supported in part by El Paso MSP grant funded by NSF

Prior Knowledge and Mathematical Cognition

Hamide Dogan-Dunlap

Fan Chen

Cristina Torres

Prior Knowledge and Mathematical Cognition

• Common language

• Concept image

• Data

• Examples

• Implications/Remarks

• students struggle to make sense of new information and their common language knowledge of many mathematics concepts.

• “multiply”, meaning increase in common language, may dominate students’ cognitive processes so much so that they may struggle or not understand why multiplying fractions results in a smaller value/quantity.

(Kaput, 1989; Tobias, 1993).

(Vinner, 1990)

“Concept image is a non-verbal entity associated in our mind with the concept name. It can be visual representationof the concept…It can be a collection of impressions or experiences.”

Concept DefinitionConcept Image

“..to acquire concept means to form a concept image for its name. To understand a concept means to have a concept image for it”

“Concept image is shaped by common experience, typical examples, class prototypes..”

(Vinner, 1990).

Concept Maps

• External representations of concept images/knowledge structures.

• Research tools.

(Kinchin and Hay, 2000; Novak and Gowin, 1984; Williams, 1998; Bolte, 1998, 1999;

McGowen and Tall, 1999; Williams, 1998)

Function

Students’ conception:1. Algebraic term, a formula, an equation2. Should be given by one rule3. Graph should be regular and systemic4. One-to-one correspondence5. Correspondence constituting the function should be

systemic: should be established by a rule. An arbitrary correspondence is not considered a function

(Bills, 2001; Carlson, 1998; Selden and Selden, 1990; Trigueros and Ursini, 2003; Vinner, 1990; Williams, 1998).

Research

Students’ cognitive tendencies in acquiring mathematical meaning of function concept.

Subjects

• Two Intermediate Algebra classes.

• Majority Hispanic: Speaking both Spanish and English

• Two sections with two instructors.

• Traditional lecture style.

Data

• Concept maps (assessed, required): 1. In class about 20 minutes:one section.

2. Take home (samples for the presentation): second section.

• Essays: As part of concept maps.

• Definitions: As part of concept maps.

• Post test/Questionnaires: start and end of semester.

• Interviews: At the end of the semester: Hour and a half.

Investigated

Effect of prior knowledge dominated by common language meaning on cognitive processes in acquiring mathematical meaning

Excerpt from student V’s interview

……

I. Could you define function in words?

V. Function is something that flows, something that works together.

I. …. In mathematical setting?

V. Function would be something where all of the numbers plugged in.. All of the numbers that were plugged in would create something that was consistent like a graph It would be consistent.

Student D’s concept map

Student D’s definition and essay

Excerpt from Student D’s Interview

…. I. I see so “answer” [referring to the term on her concept map] will cover

quadratics? D. It will cover everything…. D. ….it is because over all these are just examples. Over all function is a, is

an answer, is a key. I. Is an answer to? D. like, like a problem.

I. …would you mind telling me what I need to know about function?D. simple way f(x) …..D. The simpler way to teach function would be [Writes f(x)=1(x)+5

f(x)=1] another word this is a key. You. It is a replacement.I. Replacement to?D. f(x) replace it. f(x) equals 1 so you replace it with the x [pointing to x in

1(x)+5] ….. I. Is this [h(x)=5x+1] a function? D. Ya ohh no• WhyD. Not till you have x equaling something…

Cont.

I. This [f(x)=5, x=1] is a function?

D. No because there is no x there [pointing f(x)=5].

Post-test-Student D

Question: Given f(1)=2 and f(-1)=3 of a linear function. Find f(5).

Response:

I can not solve it because I don’t see a problem to solve I see what (f) equals but I don’t see what you plug it in..

Concluding Remarks

Concept Image Concept Definition

Language Meaning

Concluding Remarks

Students’ function conception reported in literature may in part be due to the effect of common language meaning on concept images.

– The formula/equation based conception of function.

– Correspondence constituting the function should be systemic: should be established by a rule. An arbitrary correspondence is not considered a function.

Concluding Remarks

The typical examples of computers, juice makers and function machine to describe correspondence/relationship/function may also strengthen students’ common language-based concept images.

• REMARKS/QUESTIONS?

Textbook definitionElementary and Intermediate Algebra 1st edition by A. R. Angel

A function is a correspondence between a first set of elements, the domain, and a second set of elements, the range, such that each element of the domain corresponds to exactly one element in the range