this database of examples has yet to be constructed. Suggestions? Mail me.

"One simple but often effective use of ICT involves the exploitation of the ability to *drag and drop* on the screen."

"This question focuses on the pupils' understanding of the structure of the flower, rather than on the 'naming of parts'."

"This question focuses on the pupils' understanding of the structure of the flower, rather than on the 'naming of parts'."

Tandi Clausen-May (2001). pdf

TESTING ERROR CORRECTION SKILL

Dear Ralph,

This kind of error correction is a section of the English test, one of five or six tests for admission to Chinese universities. Liying Cheng and Luxia Qi (2006). Description and Examination of the National Matriculation English Test*Language Assessment Quarterly: An International Journal, 3*, 53-70.

Dear Ralph,

I'm a newcomer here of a small town. I would | 76. ________________ |

describe myself as shy and quietly. Before my classmates, | 77. ________________ |

- - - - - | - - - - - |

seem to work. Can you tel me about what I should do? | 85. ________________ |

This kind of error correction is a section of the English test, one of five or six tests for admission to Chinese universities. Liying Cheng and Luxia Qi (2006). Description and Examination of the National Matriculation English Test

*logical and mathematical relations*

*modeling*

*maps, graphics, statistics*

Kirsten R. Butcher (2006). Learning From Text With Diagrams: Promoting Mental Model Development and Inference Generation. *Journal of Educational Psychology. 98(1)*, 182-197. abstract

Patrick B. Kohl and Noah D. Finkelstein (2006). Effect of instructional environment on physics studentsÕ representational skills.

**abstract**In a recent study we showed that physics studentsÕ problem-solving performance can depend strongly on problem representation, and that giving students a choice of problem representation can have a significant impact on their performance P. B. Kohl and N. D. Finklestein, Phys. Rev. ST. Phys. Educ. Res. 1, 010104 2005 In this paper, we continue that study in an attempt to separate the effect of instructional technique from the effect of content area. We determine that students in a reform-style introductory physics course are learning a broader set of representational skills than those in a more traditional course. We also analyze the representations used in each course studied and find that the reformed course makes use of a richer set of representations than the traditional course and also makes more frequent use of multiple representations. We infer that this difference in instruction is the source of the broader student skills. These results provide insight into how macrolevel features of a course can influence student skills, complementary to the microlevel picture provided by the first study.

The intersting thing in the Kohl Finkelstein article is the different ways to represent some state of the world, in this case a physics problem.

*summarize, abstract, give concrete form*

Alan Blackwell (WWW 2000?). Diagrammatic reasoning and system visualization. Computer Laboratory Cambridge University.pdf

- Be sure to take a look at this work. If you are in education, you are in diagrammatic reasoning.

FROM ABSTRACT TO CONCRETE / FROM RECALL TO APPLICATION

Which of the following presents as chronic (longer than 3 months) airspace disease on a chest radiograph?

A 30-year-old-man presented with a 4-month history of dyspnea, low-grade fever, cough and fatigue. Given the following chest radiograph [photo], what is the most likely diagnosis?

Collins, 2006

Which of the following presents as chronic (longer than 3 months) airspace disease on a chest radiograph?

- Streptococcal pneumonia
- Adult respiratory distress syndrome
- Pulmonary edema
- Pulmonary alveolar proteinosis

A 30-year-old-man presented with a 4-month history of dyspnea, low-grade fever, cough and fatigue. Given the following chest radiograph [photo], what is the most likely diagnosis?

- Adult respiratory distress syndrome
- Pulmonary edema
- Streptococcal pneumonia
- Pulmonary alveolar proteinosis

Collins, 2006

Collins (2006): "Vignettes do not have to be long to be effective, and should avoid verbosity, extraneous material and 'red herrings.'"

R. Bareiss, B. Porter and R. Holte (1990). Concept Learning and Heuristic Classification in Weak-Theory Domains, Artificial Intelligence Journal, v45 (nos. 1-2), pp. 229-264. postscript

**abstract**This paper describes a successful approach to concept learning for heuristic classification. Almost all current programs for this task create or use explicit, abstract generalizations. These programs are largely ineffective for domains with weak or intractable theories. An exemplar-based approach is suitable for domains with inadequate theories but raises two additional problems: determining similarity and indexing exemplars. Our approach extends the exemplar-based approach with solutions to these problems. An implementation of our approach, called Protos, has been applied to the domain of clinical audiology. After reasonable training, Protos achieved a competence level equaling that of human experts and far surpassing that of other machine learning programs. Additionally, an "ablation study" has identified the aspects of Protos that are primarily responsible for its success.

The cognitive psychology seeing on schemes and algorithms is known under the label 'scripts' in the original work by Schank and Abelson (1977).

A special case, maybe, are algorithms in arithmetics, as for example the long division (Lee, 2007).

The problems with word problems do not properly belong in this paragraph. They are of a more general nature, touching on most of the design issues in achievement testing. In due time I will rework the material mentioned below. Sorry, much or most of the material has been moved now to a special page wordproblems.htm, collecting the literature regarding word problems and their abuses.

There are 26 sheep and 10 goats on a ship. How old is the captain?

The example item is authored by IREM Grenoble 1980, and used in an attempt to empirically show the phenomenon of 'suspension of sense-making' by pupils solving word problems in mathematics. This approach is not new, Waterink had something to say on it, see Leen (1961, p. 131 ff). An important publication on the topic is:

Lieven Verschaffel, Brian Greer, Erik De Corte (2000).**Making sense of word problems.**. Swets & Zeitlinger. http://beta.springerlink.com/content/h053465217792u77/fulltext.pdf [Dead link? May 2009] review by Christoph Selter (he cites the 'age-of-the-captain' problem from the book. *Educational Studies in Mathematics, 42*, 211-213. For annotations to the Verschaffel et al. book see wordproblems.htm

The example item is authored by IREM Grenoble 1980, and used in an attempt to empirically show the phenomenon of 'suspension of sense-making' by pupils solving word problems in mathematics. This approach is not new, Waterink had something to say on it, see Leen (1961, p. 131 ff). An important publication on the topic is:

Lieven Verschaffel, Brian Greer, Erik De Corte (2000).

The point is: the wording of many so-called word problems is fake, teaching children to disregard the words, and start manipulating the numbers as soon as possible. The 'how-old-is-the-captain' question is not a trick question, but children should understand - and should have been taught - that some problems simply do not have a realistic answer, and should say so.

If there is any indication that pupils tend to answer word problems without even trying to make sense of the word problem, then these word problems should not be used in assessment, or even in instruction, because they fail the construct validity criterion. This will pose a serious problem for much of mathematics teaching. I suspect, however, that the same phenomenon is present in many other disciplines, albeit not that pronounced as in mathematics. Do not expect university level tests to be clean from the phenomenon of 'suspension of sense-making.'

A. Leen (1961). **De ontwikkeling van het rekenonderwijs op de lagere school in de 19e en het begin van de 20ste eeuw.** Groningen; Wolters. Proefschrift Vrije Universiteit Amsterdam.

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. scan

- See also Sfard's homepage
**abstract**This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways:*structurally*—as objects, and*operationally*—as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps:

*interiorization*,*condensation*, and*reification*. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students.

Maak een tekening van zes figuren. Plaats ten minste een van de personen op de voorgrond, evenzo ten minste een persoon op de achtergrond, en tenminste een in een tussenpositie. De relatie tussen de figuren moet de regels van het *rechtlijnig perspectief* weerspiegelen.

[Wilson, in: Bloom, Hastings en Madaus, 1971, blz. 550]

[Wilson, in: Bloom, Hastings en Madaus, 1971, blz. 550]

I will use physics as the paradigm science here. For example Gerald Holton (1953), a textbook directed to the general student at the undergraduate level. Within that field, I will concentrate on Newton's laws of motion. Of special importance will be the folk physics problem: how to deal with the common sense ideas about physics concepts, assuming one sees there is a real instructional problem here, and therefore the design of achievement test items should recognize this also. I would not mention the folk physics problem in this paragraph, if I did not have the strong conviction that common sense ideas present a serious problem in all disciplines, not alone in physics.

How about the social sciences? Do they have laws comparable to those in physics? I will have to tackle that one. A big problem here is the concept of measurement, and how that is handled by psychologists and other social scientists; I will follow Joel Michell (1999) here.

Gerald Holton (1953). **Introduction to concepts and theories in physical science.** Cambridge, Mass.: Addison-Wesley.

Joel Michell (1999). *Measurement in psychology. A critical history of a methodological concept*. Cambridge University Press.

NEWTON's AXIOMS, or LAWS OF MOTION

Isaac Newton (1686/1729/1934).**Principia: Sir Isaac Newton's Mathematical Principles of Natural Philosophy & His System of the World**. p. 13.

- Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
- The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
- To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Isaac Newton (1686/1729/1934).

Newton's first law of motion looks quite simple, is one of the most important or the most important physical law ever, is highly abstract even though it does not look so, goes against the grain of the common sense conception of motion as well as of Aristotle ideas about motion (Newton does not say so, though). Here the item designer has complicated content at hand that is extremely difficult to handle adequately. Most of the time it isn't handled adequately, though. Obviously, it is no use at all to ask for the definition, be it in Latin, the first English translation's version of 1729, the Dover version, or any formulation equivalent in meaning (whatever that may be). Is it useful to ask for mathematical manipulations, be they geometrical (Newton's preferred method) or algebraic? There is some controversy on this question. On the one hand, doing the mathematics evidently is not equal do doing physics, but then what is the meaning of 'doing physics' here? In educational courses more often than not it is the mathematics that counts, risking to pass students clever enough to do the mathematics, while not understanding the physics itself. The item designer will have to take a stand here: it should be the understanding of the physical principles that counts, not mathematical wizardry. That, simply, is a question of * validity*. In due course I will collect lots of sources on the points made here, including validity, and the methods available to go for the 'real' understanding of the physics involved, contrasted against (the same person's earlier) common sense understanding (f.e. Halloun and Hestenes, 1985 pdf) or the superficial understanding evidenced by the correct manipulation of givens, formulas and the calculus (f.e. Tuminaro, 2004). Dijksterhuis (1969) is an important source from philosophy of science; for Newton's physics he uses the Dutch publication by Beth (1932), but Mach's (1893/1960) work (and a whole library following this up) will do as well.

To this day every student of elementary physics has to struggle with the same errors and misconceptions which then had to be overcome, and on a reduced scale, in the teaching of this branch of knowledge in schools, history repeats itself every year. The reason is obvious: Aristotle merely formulated the most commonplace experiences in the matter of motion as universal scientific propositions, whereas classical mechanics, with its principle of inertia and its proportionality of force and acceleration, makes assertions which not only are never confirmed by everyday experience, but whose direct experimental verification is impossible .... (p. 30).

Champagne, Gunstone and Klopfer (1985, p. 62), citing from E. J. Dijksterhuis (1951/1969).**The mechanization of the world picture.** London: Oxford University Press.

Champagne, Gunstone and Klopfer (1985, p. 62), citing from E. J. Dijksterhuis (1951/1969).

The Physics Classroom and Mathsoft Engineering & Education (2004). **A high school physics tutorial: Newton's Laws**. html

J. H. Mandleberg (1952). **Physical chemistry made plain. An aid for intermediate students and others.** London: Cleaver-Hume Press.

- Makes one wonder: lots of attention for mathematical workouts ("with the stress upon alculation"); examples of given methods (instead of problems presented, asking what methods to use for solving them; no mention of common misconceptions and how to recognize them for yourself. This much I can readily see, not being a physicist or chemist myself.

David Hammer (2000). Student resources for learning introductory physics. *American Journal of Physics, Physics Education Research Supplement, 68*, 52-59. html

**abstract**With good reason, physics education research has focussed almost exclusively on student difficulties and misconceptions. This work has been productive for curriculum development as well as in motivating the physics teaching community to examine and reconsider methods and assumptions, but it is limited in what it can tell us about student knowledge and learning. This article reviews perspectives on student resources for learning, with an emphasis on the practical benefits to be gained for instruction.

David Hammer and Andrew Elby (2003). Tapping epistemological resources for learning physics. *Journal of the Learning Sciences, 12*, 53-90. paper pdf

- The class exercises in this research are about Newton's second and third law.
**abstract**Research on personal epistemologies has begun to consider ontology: Do naive epistemologies take the form of stable, unitary beliefs or of fine-grained, context-sensitive resources? Debates such as this regarding subtleties of cognitive theory, however, may be difficult to connect to everyday instructional practice. Our purpose in this article is to make that connection. We first review reasons for supporting the latter account, of naive epistemologies as made up of fine-grained, context-sensitive resources; as part of this argument we note that familiar strategies and curricula tacitly ascribe epistemological resources to students. We then present several strategies designed more explicitly to help students tap those resources for learning introductory physics. Finally, we reflect on this work as an example of interplay between two modes of inquiry into student thinking, that of instruction and that of formal research on learning.

Antti Savinainen (2004). **High School Students' Conceptual Coherence of Qualitative Knowledge in the Case of the Force Concept**. Dissertation, the Faculty of Science of the University of Joensuu. pdf

- Antti Savinainen, Philip Scott and Jouni Viiri (2005). Using a bridging representation and social interactions to foster conceptual change: Designing and evaluating an instructional sequence for Newton's third law.
*Science education, 89*, 175-195. pdf " This paper offers an account of, and findings from, an approach to designing and evaluating an instructional sequence set in the context of Newton's third law. The design of the sequence draws upon conceptual change theory and the concept of the 'bridging analogy' is extended to introduce the notion of a 'bridging representation'."

David Hestenes, Malcolm Wells, and Gregg Swackhamer (1992). Force Concept Inventory. *The Physics Teacher, Vol. 30*, 141-158. pdf. This article describes the Inventory, but it does not show specific items from the instrument.

- "Specifically, it has been established that1 (1) commonsense beliefs about motion and force are incompatible with Newtonian concepts in most respects, conventional physics instruction produces little change in these beliefs, and this result is independent of the instructor and the mode of instruction. The implications could not be more serious. Since the students have evidently not learned the most basic Newtonian concepts, they must have failed to comprehend most of the material in the course. They have been forced to cope with the subject by rote memorization of isolated fragments and by carrying out meaningless tasks. No wonder so many are repelled!"
- The Force Concept Inventory is available as pdf file, but a password is needed to open the file, see http://modeling.asu.edu/R&E/Research.html for the email address for the request.
- See also: Findings of the Modeling Workshop Project (1994-00) pdf This is one section in the Final Report submitted to the National Science Foundation in fall 2000 for the Teacher Enhancement grant entitled Modeling Instruction in High School Physics. David Hestenes, Professor of Physics at Arizona State University, was Principal Investigator.
- See also the Modeling Workshop site http://modeling.asu.edu/R&E/Research.html for other rearch reports.
- N. Sanjay Rebello, Dean A. Zollman, Alicia R. Allbaugh, Paula V. Engelhardt, Kara E. Gray, Zdeslav Hrepic, and Salomon F. Itza-Ortiz (2005). Dynamic Transfer: A Perspective from Physics Education Research. In J. P. Mestre (2005).
**Transfer of learning: from a modern multidisciplinary perspective**p. 217-250. San Francisco: Sage. an earlier paper pdf- I mention this chapter here because it shows some items from the Force Concept Inventory, an instrument to assess mental models about forces.

Zdeslav Hrepic (2004). **Development of a real-time assessment of students' mental models of sound propagation.** Dissertation Kansas State University. pdf

- For materials etc. see his website http://perg.phys.ksu.edu/role/sound/index.html

Alicia R. Allbaugh (2003). **The problem-context dependence of students' application of Newton's Second Law.** Dissertation Kansas State University. pdf

**abstract**Previous research has indicated that improved knowledge organization allows experts to solve problems in a larger variety of contextual settings. In addition, it has been suggested that contextual appreciation is a form of learning ignored by much instruction. To that end, this study investigated students' understanding and application of Newton's Second Law (F=ma) in scenarios differing from those used in instruction of the concept. Instruction in these other contextual arenas, for example electrostatics, does not necessarily include Newton's laws explicitly. Instructors tacitly assumed that the student already has learned the concept fully from previous instruction on the topic.

The study used a qualitative design in a constructivist framework. Students were asked questions regarding that concept in a series of six interviews that spanned several topics in a two-semester, calculus-based introductory physics course. No student was consistent with respect to the application of Newton's Second Law throughout the entire course. However, student responses from these interviews fell into clear categories and themes emerged.

These categories revealed new contextually dependent misconceptions for Newton's second law. Additionally, student responses were clearly affected by the question contextual scenario for the following areas: Rotational Motion, Changing Mass Propulsion, Electric Charges, Electric and Magnetic Fields, Charge with Velocity.

RELATED RATES PROBLEMS IN CALCULUS

A boy is walking away from a lamppost. How fast is his shadow moving? A ladder is resting against a wall. If the base is moved out from the wall, how fast is the top of the ladder moving down the wall?

Bill Austin, Don Barry and David Berman (2000). The Lengthening Shadow: The Story of Related Rates.*Mathematics Magazine, 73*, 3-12. pdf

A boy is walking away from a lamppost. How fast is his shadow moving? A ladder is resting against a wall. If the base is moved out from the wall, how fast is the top of the ladder moving down the wall?

Such 'related rates problems' are old chestnuts of introductory calculus, used both to show the derivative as a rate of change and to illustrate implicit differentiation. Now that some 'reform' texts [Callahan and Hoffman (1995), Hughes-Hallett et. al. (1994)] have broken the tradition of devoting a section to related rates, it is of interest to note that these problems originated in calculus reform movements of the 19th century.

Bill Austin, Don Barry and David Berman (2000). The Lengthening Shadow: The Story of Related Rates.

I think arithmetic, or the calculus, is in some ways an ideal discipline for the researcher looking for item design principles. Therefore the Austin, Barry and Berman article is intriguing. I will have to look up more of this kind of research, not only historical studies, but instructional science studies as well.

The Fluxionary or Differential and Integral Calculus has within these few years become almost entirely a science of symbols and mere algebraic formulae, with scarcely any illustration or practical application. Clothed as it is in a transcendental dress, the ordinary student is afraid to approach it; and even many of those whose resources allow them to repair to the Universities do not appear to derive all the advantages which might be expected from the study of this interesting branch of mathematical science.

William Ritchie (1836).

And that is the point, isn't it: natural science ins't a science of symbols, and even the calculus itself shouldn't be one.

5. A stone dropped into still water produces a series of continually enlarging concentric circles; it is required to find the rate per second at which the area of one of them is enlarging, when its diameter is 12 inches, supposing the wave to be then receding from the centre at the rate of 3 inches per second

6. One end of a ball of thread, is fastened to the top of a pole, 35 feet high; a person, carrying the ball, starts from the bottom, at the rate of 4 miles per hour, allowing the thread to unwind as he advances; at what rate is it unwinding, when the person is passing a point, 40 feet distant from the bottom of the pole; the height of the ball being 5 feet? . . .

12. A ladder 20 feet long reclines against a wall, the bottom of the ladder being 8 feet distant from the bottom of the wall; when in this position, a man begins to pull the lower extremity along the ground, at the rate of 2 feet per second; at what rate does the other extremity begin to descend along the face of the wall? . . .

13. A man whose height is 6 feet, walks from under a lamp post, at the rate of 3 miles per hour, at what rate is the extremity of his shadow travelling, supposing the height of the light to be 10 feet above the ground?

Problems from James Connell (1844).**The Elements of the Differential and Integral Calculus**. London: Longman, Brown, Green, and Longman, as cited in Austin, Barry and Berman (2000) pdf.

6. One end of a ball of thread, is fastened to the top of a pole, 35 feet high; a person, carrying the ball, starts from the bottom, at the rate of 4 miles per hour, allowing the thread to unwind as he advances; at what rate is it unwinding, when the person is passing a point, 40 feet distant from the bottom of the pole; the height of the ball being 5 feet? . . .

12. A ladder 20 feet long reclines against a wall, the bottom of the ladder being 8 feet distant from the bottom of the wall; when in this position, a man begins to pull the lower extremity along the ground, at the rate of 2 feet per second; at what rate does the other extremity begin to descend along the face of the wall? . . .

13. A man whose height is 6 feet, walks from under a lamp post, at the rate of 3 miles per hour, at what rate is the extremity of his shadow travelling, supposing the height of the light to be 10 feet above the ground?

Problems from James Connell (1844).

As Austin, Barry and Berman point out, these problems illustrate some calculus concepts, and if they do not resemble Ritchie's problems, they "are new and original and many remain in our textbook."

B. Sherin (2001). How students understand physics equations. *Cognition and instruction, 19* 479-541. http://www.sesp.northwestern.edu/publications/141061094744ad835da3692.pdf [Dead link? May 2009]

**abstract**What does it mean to understand a physics equation? The use of formal expressions in physics is not just a matter of the rigorous and routinized application of principles, followed by the formal manipulation of expressions to obtain an answer. Rather, successful students learn to understand what equations say in a fundamental sense; they have a feel for expressions, and this guides their work. More specifically, students learn to understand physics equations in terms of a vocabulary of elements that I call symbolic forms. Each symbolic form associates a simple conceptual schema with a pattern of symbols in an equation. This hypothesis has implications for how we should understand what must be taught and learned in physics classrooms. From the point of view of improving instruction, it is absolutely critical to acknowledge that physics expertise involves this more flexible and generative understanding of equations, and our instruction should be geared toward helping students to acquire this understanding. The work described here is based on an analysis of a corpus of videotapes in which university students solve physics problems.

ARITHMETIC

A swallow once invited a snail to dinner. He lived one league from the place, and the snail travelled at the rate of one inch a day. How long would it be before he dined? An old man met a child. "Hello," he said, "I hope that you will live as long as you have lived already, and the same period again, and then three times as much as those two periods together. Then, if God lets you live one year more, you will be a hundred years old." How old was he?

From a pedagogic work on arithmetic, by Bede (672-735), as cited in Frank Davies (1973).**Teaching reading in early England** (p. 90). Pitman.

A swallow once invited a snail to dinner. He lived one league from the place, and the snail travelled at the rate of one inch a day. How long would it be before he dined? An old man met a child. "Hello," he said, "I hope that you will live as long as you have lived already, and the same period again, and then three times as much as those two periods together. Then, if God lets you live one year more, you will be a hundred years old." How old was he?

From a pedagogic work on arithmetic, by Bede (672-735), as cited in Frank Davies (1973).

19th CENTURY MATHEMATICS in the EDUCATIONAL TIMES

Cited from Delve, 2003, p. 172

He [Whewell] had a condescending attitude toward the teaching of arithmetic in schools and promoted the use of rules without understanding.

De Morgan was a celebrated historian of mathematics and thought deeply about how mathematics should be taught. He admired the French higher education system where students were made to reflect on the subject, instead of merely copying out exercise after exercise, as was the case at Cambridge. The French system produced research mathematicians, and Cambridge by and large did not.

... Wharton found De Morgan's textbooks had too much discussion of underlying principles. Wharton differed from De Morgan in another fundamental way Wharton actively supported teaching with examples, whereas De Morgan preferred the approach of French textbooks with no examples in at all. Despite these differences, there was a consensus among the participants that teaching arithmetic using unexplained rules was quite unacceptable and needed to be stopped.

In rejecting the rote teaching of arithmetic, Wharton, Wilson, and Parker found themselves in opposition to Whewell's system. Their convictions would eventually lead to entrance examinations to Oxford and Cambridge intended to ensure an even standard of mathematics education in the schools.

De Morgan was a celebrated historian of mathematics and thought deeply about how mathematics should be taught. He admired the French higher education system where students were made to reflect on the subject, instead of merely copying out exercise after exercise, as was the case at Cambridge. The French system produced research mathematicians, and Cambridge by and large did not.

... Wharton found De Morgan's textbooks had too much discussion of underlying principles. Wharton differed from De Morgan in another fundamental way Wharton actively supported teaching with examples, whereas De Morgan preferred the approach of French textbooks with no examples in at all. Despite these differences, there was a consensus among the participants that teaching arithmetic using unexplained rules was quite unacceptable and needed to be stopped.

In rejecting the rote teaching of arithmetic, Wharton, Wilson, and Parker found themselves in opposition to Whewell's system. Their convictions would eventually lead to entrance examinations to Oxford and Cambridge intended to ensure an even standard of mathematics education in the schools.

Cited from Delve, 2003, p. 172

JOHN HIND 1831 versus AUGUSTUS DE MORGAN 1837

The *Limits* of a quantity which admits of change in its magnitude, are those magnitudes between which all the values that it can have during all its changes are comprised; beyond which it can never pass; and fro which it may me may to differ by quantities less than any that can be assigned in finite terms.

This is the first sentence in John Hind's (1831)**The principles of differential calculus with its application to curves and curve structures designed for the use of students in the university.**

If the mathematical sciences were cultivated wholly for their practical utility, as it is called, meaning their application to the formation and management of all the mechanism by which the arts of life are advanced, it would not be necessary to consider any magnitude as having existence at all, unless it were sufficiently great to be either useful or noxious to some object connected with some application in question."

This is the first sentence in Augustus De Morgan (1837)**The Differential and Integral Calculus. Containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals - with applications to algebra, plane geometry, solid geometry, and mechanics. ** (reprint: Elibron)

This is the first sentence in John Hind's (1831)

If the mathematical sciences were cultivated wholly for their practical utility, as it is called, meaning their application to the formation and management of all the mechanism by which the arts of life are advanced, it would not be necessary to consider any magnitude as having existence at all, unless it were sufficiently great to be either useful or noxious to some object connected with some application in question."

This is the first sentence in Augustus De Morgan (1837)

John Hind does not think it necessary to give any motivation at all, an attitude that I personally experienced still in 1973: my professor simply refused to provide a motivation for the mathematical analysis he was going to teach for one year, being explicitly asked to provide one by his students. Hinde probably would have answered, if directly asked for it, that mathematics is meant to train the mind, the common attitude at the time. Augustus de Morgan, on the contrary, thinks motivation and explanation of the essence. This is a crucial moment in the history of mathematics - at Cambridge University that is.

Smith favoured natural philosophy over pure mathematics

June Barrow-Green (1999). 'A corrective to the spirit of too exclusively pure mathematics': Robert Smith (1689-1768) and his prizes at the Cambridge University. Annals of Science, 56, 271-316.

On the one hand, the Tripos was a problem-solving marathon second to none. Its multitude of papers contained more questions than could be solved in the allotted time and success depended more on having the mechanical ability to solve problems as rapidly as possible than on having a clear understanding of the theory. On the other hand, the Smith's Prize examination consisted of only a few papers and was generally aimed at soliciting a more thoughtful or philosophical approach to the questions asked. (...) By fostering an interest in the study of applied mathematics, the competition [Smtih's Prize exam] played a significant part in promoting the remarkable achievements in mathematical physics that characterized Cambridge mathematics during the second half of the nineteenth century. (p. 272)

June Barrow-Green (1999). 'A corrective to the spirit of too exclusively pure mathematics': Robert Smith (1689-1768) and his prizes at the Cambridge University. Annals of Science, 56, 271-316.

This seems to be a perennial problem. It may result, and often does, in bad design of test questions: worded problems that really function as mathematical problems only, while the intention of the item designer might have been that pupils first build a model of the problem situation, and only then translate the model into formulas that will allow a specific solution.

H. J. E. Beth (1932). **Newton's 'Principia.' deel I, II.** Groningen: Noordhoff. [in Dutch]

Benjamin S. Bloom, J. Thomas Hastings and George F. Madaus (Eds) (1971). **Handbook on formative and summative evaluation of student learning**. London: McGraw-Hill.

Audrey B. Champagne, Richard F. Gunstone and Leopold E. Klopfer (1985). Instructional consequences of students' knowledge about physical phenomena. In Leo H. T. West and A. Leon Lines: **Cognitive structure and conceptual change** (pp. 61-90). Academic Press.

- Susan R. Singer, Margaret L. Hilton, and Heidi A. Schweingruber (Eds) (2005).
**America's Lab Report: Investigations in High School Science**.- . The book itself may be read on NAP html. The book is not about mental models, but shuld give a fair picture of the kinds of fysics teaching nowadays in the US.

- Senta A. Raizen, Joan B. Baron, Audrey B. Champagne, and others. Assessment in Science Education: The Middle Years. Washington, D.C.: National Center for Improving Science Education, 1990. 129 pp.

Tandi Clausen-May (2001). An approach to test development. nfer

- chapter 1 'Looking ahead - ICT-based assessment.' downloadable pdf, nice examples of drag-and-drop questions on the computer, see above

Jannette Collins (2006). Education techniques for lifelong learning: writing multiple-choice questions for continuing medical education activities and self-assessment modules. *Radiographics, Mar-Apr 26(2)*, 543-51. http://www.arrs.org/StaticContent/pdf/ajr/pdf.cfm?theFile=ajrWritingMultipleChoiceHandout.pdf [Dead link? May 2009]

- "The article provides an overview of established guidelines for writing effective MCQs, a discussion of writing appropriate educational objectives and MCQs that match those objectives, and a brief review of item analysis."

Janet Delve

(2003). The College of Preceptors and the Educational Times: Changes for British mathematics education in the mid-nineteenth century. *Historia Mathematica 30*, 140-172. pdf

**abstract**Founded in Britain in 1846 to standardize the teaching profession, the College of Preceptors is little known today. The College was closely linked to the Educational Times (hereafter ET), a journal of ñEducation, Science and Literatureî launched in 1847. This paper examines in detail a sample of College examinations, articles on mathematics education, and reviews of mathematics textbooks that appeared in the ET. Key figures in the mathematical discussion were William Whewell, Augustus De Morgan, and Thomas Tate. The paper shows how the discourse on mathematics education led to the introduction of entrance examinations for Oxford and Cambridge Universities.

E. J. Dijksterhuis (1951/1969). **The mechanization of the world picture.** London: Oxford University Press.

Ibrahim Abou Halloun and David Hestenes (1985a). The initial knowledge state of college physics students. *Am. J. Phys. 53 (11)* 1043-1048. pdf. And

Ibrahim Abou Halloun and David Hestenes (1985b). Common sense concepts about motion. *Am. J. Phys. 53 (11)*, 1056-1065. pdf.

**abstract**An instrument to assess the basic knowledge state of students taking a first course in physics has been designed and validated. Measurements with the instrument show that the student's initial qualitative, common sense beliefs about motion and causes has a large effect on performance in physics, but conventional instruction induces only a small change in those beliefs.- David Hestenes, Malcolm Wells, and Gregg Swackhamer (1992). Force Concept Inventory.
*The Physics Teacher, 30*, 141-158. pdf - David Hestenes and Ibrahim Halloun (1992). Interpreting the Force Concept Inventory.
*The Physics Teacher, 30*, 502-506. pdf - Richard Hake (1998). Interactive-engagement vs. traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses
*American Journal of Physics, 66*,64-74 pdf - Antti Savinainen1 and Philip Scott (2002). The Force Concept Inventory: a tool for monitoring student learning.
*Physics Education, 37*, 45-52. pdf - Antti Savinainen1 and Philip Scott (2002). Using the Force Concept Inventory to monitor student learning and to plan teaching.
*Physics Education, 37*, 53-58. pdf

Ji-Eun Lee (2007). Making sense of the traditional long division. *Journal of Mathematical Behavior 26,* 48Ð59.

Ernst Mach (1893/1960). **The science of mechanics: A critical and historical account of its development.** La Salle: Open Court. Translated by Thomas J. McCormack.

Roger C. Schank and Robert P. Abelson (1977). **Scripts, plans, goals and understandingÊ: an inquiry into human knowledge structures**. Erlbaum.

Jonathan Tuminaro (2004). **A cognitive framework for analyzing and describing introductory students' use and understanding of mathematics in physics.** Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park. pdf

Roman Frigg and Stephan Hartmann (2006). Models in science. In **Stanford Encyclopedia of Philosophy.** html

Stephan Hartmann (2005) The World as a Process: Simulations in the Natural and Social Sciences. pdf

Carl G. Hempel (1952/1972). **Fundamentals of concept formation in empirical science**. London: The University Of Chicago Press, 1972.

Brenda R. J. Jansen and Han L. J. van der Maas (2002). The Development of Children's Rule Use on the Balance Scale Task. *Journal of Experimental Child Psychology 81,* 383-416. pdf

**abstract**Cognitive development can be characterized by a sequence of increasingly complex rules or strategies for solving problems. Our work focuses on the development of children's proportional reasoning, assessed by the balance scale task using Siegler's (1976, 1981) rule assessment methodology. We studied whether children use rules, whether children of different ages use qualitatively different rules, and whether rules are used consistently. Nonverbal balance scale problems were administered to 805 participants between 5 and 19 years of age. Latent class analyses indicate that children use rules, that children of different ages use different rules, and that both consistent and inconsistent use of rules occurs. A model for the development of reasoning about the balance scale task is proposed. The model is a restricted form of the overlapping waves model (Siegler, 1996) and predicts both discontinuous and gradual transitions between rules.

Richard Lesh, Mark Hoover and Anthony E. Kelly (nd). Equity, Assessment, and Thinking Mathematically: Principles for the Design of Model-Eliciting Activities. Rational Number project. html

Levy, S.T., Mioduser, D., & Talis, V. (2006 in preparation). Episodes to Scripts to Rules: Concrete-abstractions in kindergarten children's construction of robotic control rules pdf

**abstract**This study explores young children's evolving understanding of the rules underlying an adapting robot's behavior in a progression of tasks sequenced for increasing complexity. The study was conducted individually with six children, aged 5-9' (3') along six sessions that included tasks, ordered by increasing difficulty. We developed and used a robotic control interface. Each session included a description and a construction task. To capture the children's changing knowledge representations, we have employed a framework that highlights the differences in generality between episodes, a description of a unique sequence of events, scripts, which include temporally-structured repeating patterns and rules, atemporal associations between environmental conditions and the robot's actions. Our data unravels the progression through which rules are constructed. From an action-focused episode, scripts are triggered by specific conditions. The scripts promote a search for rules in two ways. The invariance of the embedded scripts suggests that more general constructs may be searched for. Including environmental conditions move the reasoner away from noticing only a sequence of actions to including conditions in her representation. These two pave the way to noticing the co-variance of environmental conditions with robot actions, from which a-temporal rules may be abstracted. We have supported the children's reasoning by helping them attend to relevant features, and compared their spontaneous and supported descriptions. We elaborate on the role of 'concrete-abstractions' in the interaction between cognitive schemas and object-embedded abstract schemas, for the evolving understanding of the controlled robot.

J. E. Mezzick en H. Solomon (1980). Taxonomy and behaviorial science. London, Academic Press.

M. Macdonald-Ross (1979). Scientific diagrams and the generation of plausible hypotheses: an essay of the history of ideas. *Instructional Science, 8*, 233-234.

**from the abstract**... a contribution to the study of the history, structure and functions of scientific diagrams. The author considers the role of scientific diagrams in the structure of scientific communication, their perception and interpretation by the reader.

The author surveys the main conceptual issues, and proposes that these specialized graphic forms have played an important part in the origin of scientific hypotheses.- William Winn (1991). Learning from maps and diagrams.
*Educational Psychology Review, 3*, 211-247.**abstract**This review of learning from maps and diagrams consists of two sections. The first section presents a theoretical framework for learning from maps and diagrams. The case is made that the symbol systems of maps and diagrams are sufficiently similar for them to be considered together. The theoretical framework is built around what is known of pre-attentive and top-down psychological processes. It accounts for the way people discriminate between symbols used in maps and diagrams and how they group them into clusters. The second section comprises a review of psychological and instructional research. This research provides support for a number of hypotheses arising from the theoretical framework. Many of these are based on the notion that maps and diagrams communicate a considerable amount of information by the way in which components are placed relative to each other and to the frame surrounding them. Evidence that configuration and discrimination are fundamental to learning from maps and diagrams is summarized in 10 concluding points.

Margaret Morrison (). Models as representational structures. Paper presented in: Nancy Cartwright's Philosophy of Science. An International Workshop, December 16-17, 2002 pdf

E. A. Murphy (1976). **The logic of medicine**. London: Johns Hopkins University Press.

A. P. van Leeuwen (1941). **2 x 2 = 5. Merkwaardige uitkomsten van cijfer- en getallencombinaties.** Rekenkundige sels, vreemde vraagstukken, goocheltoeren met cjfers, toovervierkanten, paradoxen enz. Amsterdam: Becht. Aad van Leeuwen

William E. Becker and William H. Greene (2001). Teaching Statistics and Econometrics to Undergraduates. *Journal of Economic PerspectivesÑVolume 15*, 169-182. pdf

- "In two national surveys, Becker and Watts (1996, 2001) found that problem sets are used more heavily in statistics and econometrics courses than in other undergraduate economics courses. Surprisingly, however, those applications are rarely based on events reported in financial newspapers, business magazines or scholarly journals in economics. Instead, they appear to be contrived situations with made-up data, as in textbooks that are characteristic of the chalk-and-talk teaching methods predominant across the undergraduate curriculum in economics."
- "Descriptions of actual situations and easily retrievable data are abundant today. There are at least three sources for vivid examples to engage students: history, news and the classroom itself. Stephen Stigler, in his work on the history of statistics, has pointed out a number of episodes that can make vivid classroom examples. For example, Stigler (1999, pp. 13Ð41) describes an exchange between statistician Karl Pearson and three renowned economistsÑAlfred Marshall, John M. Keynes and Arthur C. PigouÑabout a study Pearson did that suggested that alcoholism is not inherited and does not wreak havoc on offspring. The debate was widely covered by major newspapers in 1910 when, then as now, the reasons for and consequences of alcohol consumption were a hotly debated issue."
- "Becker (1998) provides numerous examples of how the news media can be used to teach quantitative methods in an economics context."

proefjes.nl (plaatje - cartoon van Galileo? - op proefjes.nl)

The Diagrammatic Reasoning Site http://www.cs.hartford.edu/~anderson/

**Galileo Galilei's Notes on Motion,** integral text in Latin as well as in English, plus electronic representation of the manuscript. Biblioteca Nazionale Centrale, Florence - Istituto e Museo di Storia della Scienza, Florence - Max Planck Institute for the History of Science, Berlin. site.

- I am sure this book offers many ideas for the designer of items on motion and force. You will yourself, of course, have to relate the material to the corpus of classical - Newtonian - physics.

American Translators Association. **Translation: Getting it right.** pdf

Robert Rynasiewicz (www 2004). Newton's Views on Space, Time, and Motion. Stanford Encyclopedia of Philosophy. html

- Bernard le Bovier de Fontenelle (1728). The Life of Sir Isaac Newton; with an Account of his Writings. Transcribed by David R. Wilkins (www 2002) html
- The Newton Project . "Our goal is to make all Newton's writingd freely available online." site
- Isaac Newton (1729, Motte's translation) Axioms, or Laws of Motion (Principia). html

May 23, 2007 \ contact ben at at at benwilbrink.nl
http://www.benwilbrink.nl/projecten/06examples5.htm