White & Mitchelmore’s paper“Conceptual Knowledge in

Introductory Calculus”(JRME, 1996)

Synopsis

Annie SeldenDepartment of Mathematical Sciences

New Mexico State University

Tucson Calculus WorkshopApril 18, 2009

2

Some Author Definitions

• W&M use abstracting for “the process of identifying certain invariant properties in a set of varying inputs.” (Skemp,1986)

• One goes from generalizing synthesizing abstracting. (Dreyfus, 1991)

3

• Abstracting is a move to a higher cognitive plane. One goes from interiorization condensation reification. (Sfard, 1991, 1992)

• The first two are operational because they are process-oriented. Reification is the “leap” from an operational mode to a structural mode, in which a process becomes an object. Dubinsky (1991) calls this leap from dynamic process to static object encapsulation.

4

Examples

• Differentiating is a process that students learn to do. Differentiation in the sense of a differentiation operator is an object.

• Integrating is a process that students learn to carry out. Integration is an object.

• Often the symbolism refers to both. In algebra, 2 + 3x can be considered as the process of adding 2 to the product of 3 and x. It can also be considered as the object that is the result of that process, so that 2 + 3x is a “thing” (object) one can work on.

5

Procepts

• Tall has introduced the term procept for the amalgam of process, resultant object, and the common symbol used to represent both. Gray & Tall (1994) hypothesize that successful mathematical thinkers can think proceptually, i.e., can deal comfortably with symbols as either processes or objects.

6

• Once an abstraction has occurred, the generalizing synthesizing abstracting sequence can be repeated. It is a feature of advanced concepts that they are often based on several repetitions of the sequence.

• Each repetition leads to a higher order of abstraction and further away from “primary concepts,” those that are formed from direct experience. (Skemp, 1986)

7

Different Contexts of Calculus Problems

• The context of a calculus application problem may be a realistic or artificial “real world” problem situation, or it may be an abstract, mathematical context at a lower level of abstraction than the calculus concept to be applied.

• W&M only consider problems that can be solved using algebra and symbolic calculus.

8

Solving Application Problems

1. First translate the situation from the context to the abstract level of calculus. This calls on conceptual knowledge because one needs to identify appropriate calculus concepts (such as derivative) and the relationships between them.

2. Next, solve the abstract (mathematical) problem. This may require only procedural knowledge.

3. Finally, translate the solution back to the original context, which may require the same conceptual understanding as the first step. (Tall, 1991a)

9

Some more author definitions

• W&M refer to the definition of new variables and the symbolic expression of relations between them as (algebraic) modeling.

• The selection of a calculus concept and its expression in symbolic form they term symbolization.

• For them, modeling and symbolization together constitute translation.

10

• The use of symbols to represent changing quantities is crucial.

• In research on the teaching and learning of algebra, it has been observed that the meaning of the letters is often neglected so many students only learn manipulation rules without reference to the meaning of the expressions involved. (Kuchmann, 1981; Eisenberg, 1991; Wagner, 1981; Kieran, 1989; Booth; 1989)

11

Justification for the Study

W&M say,

• “It is a matter of some interest to find out whether students who aspire to the advanced mathematical thinking involved in calculus have an adequate concept of variable [italics mine].”

12

The Study

• The sample consisted of 40 1st-year, full-time university math students. A prerequisite for entry to the univ. math program was a satisfactory result in the final h.s. exam for a math course that had a large component of calculus.

• None of the students finished in the top 10% in the exam; most were between the 50th and 80th percentiles.

13

• Conceptual calculus was taught by W for 4 hrs/wk for 6 wks, as half a semester course, following Burns (1992) in which rates of change are investigated using graphs of physical situations.

• The secant was average rate of change, the tangent instantaneous rate of change, and the derivative was defined as the instantaneous rate of change.

14

• A preliminary study had suggested the crucial step in successfully solving calculus application problems was identifying an appropriate derivate.

• Items1,2,3,4 were constructed in four versions (A,B,C,D) so the manipulation in each version was essentially the same. [See Table 1 (Handout).] Each version involved successively less translation. Version A required translating all rates to appropriate symbolic derivatives, whereas version D had all information in symbolic form.

15

• Table 1 (Handout) gives Items 1-4 across and Versions A-D down.

• The four versions allowed the translation steps in each item to be isolated.

• Algebraic modeling was not required in Item 1; it was rather obvious in Item 2A; it was substantial in Items 3 and 4 in both versions A and B.

16

Procedure

• The 40 students were tested 4 times – before, during, immediately after, and 6 weeks after the calculus course. They were divided into 4 approximately parallel groups of 10, based on their algebra performance the previous semester.

• The students were unaware of these groupings.

17

• 4 tests of 4 questions each were constructed, containing one version of each of the 4 items.

• Each version of each item was on 1 and only 1 test.

• Each test had only one question in each version. For example, one test might have items 1A, 2B, 3C, and 4D.

• Tests were given in cyclic fashion to each of the 4 groups over the 4 data collections.

18

Interviews

• Interviews were conducted to clarify and expand on written responses.

• 4 students per group were selected at the start of the research – they were interviewed within 3 days of each of the written data collections.

• Interviews established that students were unaware they were answering different versions of the same 4 items each time.

19

Results

Table 2 -- # of correct responses Time 1 Time 2 Time 3 Time 4Item A B C D A B C D A B C D A B C D 1 0 0 1 1 0 1 1 1 1 2 2 3 0 3 3 1

2 1 1 2 4 0 2 7 4 4 9 6 7 5 4 6 7

3 0 1 1 2 1 1 2 2 1 3 4 6 2 2 2 6

4 0 0 0 7 1 1 0 9 2 1 5 8 1 1 3 10TOTAL 1 2 7 14 2 5 10 16 8 15 17 24 8 10 14 24out of 40

20

Authors’ Interpretation• Performance on version A at Time 1

shows the students could not initially apply their knowledge of calculus, although they had similar items in high school.

• The general pattern of difficulties for across the 4 versions confirms they were correctly ordered in terms of the amount of translation (difficulty?) required to solve the items.

21

• The improvement in # of correct responses at Times 3 and 4 (after instruction) was substantial, but was only slightly more than 50% for Item 2.

• The # who improved suggests that teaching was a positive factor.

• There follows a discussion of Items 1 & 2 (rates of change) and Items 3 & 4 (maximization). I concentrate on the former.

22

Rates of Change

Table 3 # correct symbolizations and correct solutions (in parentheses) to Versions A & B of Items 1 & 2_________________________

Item Time 1 Time 2 Time 3 Time 4

1 0(0) 1(1) 3 (3) 4 (3)

2 4(2) 2(2) 18(13) 18 (9)

TOTAL 4(2) 3(3) 21(16) 22(12)out of 40

23

• Items 1 & 2 required only trivial modeling and symbolization was required in only Versions A and B.

• In the more complex Item 1, few were ever able to correctly symbolize, but those who did were almost always correct.

• In the less complex Item 2, during Times 1 and 2, almost all could correctly symbolize, but only slightly more than 60% could get a correct solution.

24

Dominant Errors

• In Item 1, the dominant error was to substitute v = ½ c before differentiating. (No surprise here.)

• In Item 2, the dominant error was students’ inability to correctly use V = 64.

Examples: Student 3 left the answer at -6x2.Student 4 gave 2 answers, one for V=x3 and

one for V=64.

25

When interviewed,

• Student 5 said: The V=x3 and V=64 at the same time confused me. I didn’t know which one to use.

• Student 6 said: Is the 64 the starting volume?

26

Table 4 -- # responses out of 40 showing the dominant error in Items 1 & 2 (all Versions).

________________________________________Item Time 1 Time 2 Time 3 Time 4 1 21 17 8 8_ 2 7 7 11 15_______

• The decrease in dominant error in Item 1 (substituting before differentiating) and the increase in Item 2 seemed to result from instruction as the students became more aware of the need for a derivative.

27

• In Item 1, most students symbolized an incorrect derivative in the last two data collections instead of substituting first.

• In Item 2, more students symbolized the correct derivative in the last two times, providing more opportunities for making the dominant error (substituting before differentiating).

28

Authors’ Discussion of Items 1 & 2

• The main problem seems to have been an underdeveloped concept of variable.

• Other errors suggested this confusion was part of a manipulation focus, in which students based decisions about which procedure to apply on the given symbols and ignored the meanings behind the symbols.

29

Examples of Manipulation Focus

When interviewed,Student 7 said: I couldn’t see how to get the

t’s out of the v’s.Student 8 said: You have to differentiate,

but there is a v and a c, and they’re both given. I don’t know which one to use.

Student 9 said: There is a change so I thought of dm/dv because v was the only variable there.

30

• Being able to symbolize derivatives involves forming relationships between concepts and should be indicative of conceptual knowledge.

• Although Item 2 is non-complex, the formulations of relationships can be on visible symbols alone and does not require a sound conceptual base.

• There is a similar analysis of the 2 maximization problems, which I will skip.

31

W&M’s General Discussion

• Responses to the 4 items strongly suggest a major source of students’ difficulties in applying calculus lies in an underdeveloped concept of variable.

• Students frequently treat variables as symbols to be manipulated rather than quantities to be related.

• Students have a manipulation focus.

32

Examples of a Manipulation Focus

• Failure to distinguish a general relationship from a specific value (e.g., the difficulty with V=64 in Item 2).

• Searching for symbols to which to apply known procedures regardless of what the symbols refer to (e.g., substituting first in Item 1).

• Remembering procedures solely in terms of symbols used when they were 1st learned (e.g., the x, y syndrome in Item 4).

33

• Students showing a manipulation focus have a concept of variable limited to algebraic symbols – they have learned to operate with symbols without regard to their contextual meaning. W&M call such concepts abstract-apart.

(Mitchelmore, 1994; Mitchelmore & White; 1995;

White, 1992; White & Mitchelmore, 1992)

34

• Tall (1991a) calls the accumulation of new rules, learned by rote and added to existing knowledge without any attempt to integrate the rules with old ideas, disjunctive generalization. This agrees with W&M’s abstract apart.

• The important characteristic is that abstract-apart ideas are formed without any true abstraction, or even without any generalization.

35

• W&M call concepts that are formed by a generalizing synthesizing abstracting sequence abstract-general.

• Only abstract-general ideas can be linked to conceptual knowledge.

• In practice, there is a continuum, rather than a dichotomy between abstract-apart & abstract-general.

• Abstract-apart concepts are rarely completely devoid of meaning.

36

• The generality of an abstract-general concept depends on the variety of contexts from which it has been abstracted.

• The two extremes are often confused and W&M believe it is important to distinguish them.

• Most readers would probably say that the D versions of Items 1-4 are abstract, by which they mean these items are removed from reality.

37

• The two different views of abstraction lead to different views of what constitutes mathematics, and hence, how it should be taught.

• When symbols represent abstract-apart concepts, as in the manipulation focus, they are not related to any mathematical objects that could give them meaning.

• Relationships between symbols are superficial, i.e., they are based only on what the symbols look like; and learned rules can only be applied on the basis of the visible symbols available,

38

• An abstract-apart concept might be adequate to deal with routine symbolic procedures

• But symbolizing a rate of change in complex problem situations requires the existence of abstract-general concepts because the symbols used to represent general variables and derivatives have to be related to the specific variables and the rate of change that occur in that situation.

39

• Students prefer to learn in an abstract-apart fashion and become comfortable with decontextualized problems.

• Abstract-apart ideas are easier to learn because they are limited to a purely symbolic context (sometimes only x and y).

• All decontextualized problems can look very similar to students, and the appropriate procedures are easy to formulate.

40

• Success in such a narrow context can lead to a sense of satisfaction.

• By contrast, learning abstract-general concepts requires the formation of links among a wide variety of superficially different contexts.

• This takes longer and is more intellectually demanding, but the learned relationships can be used to solve more diverse problems.

41

• This study illustrates the significance of encapsulation (Dubinsky, 1991) or reification (Sfard, 1991, 1991) and the importance of thinking proceptually (Gray & Tall, 1994) in the formation of abstract-general concepts.

• Students who can use defined variables but cannot identify and define their own variables are using symbols to express the process of relating one variable to another.

42

• Such students are still at the condensation phase of developing their concept of variable.

• BUT, students who can create variables to solve complex problems have reified variables.

• Student comments for Item 3 such as: “I knew I wanted time and had some distances and speeds. So I looked for a connection and got time and distances so gave x for the distance.” show that the student was using variables as a tool/object.

43

• Such students never made manipulation errors, which is consistent with the view that reification can only occur after extensive successful experience using variables in the operational mode.

44

W&M’s Implications

• In this study, the only detectable result of 24 hrs of instruction that were intended to make the concept of rate of change more meaningful was an increase of manipulation-focus errors in symbolizing a derivative.

• Most of the students had an abstract-apart concept of variable that blocked meaningful learning of calculus.

45

• These findings parallel those reported in school algebra research.

(Booth, 1989; Eisenberg, 1991; Kieran, 1989, Kuchemann, 1982)

• M&W find this a “most disappointing result.”

• They conclude that a prerequisite to successful study of calculus is an abstract-general concept of variable at or near the point of reification.

46

• Even a concept-oriented calculus course is unlikely to be successful without this foundation (of variable).

• Students probably need to spend a considerable amount of time using algebra to manipulate relations before they can achieve a mature concept of variable. (White & Mitchelmore, 1993)

47

• It is unrealistic to attempt to provide remedial activities within a calculus course for students with an abstract-apart concept of variable.

• Either entrance requirements for calculus should be more stringent in terms of variable understanding, or an appropriate precalculus course should be offered.

48

THE END

Comments/Discussion?