to information in the possession of the student. Not exactly the same as the transfer problem. And what is the relation to strategy training on improving access to knowledge?

Kenneth L. Watson & Michael J. Lawson (1995). Improving access to knowledge: the effect of strategy training for question answering in high school geography.

*British Journal of Educational Psychology, 65*, 97-112. abstract

WP). Of course, word problems (WP) have always been used in math education. Lots of research (see here), yet strangely irrelevant to using word problems in high stakes testing (exit exams) situations. How is that?

BUS). The busing problem (BUS). Highly popular among reform educationalists, especially so in the Netherlands. 36 passenger places in every bus. How many buses are needed for 1128 passengers to Mickey Mouse Land? What do you count as a correct answer? In Dutch: see here. [It is merely an authoritarian convention, and not mathematics, to only count the integer number ‘32’ as a correct answer]. In NAEP it is not damaging to students. In the Netherlands, students will have to answer questions like this one in exit exams, very high stakes to them.

Alan H. Schoenfeld (2007). What Is Mathematical Proficiency and How Can It Be Assessed? In Alan H. Schoenfeld (Ed.) (2007).

**Assessing mathematical proficiency**(59-73). Cambridge University Press. p. 69-70. pdf

PP). Are there any positive points (PP) at all?

D1) What is the demarcation between funny math problems and true math problems? Barry Garelick gives a hint I will have to follow up. I call this #1, because there are more questions of demarcation (such as Thijssen *De Examenidioot* on *denksommen* (what might be an English equivalent?) [Treffers, last note in his last book, on *denksommen* in *rekentoets*],

Barry Garelick (2010). The separate path and the well traveled road. webpage

TC) True constructs (TC) are important in psychological testing as well as in examinations [Dutch: *constructzuiverheid*] (do not use calculators in arithmetics tests; distinguish between math modeling - math calculation - interpretation of results; beware of cognitive overload)

f.e. this random find (interesting for its references mainly): Benjamin Euen, Heike Wendt & Wilfried Bos (2015). Do TIMSS and PIRLS measure only a single general cognitive ability? ECER EERA web page

LC). Using new kinds of problem situations in fact will test students on learning capabilities (LC) (among other things). This observation follows from Stellan Ohlsson’s 2011 theory of learning: starting with the abstract rule, specializing that rule to fit concrete situations. On the difference between known and new kinds of problem situations see also the dissertation by Anton J. H. Boonen (2015). Comprehend, visualize & calculate. Solving mathematical word problems in contemporary math education. Amsterdam: Vrije Universiteit. I take issue with Boonen on the difference he makes.

GP). Dutch educators, f.e. Anne van Streun and the math test-3F team (Victor Schmidt, pres.), define ‘math in everyday situations’ as problem solving requiring a problem solving scheme attributed to George Polya (GP). This might be a huge misconception. It is not at all clear to Artificial Intelligence scientists like Allen Newell, student of Polya, what exactly is the theory of Polya. See Newell 1983 Technical Report version is free access; scan 21 Mb of the book chapter.

RT). Retrieval (RT) of relevant information, even if well known, might be problematic.

Greg A. Perfetto, John D. Bransford and Jeffery J. Franks (1983). Constraints on access in a problem-solving context.

*Memory & Cognition, 11*, 24-31. pdf. Cited 207 times.

The general point is: testing for problem solving in examinations is tricky business if the relevant *productions* [for this technical term see the literature based on the work of Allen Newell & Herbert A. Simon] have not been specifically exercised. For Dutch readers: Wilbrink Toetsvragen ontwerpen, hoofdstuk Problemen stellen

- Finally, the finding that access to relevant information is strongly influenced by whether one is informed or uninformed has important implications for educational practice. Many classroom tests designed to assess learning seem to be episodic in nature. That is, students know that the correct answers to problems were provided by a particular professor, set of readings, and so forth. They are therefore informed about possible sources of information that are relevant to the test question they receive. Imagine that a student performs very well on a test such as this yet, when leaving the classroom, confronts everyday problems in which information from the course would be useful. Will the student spontaneously utilize this information? The results of the present experiments suggest that the answer may weIl be no. It seems clear that theories of learning and instruction must address the question of how information can become accessible under conditions in which students are not explicitly informed about the particular acquisition context that is relevant to the problems they confront.
Perfetto, Bransford, Franks, 1983, cs p. 31

Bransford and Franks here suggest that transfer is an important problem in the world as we know it, as well as in learning theory. I do not think so: in vocational situations the problems typically are of very well known kinds. The same will be true of problems in daily life (grocery shopping etcetera). The Bransford and Franks argument might well be false, as it has always been (see the Spencer quote at the top of this web page). Then the question becomes: what might the demarcation be between typical and a-typical problem situations?

WA). There is somewhat more than a familiy resemblance with *applied math* as implemented in, f.e., the Dutch courses on *Wiskunde A* (WA). The ideology behind them is roughly the same, in the Netherlands identified with the Freudenthal Institute (Realistic Math Education also called Reform Math Education, RME). The program for secondary education started in 1985. An important but today mostly forgotten document is: Cito-werkgroep (not dated, about 1987). **Wiskunde A: doelgericht toetsen. Leerdoelen en voorbeeldopgaven verzorgd door het Cito**. isbn 9001186319 [not online, in Dutch only]. It explains how Cito [the Dutch Educational Testing Service] expects to handle the many problems involved in designing exam questions on this ‘applied math’.

PA). How can students master their ‘math in stituations of daily life’? By practising on a lot of context math problems? That is probably a very inefficient way. The parallel here is with learning to write. Scardamalia 1981 ‘How children cope with the cognitive demands of writing’, p. 100-101:

West (1967) reports that studies on teaching methods consistently show that *practice alone* (PA) is not very effective. Careful reading of essays combined with analytic discussions of ideas, presentation of functional writing assessments, and intensive evaluation were effective strategies when combined with practice; but practice alone was never found to be the most effective teaching strategy. I believe such findings point to a serious fallacy in the whole conventional approach to writing—the belief that when students engage in complex learning tasks they are actually engaged in all the complex problems of writing. Stollard’s research suggests that this is not true, as does much of our recent rsearch on the thinking that children bring to their writing. Papers presented at the 1980 AERA meetings (“Knowledge Children Have but Don’t Use in Their Writing”) address this topic directly.

Marlene Scardamalia: 'How Children Cope with the Cognitive Demands of Writing' in Carl H. Frederiksen and Joseph F. Dominic (Eds.) (1981).

**Writing: The nature, development, and teaching of written communication. Volume 2: Process, development and communication.**Erlbaum. isbn 0898591589.

AMB). Paper tests are highly artificial. How could they ever be valid tests of the propensity or capability of using math in everyday situations? The information available to the candidates is wildly different from that what they encounter in those supposedly mathematical daily life situations. One of the problems is: in daily life situations come without necessarily being labelled as problems to be solved mathematically. It is difficult to say of situations in daily life that the information available is ambiguous: after all, it is the only information there is, one has to cope with that. The test situation is totally different: it is artificial, and made up by item writers wanting you, the testee, to see the available information in a particular way, not in other ways. A good working hypothesis now is: context problems (such as PISA Math items) *inherently* are ambiguous (AMB) to testees (and analysts such as your reporter ;-). *Can one fill a 70 L wheelbarrow with exactly 70 L sand? The Dutch education secretary Sander Dekker told our Parliament he can! Not just once, but even 65 times in a row!* The prediction therefore is: ambiguities will be revealed by letting testees report (in one way or another, from verbal report, scratch paper, eye tracking (Gerdineke van Silfhout, dissertation; site), to fMRI-scan) on their attack on individual problems. Not every possible ambiguity is fatal for the valid use of the item in exams or other high stakes tests, of course. Failing an exam on a score just below the cutting point, however, is exactly the kind of situation where any ambiguity at all is unacceptable. The criterion: Court of Law (Dutch study by Job Cohen, on the rights of students in higher education in the Netherlands. By the way, Job Cohen also was mayor of Amsterdam).

AC). The testpsychology of ‘Assessment Centers’ (AC) is exactly the model that is relevant to the development of math tests (such as PISA Math) that pretend to be valid representations or at least valid predictors of behaviour in situations in daily life being in one way or another characteristically mathematical.

chapter 7 in the (Dutch) report http://benwilbrink.nl/publicaties/90SelectieNPA.htm. English info: https://en.wikipedia.org/wiki/Assessment_center; the literature is tricky: most of it is biased towards money making with AC’s, almost no AC’s being developed according to valid methodological rules.

SIT). There are situations, and situations. Situations one of a kind, and situations of different kinds. A situation here is, of course, a problem situation. Don’t worry, the semantics of reform math is full of surprises. A characteristic of valid examinations is that they test for mastery of the kinds of problem situations that students have been exercising in the past few weeks, months or years. A characteristic of psychological tests, such as intelligence tests, is that they confront the testees with problem situations that supposedly are entirely *new* to them — or thoroughly and equally well known to all testees. See also Job Cohen 1981 on the crucial distinction between *same* kind and *new* kind of problem situation. Where does PISA Math stand? Or the exams based on the idea of testing for use of math in situations of daily life? At issue is that the reform math literature does not recognize any distinction here, at least as far as I know that literature, or it must be that the ideal is to present exclusively *new* kinds op problem situations.

In Dutch: the distinction between summative tests in education, and psychological tests Wilbrink 1986.

See also the dissertation of Anton Boonen on the theme of new kinds of contexts in math problems.

QL). Surely knowing one’s math must impact on quality of life (QL). This requires longitudinal cohort research, and it has been done!

Reyna and others (2009). How Numeracy Influences Risk Comprehension and Medical Decision Making.

*Psychological Bulletin, 135*, 943-973. Open access http://goo.gl/JuJ1zD ]

EC). Education itself is an important part of daily life. Educational careers (EC) surely will depend on the quality of math education in the early years.

Siegler and others http://www.psy.cmu.edu/~siegler/Siegler-etal-inpressPsySci.pdf

S). People typically do not tend to solve their problems in daily life in rational ways, but so as to get preliminary answers that are perfectly satisfactory to them. Herbert Simon called it ‘satisficing’ (S).

Herbert A. Simon, in: Rational choice and the structure of the environment.

*Psychological Review*, 1956, 63, 129-138 (reprinted in his**Models of thought**and in**Models of man, social and rational**. Wiley, 1957.

DM). In general: daily decision making (DM) is not rational [Daniel Kahneman **Thinking fast and slow.** ] Expect also school children to behave ‘not rational’ in the sense of Simon, or of Kahneman. F.e., do not expect students to use shortcuts in solving arithmetics problems if straight calculation surely will give the right answer [a.o. research by Joke Torbeyns]

EXP). Expertness (EXP). Typically the math needed on a daily basis in vocational and daily life will eventually be mastered to a high degree of expertise.

Anders K. Ericsson, part of the Anderson, Reder, Simon chapter http://goo.gl/6ULfY4 ; but also Jean Lave info favorite of situationist thinkers about math education.

VT). vocational training (VT) includes training in the specific math needed in that vocation. Pick up any course book for technical vocations (I favour the old ones, for flight mechanics in the early forties, or the paper printing industry). In reform math countries/provinces vocational institutions (nursing, teaching, administration) nowadays will have a very difficult time bringing their students’ arithmetics fluency up to the level that minimally is required for the responsible professional.

H). Historical (H): Dutch master van Pelt systematically took inventories of the kind of math being used by housewives, masons, carpenters, etcetera, exercised those with his grade six pupils, and took care every pupil left school with a copybook of those exercises, for later consultation. [see the first pages in Van Pelt 1903 (in Dutch) pdf] What can we, a century later, still learn from him?

CM). Cost of mistakes (CM). Lack of number sense may cost one dearly. In the Netherlands a nurse was convicted to prison sentence for wrongful death. She mistook the prescription of 0.30 mg for ten times 0.3 mg; the mistake proved fatal to her patient. Also in the Netherlands prime minister Mark Rutte did not remember correctly, after returning from Brussels, a critical amount of 90 billion euro, instead reporting it to be 50 billion. The question now becoming: are there systematic surveys of number sense failures and their incurred costs? For example: the number of mistakes made by nurses in their daily routine: according to a Dutch investigation that number is very high, and it can be reduced significantly by pharmacists taking over critical calculations in medication preparation. Nursing is typically the kind of vocation that education should adequately prepare the students for, regarding number sense and language mastery.

PC). ‘Math in daily life’ is a poor concept (PC). At the very least one should distinguish 1) daily use such as calculations/estimations of time, distances, amounts; 2) the rare occasion some significant decision has to be made, based on calculations: booking a vacation, buying insurance, renting a house; 3) the routine but possibly complex math in vocational practice. The first can be researched, f.e., with techniques used by Mihaly Csikszentmihalyi in his research of flow experience. The second can be researched in the psychological lab, or simply in real life situations. The third should be quite interesting to the math researcher: in many vocational situations correct calculations are crucial, mistakes might come at a cost, so there must an extensive literature covering math use in vocational situations. I’ll have to search for that literature (ergonomics, for example?).

MT). Allocating students to different math tracks (MT) on the premise of the kind of vocation they will probably have in ten to twenty years is highly problematic morally. This is at issue in the Netherlands, where time and again different tracks have been created for the mathematically strong versus weak students. The weaker tracks emphasize using math in problem situations of daily life, the stronger tracks emphasizing using math is math type problems.

SC). Instead of seeking the fundaments in what is known scientifically, protagonists of ‘math in daily situations’ rely on two somewhat antagonistic ideologies: constructivism, and the more recent situationism. [ Anderson, Reder and Simon sounded alarm bells on this development at the end of last century, f.e. http://goo.gl/12hnpD ]

TR). The idea of transfer (TR) lies at the basis of much (fuzzy psychological) thinking in reform math. The basic (but naive) observation is that many people do not seem to use their knowledge of math in situations where it evidently would be applicable. We know already, from the above, that human decision making need not be rational in the school sense; it is no wonder people do not take the trouble to calculate their options if a satisficing approach already solved their decision problem. In psychology itself transfer is not quite well understood. Stellan Ohlsson (2011, **Deep Learning. How the Mind Overrides Experience**) thinks transfer is nothing else but learning itself, and I think he has strong arguments for his position.

AM). Anecdotal materials (AM), such as this Albertan piece, the kind of stuff one hears at birthday parties, from employers, from desperate parents, about their Johnny not knowing any more how to add. Surely Johnny, if he doesn’t now know how to mentally add, won’t be able to mentally add in daily life, *during his lifetime* (what would be the cost of that?). Even assuming these anecdotes to be valid, they have a signaling value only. Some cases might count as *critical incidents*: if valid, they are proof of something being seriously amiss. For example: a teacher declaring, for national television (the Netherlands: Brandpunt), ‘In this school I am forbidden to teach long division’.

RE). Reverse Engineering (RE): given test or exam questions presumably testing for the use of math in situations of daily life, is it possible that the situation imagined truly is a situation that occurs in real life daily, or once in a lifetime, as well as a situation that education should prepare its students for? Most items from PISA Math and comparable tests such as the Dutch *rekentoets* are fantastic, contrived, trivial, not mathematic at all; meaning there is no real world corresponding to most of these contrived math test items.

CLT). Another approach will be along the lines of psychological theories such as Sweller’s Cognitive Load Theory (CLT) or simply using Miller’s 7 plus or minus chunks capacity of working memory — for children even (substantially) less than 7 chunks. Research literature: see here. It will be immediately evident that confronting students with situations they have no experience with, can not possibly be valid for the use of math in that kind of situation later in life. Etcetera.

E). The E of Exercise. The amazing observation of the outsider in the math education debate: reform protagonists really do think investing in fluency in the basic algorithms of arithmetics is wasting time and money. I can’t remember even one protagonist acknowledging that fluency in multiplication and division of large numbers necessarily implies fluency in the basic operations on single digit numbers, and probably also otherwise a high degree of number sense. It is this number sense that is of lifetime importance regarding one’s health in the realms of life that really count: health, also financial health. The research mentioned, by Reyna and others, already hinted in that direction. More research would be very welcome.

CA). That number sense can be made less abstract by operationalising it in models of cognitive architecture (CA), such as Anderson’ ACT-R model, or Newell’s SOAR. The next step then is meticulously researching what is happening in the brain while people tackle their everyday problems involving number sense in one way or another. This kind of research already has been done by Lebiere and Anderson (a cognitive specification for solving simple algebraic equations, subsequently using fMRI-scanning to follow the brain-events from the young problem solvers), it only has to be specialised to number sense implicated in daily functioning of people.

onderstaand is integraal overgenomen uit model.htm, en moet ik nog selecteren op relevantie:

Timothy Gowers (2002). **Mathematics. A Very Short Introduction.** Oxford University Press.

Gowers is Fields Medal winner. Leuk boekje, maar ook wel van belang: hfdst 1 over modellen, 2 over abstracties, 3 over bewijzen. Voor het modelleren van een probleemstelling moet er nogal wat worden vereenvoudigd. Pas dat eens toe op de alledaagse problemen in de contextopgaven zoals die in veel rekentoetsen zijn te vinden! Dat is een oefening die ik maar eens moet gaan doen; dus niet denken vanuit wat de ontwerper van de vraag wel zal hebben bedoeld, maar serieus naar de gegeven context kijken. Hoeveel veronderstellingen zitten daar impliciet al in, enzovoort.

Andrzej Sokolowski, Bugrahan Yalvac & Cathleen Loving (2011): Science modelling in pre-calculus: how to make mathematics problems contextually meaningful, *International Journal of Mathematical Education in Science and Technology, 42*, 283-297. abstract

Dit artikel adresseert het probleem dat het toepassen van wiskundige kennis bij opgaven in de zaakvakken vaak neerkomt op het oppervlakkig manipuleren van de juiste formules. Draai het om: bouw de instructie op rond opgaven waarbij op een niet-triviale wijze vanuit de beschikbare of te verzamelen gegevens een geschikt wiskundig model moet worden opgebouwd.

Dédé de Haan (2001). Praktische opdrachten bij wiskunde: verslag van een onderzoek. *Nieuwe Wiskrant 20*-3/maart pdf

L. N. Tronsky (2005). Strategy use, the development of automaticity, and working memory involvement in complex multiplication. *Memory and Cognition, 33*, 927-940.
free access

- Most theories and models of arithmetic processing have been constructed with the implicit or explicit assumption that adults retrieve most, if not all, basic arithmetic facts from long-term memory (Ashcraft & Battaglia, 1978; Butterworth, Zorzi, Girelli, & Jonckheere, 2001; Campbell & Oliphant, 1992; De Rammelaere, Stuyven, & Vandierendonck, 1999, 2001; Koshmider & Ashcraft, 1991; Lemaire, Abdi, & Fayol, 1996). In recent years, there has been a proliferation of research in which strategy use and development in children’s and adults’ simple mental arithmetic problem solving have been examined (e.g., Geary, 1996; Hecht, 2002; Kirk & Ashcraft, 2001; LeFevre, Bisanz, et al., 1996; LeFevre, Sadesky, & Bisanz, 1996; Lemaire & Siegler, 1995; Seyler, Kirk, & Ashcraft, 2003; Siegler, 1988). One of the more interesting findings from this research is that a large percentage of adults continue to use strategies to solve basic addition, subtraction, multiplication, and division problems (e.g., Campbell & Xue, 2001; LeFevre, Bisanz, et al., 1996; LeFevre, Sadesky, & Bisanz, 1996; Seyler et al., 2003; Tronsky & Shneyer, 2004). This widespread finding has made it necessary to reexamine some of the empirical effects in basic arithmetic research in order to clarify and qualify some of those findings, so that more accurate models of arithmetical cognition can be developed; research related to this has just begun, at least in the domain of simple arithmetic (Campbell & Xue, 2001; Hecht, 2002; Seyler et al., 2003; Tronsky, Anderson, & McManus, 2005).
Investigations of adults’ complex mental arithmetic skills also have become more numerous, especially recently (e.g., Ashcraft, Donley, Halas, & Vakali, 1992; Fürst & Hitch, 2000; Geary, Frensch, & Wiley, 1993; Logie, Gilhooly, & Wynn, 1994; Seitz & Schumann- Hengsteler, 2000; Trbovich & LeFevre, 2003). At present, few researchers have examined the strategies that adults use to solve these problems (with the exception of Geary et al., 1993), how the use of strategies impacts working memory (WM), and how the development of automaticity impacts WM involvement. The purpose of the present investigation is to examine adults’ initial strategy use in complex mental multiplication and corresponding WM involvement, to document how problemsolving processes and WM involvement change with practice, and to examine the factors that govern performance after retrieval from long-term memory (automaticity) has been established. In order to set the context for the present investigation, it is necessary to provide a review of theory and research related to strategy use and development in mental arithmetic, the structure of WM, and the implications that strategy use and automaticity have for WM’s role in arithmetic.

p. 927

Lieven Verschaffel, Brian Greer and Erik de Corte (2000). **Making sense of word problems.** Lisse: Swets & Zeitlinger.

Sean P. Yee & Jonathan D. Bostic (2014). Developing a contextualization of students' mathematicalproblem solving. *The Journal of Mathematical Behavior, 36*, 1-19.
abstract

- Students experiencing rich problem-solving instruction have better problem-solvingoutcomes than peers in exercise-laden learning environments (Bostic, 2011; Lesh & Zawojewski, 2007).
Is that true? I doubt it. The references are

Bostic, J. D. (2011). The effects of teaching mathematics through problem-solving contexts on sixth-grade students’ problem-solving performance and representationuse (Unpublished doctoral dissertation). Gainesville, FL: University of Florida. pdf

Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (2nd ed., pp. 763–804). National Council of Teachers of Mathematics.

See appendix D in Bostic, for what here is called ‘math problem solving’. This kind of question is fuzzy mathematics. Bostic calls them ‘Open, complex, realistic problems were the main focus of instructional materials for intervention participants’. I do not know what it means if instructed students score better on this type of ‘math in situations of daily life’, ‘functional mathematics’, or whatever it may be called.

Authors Lesh & Zawojewski are known as constructivists. The chapter in the handbook undoubtedly is not a research report. Maybe it summarizes research? I could have a look, but would it be worth my time?

The Yea & Bostic claim cited above simpy is not true: the problem solving involved is not mathematical problem solving. The authors are perfectly aware of this problem, otherwise they would not have mentioned Polya’s work on the teaching of problem solving.

This research is not up to the standards of experimental psychology. Looks much like an attempt to re-invent the problem solving psychology of Duncker (between the world wars), in a psychologically simplistic way. As such it might be representative of most of the ‘research’ (often called ‘design research’) by protagonists of reform mathematics.

No math problem solving at all, in this article.

H. P. Bahrick & L. K. Hall (1991). Lifetime maintenance of high school mathematics content. *Journal of Experimental Psychology: General, 120*, 22-33.
abstract

Ik heb hier nog geen pdf of fotokoie van. Genoemd door Willingham (in een 2015 blog) en Anderson, Reder & Simon (2000).

Anne Anastasi (1984). Aptitude and achievement tests: The curious case of the indestructible strawperson. In Barbara S. Plake: **Social and Technical Issues in Testing. Implications for Test Construction and Usage** (129-140). Erlbaum. isbn 0898592992
pdf

on the differences (if any) of aptitude and achievement tests. Not quite the same problem as that posed by math tests using contexts from daily life, in contrast to proper math tests. Needs my attention. Very informed chapter.

Noëlle Bisseret (1979). **Education, class language and ideology.** Routledge & Kegan Paul. isbn 0710001185
info

- Ch. 1. Essentialist ideology. Its origins and its scientific form, the theory of natural aptitudes. 6-32
- Prior to the nineteenth century, the word ‘aptitude’ designated a contingent reality.
- The first half of the nineteenth century: ‘aptitude’ becomes an essential hereditary feature: birth of a new ideology justifying social inequalities.
- The second half of the nineteenth century: ‘Aptitude’ refers to a strictly biological causal process. The word ‘becomes a part of everyday language’
- The age of tests: aptitude as a measurrable reality. The science of aptitudes as the warrant of a legitimate social order.
- Scientific definitions of the concept of aptitude. A criticism of the relevance of its use in the social sciences. Permanence of a ninetenth century pattern of thought.

Ch. 4. From the theory of differences in aptitudes to the theory of differences in linguistic ‘codes’ 90-114.- From the assertion to the negation of a hierarchy between class languages
- Oppositions between class languages are given meaning by the same referent: power

Richard M. Brandt (1972/1981). **Studying behavior in natural settings.** University Press of America. isbn 081911829X

Shows what it takes do reaerch behavior in situations of daily life ;-)

Belangrijk wanneer iemand uit de boot is gevallen: herinner hem/haar eraan te ZWEMMEN (wijsheid opgedaan uit verslag SAIL Amsterdam 2015). Vergelijk nu eens: #rekentoets met ‘rekenen in alledaagse situaties’ ;-) Ik moet erbij vertellen: de hele heisa moet het transfer=probleem oplossen, dat mensen hun rekenvaardigheid niet altijd spontaan toepassen wanneer dat goed zou kunnen.

Donald Spearritt (1996). Carroll's model of cognitive abilities: educational implications. Themanummer *International Journal of Educational Research, 25* (2), 107-198.

On mathematical ability: pp 156-7.

- In terms of community perceptions, notable omissions from the list of higher-order factors or broad abilities would include concepts such as ‘mathematical ability’, ‘musical ability’ and ‘problem-solving ability’, all of which might be expected to be highly relevant to education. Carroll (1993a, p. 625) indicates why abilities such as these were not identified, using ‘mathematical ability’ as an example. Reverting to his definition of a cognitive ability as one that concerns some defined class of cognitive tasks, he argues that the range of mathematical tasks is too large and diverse to be described in terms of one mathematical ability. In his reanalyses he found that mathematical tasks called on a variety of abilities. Prominent among these were the higher-order abilities of G, Gf and Gc, as well as the lower-order abilities of I (induction), RG (sequential reasoning), RQ (quantitative reasoning), and KM (knowledge of mathematics). Some tasks also involved VZ, visualization. He concludes that
**‘mathematical ability’ ‘must be regarded as an inexact, unanalyzed popular concept that has no scientific meaning unless it is referred to the structure of abilities that compose it’**(John B. Carroll (1993a).

**Human cognitive abilities: A survey of factor-analytic studies.**Cambridge University Press. p. 627). ( . . . )An examination of Carroll’s datasets which focused on mathematical ability, namely WERD01, WEIS11,CANI01, WRIG01 and VERY01, and which also included measures of established cognitive abilities, shows that in general, the mathematics variables were loaded on a range of first-order factors such as quantitative reasoning (RQ), geometry achievement, algebraic computational facility, and ‘functional relationship,’ and on higher-order factors such as 3G [general intelligence], 2G, 2C [crystalized intelligence], 2F [fluid intelligence] and 2S [broad cognitive speediness]. A second-order factor labeled mathematics achievement occurred only in dataset WRIG21 (Wrigley, 1958), but with a relatively heavy loading (0,74) for the first-order factor,

*N*, as compared with the loading (0,53) of KM (mathematical knowledge). Overall, the evidence supports the view that the mathematical performance of students needs to be described in terms of several mathematical abilities rather than in terms of one overriding ‘mathematical ability’.pp. 156-7.

Reference: John B. Carroll (1993a).**Human cognitive abilities: A survey of factor-analytic studies.**Cambridge University Press. [UB Leiden PSYCHO C6.1.-102 & 76 A 18]

from table 4.7 p. 155:

G - General Intelligence: A third-order factor ( .. ) Involves higher order cognitive processes. I, VZ, RQ, V, CF have high loadings on factor.

Gf (2F) - Fluid Intelligence: Characterized by higher salient loadings from factors such as I (Induction) and RQ (Sequential Reasoning). Involves basic intellectual processes, e.g., manipulating abstractions, discerning logical relationships.

Gc (2C) - Crystalized Intelligence: Characterized by higher salient loadings from factors such as LD (Language Development) and V (Verbal Comprehension). Represents a broad mental ability acquired and developed thruogh education and experience.

Gs (2S) - Broad Cognitive Speediness: Factor dominates first stratum factors requiring speed of mental activity or response. Most of the stratum factors measure Attentive Speediness, i.e., ‘quickness in edintifying/distinguishing between elements of a .. pattern’ (Horn, 1988).

from table 4.2 p. 143 (Major first-order factors in domain of reasoning):

RG - Sequential Reasoning: Ability to reason and draw conclusions from given conditions or premises, with various kinds of stimulus materials—verbal, figural, etc. Tests: Linear syllogisms, Verbal reasoning.

I - Induction: Ability to educe a rule or concept or other common characteristic underlying a given set of stimulus materials. Tests: Progressive matrices, Letter series, Verbal analogies.

RQ - Quantitative Reasoning: Ability to reason inductively or deductively with material based on mathematical properties and relations. Tests: Arithmetical reasoning, Number series, Mathematical aptitude.

from table 4.4 p. 146 (Major first-order factors in domain of Visual Perception):

VZ - Visualization: Ability to apprehend and manipulate visual or spatial patterns, often involving rotation in two or three dimensions. Tests: Paper formboard, Block rotation, Paper folding tasks.

CF - Closure Flexibility: ‘Speed of detecting and disembedding a known stimulus array from a more complex array’ (Carroll, 1993a, p. 341). Tests: Hidden figures, copying.

from table 4.1 p. 140 (Frst-order factors in domain of language):

LD - Language Development: General development in spoken native language skills, not requiring reading ability. Oral or listening vocabulary tests.

V - Verbal (Printed) Language Comprehension: Genral native language development on primal tests, requiring reading ability. Reading comprehension/vocabulary tests.

August 24, 2015 \ contact ben at at at benwilbrink.nl

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