measuring mathematical sophistication in preservice elementary teachers

Measuring Mathematical Sophistication in Preservice

Elementary Teachers

Eric W. Kuennen

Jennifer Earles SzydlikUniversity of Wisconsin Oshkosh

Carol E. SeamanUniversity of North Carolina Greensboro

Mathematical Sophistication

An abundance of research suggests that preservice elementary teachers are “unsophisticated;” they often believe that doing mathematics means memorizing and applying formulas to contrived textbook exercises (Ball, 1990; Carpenter, Lindquist, Mattews & Silver, 1983; Schuck, 1996).

Mathematical Sophistication

We use mathematical sophistication to describe the result of acculturation into the community of practicing mathematicians.

In other words, a mathematically sophisticated individual has taken as her own the values and ways of knowing of the mathematical community.

Mathematical Sophistication Framework – Value #1

Mathematicians seek to understand patterns based on underlying structure.

“Seeing and revealing hidden patterns is what mathematicians do best” (Steen, 1990, p. 1).


Mathematicians make analogies by finding the same essential structure in seemingly different mathematical objects.

“Mathematics is the art of giving the same name to different things”

Poincaré


Mathematicians make and test conjectures about mathematical objects and structures.

“When you try to prove a theorem, you don’t just list the hypothesis, and then start to reason. What you do is trial and error, experimentation, guesswork”

Halmos, 1985, p. 321


Mathematicians create mental (and physical) models for and examples and non-examples of mathematical objects.

“A good stock of examples, as large as possible, is

indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one”

Halmos, in Gallian, 1998, p. 40


Mathematicians value precise mathematical definitions of objects.

“The mathematician is not concerned with the current meaning of his technical term…. The mathematical definition creates the mathematical meaning.”

Polya, 1957, p. 86


Mathematicians value an understanding of why relationships make sense.

“Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long as relations do not change. Matter is not important, only form interests them.”

Poincaré


Mathematicians value logical arguments and counterexamples as our sources of conviction.

“Proof is the idol before whom the pure mathematician tortures himself”

Eddington, 1928, p. 337


Mathematicians value precise language and have fine distinctions about language.

“Such is the advantage of a well-

constructed language that its simplified notation often becomes the source of profound theories”

Laplace


Mathematicians value symbolic representations of, and notation for, objects and ideas.

“In symbols one observes an advantage in

discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.”

Leibniz

Previous Research

Seaman and Szydlik (2007) Basic Math Skills Inventory given to preservice

elementary teachers Sample of high- and low-scoring students were

given a web-based teacher resource to apply a mathematical definition, to correct a procedural error in arithmetic, and to make sense of a story requiring the multiplication of fractions

Authors argue that the high-scoring students were more mathematically sophisticated.

MMSI: Measuring Mathematical Sophistication Instrument

Measures a person’s level of mathematical sophistication

27 Multiple-Choice ItemsMathematical content is novel, not a part

of the standard K-16 curriculumTool for assessing courses and pedagogy

Results from Fall 2008 Pilot

242220181614121086420

14

12

10

8

6

4

2

0

total

Perc

ent

MMSI Trial ScoresN = 111

Mean 11.9, Median 12, Std. Dev. 4.26

High Item/Test correlations23) You are a student learning about using lines to model

data and, after the lesson, a student raises her hand and makes a guess about how other types of functions could be used to model data. Which option best reflects your view?

A) I would probably just want to know whether she was correct or not. (16%)

B) I would probably want to spend time exploring her guess myself. (48%)

C) I would probably prefer to focus only on the material that was part of the real lesson. (7%)

D) I would probably want the instructor to figure it out and explain it to me. (29%)

High Item/Test correlations

3) Consider the following statement:

There are at most six ducks in the pond.

Assuming this statement is true, which of the following statements must also be true?

A) There is at least one duck in the pond. (20%)

B) There are six ducks in the pond. (9%)

C) Both of the above statements must be true. (24%)

D) None of the above statements must be true. (47%)


25) Suppose that you build a 3 × 3 × 3 block by gluing together 27 small cubes. Then you dip the block in paint, let it dry, and take it apart to again have 27 small cubes. How many of the small cubes will have paint on exactly three sides?

A) 4 (21%)

B) 8 (44%)

C) 9 (21%)

D) None of the above. (12.5%)


14) Zachary has created a new category of polygons which he named isolaterals. He calls a polygon an isolateral if at least two adjacent sides are the same length. (Adjacent sides share a common endpoint.) Which of the following 4-sided polygons is/are isolateral?

I) II) III)

A) I and II only (20%) B) II and III only (12.5%)

C) II only (7%) D) I and III only (61%)

2

2

2

2

6

6

22

6

8

3

3


7) The numbers 1, 6 and 12 are called hexagonal frame numbers because one dot, six dots, and twelve dots can each be arranged in the shape of a hexagon frame as follows:

one dot six dots twelve dots HF1 = 1 HF2 = 6 HF3 = 12

We say that 1 is the first hexagonal frame number (HF1). The second hexagonal frame number (HF2) is 6, and so on. What is the fourth hexagonal frame number (HF4)?

a) HF4 = 18 (59%) c) HF4 = 24 (27%)b) HF4 = 19 (12.5%) d) None of the above

(2%)


6) Consider the following statement:

When you multiply any two even numbers together, you always get an even product.

Which of these arguments convinces you that the

statement is true?

A) 6 × 2 = 12, 4 × 8 = 32, 12 × 2 = 24 and so you can see by doing examples that the products will all be even. (4%)

B) Even numbers have two as a factor. If two is a factor of both numbers then two must also be a factor of their product. (49%)

C) Both of the above are convincing. (38%)

D) None of the above is convincing. (9%)


14) The notation a ~ b means multiply together a copies of b. For example, 3 ~ 2 = 8. Which of the following is always equivalent to 9?

A) 1 ~ 9 (45%)

B) 9 ~ 1 (21%)

C) 0 ~ 9 (27%)

D) 9 ~ 0 (5%)

Future Research

Compare MMSI scores to math course performance and scores on basic mathematics inventory.

Use MMSI pre- and post-tests to assess mathematics courses for pre-service teachers.

Acknowledgements This work was supported in part by a University

of Wisconsin Oshkosh Faculty Development grant.

We acknowledge our colleague John E. Beam of the University of Wisconsin Oshkosh, for his assistance in the development of this instrument and in the collection of data for this study.

References Eddington, A. S. (1928). The nature of the physical world.

Cambridge: University Press. Gallian, J. A. (1998). Contemporary abstract algebra (4th ed.). New

York: Houghton Mifflin Company. Halmos, P. R. (1985). I want to be a mathematician. Washington:

Mathematical Association of America Spectrum. Poincaré, H. (1946). Foundations of science: Science and

hypothesis, the value of science, science and method. (G. B. Halsted, Trans.). Lancaster, PA: Science Press.

Polya, G. (1957). How to solve it; a new aspect of mathematical method. Garden City, NY: Doubleday.

Seaman, C. E. & Szydlik J. E. (2007). “Mathematical Sophistication” among preservice elementary teachers. Journal of Mathematics Teacher Education. v10 n3, 167-182.

Simmons, G. (1992). Calculus gems. New York: McGraw Hill Inc. Steen, L. A. (Ed.). (1990). On the shoulders of giants: New

approaches to numeracy. Washington, D. C.: National Academy Press.

Contact Information

Eric W. KuennenEric W. [email protected]

Jennifer E. SzydlikJennifer E. [email protected]

Carol E. [email protected]

measuring mathematical sophistication in preservice elementary teachers

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