textbook suhmesablakewhittmore

An analysis of examples in college algebra textbooks: Opportunities for student learning

Heejoo Suh (suhhj@umich.edu) Vilma Mesa (vmesa@umich.edu) Tyler Blake (tyblake@umich.edu)

Tim Whittemore (timwhitt@umich.edu) MichMATYC, Muskegon MI - October 16, 2010

Improving practice: The Implementation Cycle

(Blair, 2006, p. 15)

Exposition – Examples – Exercises

(Angel, 2008, p. 698, p. 704)

Sampling

•  Identify community colleges with large population

•  Find college algebra textbooks •  Analyze examples in three sections:

transformation of graphs, exponential functions, and logarithmic functions

Aspects of Examples

(angel, 2008, p. 698)

Coding Dimensions

Example

Cognitive Demand

Types of

Response

Supporting the

Answer

Types of

Representation

Cognitive Demand

Level of complexity of tasks (statement) •  Memorization •  Procedures Without Connections •  Procedures With Connections •  Doing Mathematics

(Stein, Smith, Henningsen, & Silver, 2000)

CD: Memorization

(McKeague, 2008, p. 34)

CD: Procedures without Connections

(Aufmann, Barker, & Nation, 2011, p. 347)

CD: Procedures with Connections

(Hornsby, Lial , & Rockswold, 2011, p. 110)

CD: Doing Mathematics

(Rockswold, 2006, pp. 411-412)

Cognitive Demand (N = 348)

0%

90%

10% 1%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

Memorization Procedures Without

Connections

Procedures With Connections

Doing Mathematics

Types of Response

Types of solutions expected (statement) •  Only answer •  Answer and Mathematical Sentence •  Answer and Graph •  Explanation or Justification •  Making a Choice

(Charalambous, Delaney, Hsu, & Mesa, 2010)

TR: Only Answer

(Levitan et al., 2008, p. 375)

TR: Mathematical Sentence

(Blitzer, 2007, p. 413)

TR: Graph

(Martin-Gay, 2009, p. 735)

TR: Explanation or Justification

(Levitan et al., 2008, pp. 367-368)

TR: Making a Choice

(McCallum, Connally, Hughes-Hallett et al., 2010, p. 319)

Types of Response (N = 348)

51%

15%

29%

6% 2% 0%

10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

Only Answer Answer and Mathematical

Sentence

Answer and Graph

Explanation or Justification

Making a Choice

Types of Representation

Representations used (statement, solution) •  Symbols •  Tables •  Graphs •  Numbers •  Verbal

Types of Representation

(Hornsby et al., 2011, p. 338)

Verbal

Graphs

Tables

Symbols

Numbers

73%

5% 10%

31%

20%

42%

12%

37%

71%

9%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

symbols tables graphs numbers verbal

Types of representation (in the statement) Types of representation (of the answer)

Types of Representation (N=348)

In the Statement Used in the Solution

Symbols Tables Graphs Numbers Verbal

Flow of Representation

From Statement to Solution: •  Single to Single •  Multiple to Single •  Single to Multiple •  Multiple to Multiple

Types of Flows

Number

Number

(Levitan, Kolman, & Shapiro, 2008, p. 179 & p. 373)

Types of Flows

Symbol

Number

Graph

(Levitan, Kolman, & Shapiro, 2008, p. 179 & p. 373)

Distribution of Flow (N = 348)

43%

17%

29%

12%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

Single to Single Multiple to Single

Single to Multiple

Multiple to Multiple

Supporting the Answer

Explicitly stated strategies (solution) •  Suggestion to Check •  Correctness •  Plausibility •  Interpretation •  Further Elaboration

(Mesa, 2010)

SA: Suggestion to Check

(Martin-Gay, 2009, p. 725)

SA: Correctness

(Sullivan, 2008, pp. 443-444)

SA: Plausibility

(Ostebee, & Zorn, 2002, p. 127)

This translates to around 200 mph—possible for an old-fashioned cannon.

SA: Interpretation

(Ostebee & Zorn, 2002, p. 127)

Around the height of the tallest human-built structure

SA: Further Elaboration

(Aufmann et al., 2011, p.364)

Supporting the Answer (N = 348)

2%

9%

0% 0%

9%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Suggestion to Check

Correctness Plausibility Interpretation Further Elaboration

Summary

•  Cognitive Demand •  Few examples make high cognitive demands

•  Types of Response •  Few examples require more than answers

•  Types of Representation •  Few examples present connections between types

of representations •  Supporting the Answer

•  Few examples help students build understanding

Implications for Practice

Cognitive Demand (N = 348)

0%

90%

10% 1%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

Memorization Procedures Without

Connections

Procedures With Connections

Doing Mathematics

More demanding problems

Types of Response (N = 348)

51%

15%

29%

6% 2% 0%

10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

Only Answer Answer and Mathematical

Sentence

Answer and Graph

Explanation or Justification

Making a Choice

More explanation and justification problems

73%

5% 10%

31%

20%

42%

12%

37%

71%

9%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

100%

symbols tables graphs numbers verbal

Types of representation (in the statement) Types of representation (of the answer)

Types of Representation (N=348)

In the Statement Used in the Solution

Symbols Tables Graphs Numbers Verbal

Use multiple representations

Supporting the Answer (N = 348)

2%

9%

0% 0%

9%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Suggestion to Check

Correctness Plausibility Interpretation Further Elaboration

Demonstrate supports

Adapting the textbook

(Hornsby et al., 2011, p. 335)

References Angel, A. R. (2008). Elementary and intermediate algebra. NJ: Pearson Prentice Hall.

Aufmann, R. N., Barker, V. C., & Nation, R. D. (2011). College algebra. Belmont, CA: Brooks/Cole.

Blair, R. (Ed.) (2006). Beyond crossroads: Implementing mathematics standards in the first two years of college. Memphis, Tennessee: American Mathematical Association of Two-Year Colleges.

Blitzer, R. (2007). College algebra. Upper Saddle River, NJ: Pearson Prentice Hall.

Charalambous, C., Delaney, S., Hsu, A., & Mesa, V. (2010). The addition and subtraction of fractions in the textbooks of three countries: A comparative analysis. Mathematical Thinking and Learning, 12(2), 117-151.

Hornsby, J., Lial, M., & Rockswold, G. (2011). A graphical approach to college algebra. Boston: Pearson Education.

Levitan, M., Kolman, B., & Shapiro, A. (2008). College algebra. Redding, CA: Best Value Textbooks Publishing.

McCallum, W. G., Hughes-Hallett, D., Davidian, A., Lovelock, D., & Shure, P. (2010). Algebra: Form and function. NJ: John Wiley & Sons.

McKeague, C. P. (2008). Intermediate algebra: Washtenaw community college edition. OH: Thompson.

Martin-Gay, E. (2009). Beginning & intermediate algebra. Upper Saddle River, NJ: Pearson Prentice Hall.

Mesa, V. (2010). Strategies for controlling the work in mathematics textbooks for introductory calculus. Research in Collegiate Mathematics Education, 16, 235-265.

Ostebee, A., & Zorn, P. (2002). Single variable calculus from graphical, numerical, and symbolic points of view. United States: Thomson Learning.

Rockswold, G. (2006). College algebra with modeling and visualization. Boston: Pearson Education.

Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. A. (2000). Implementing standards-based mathematics I nstruction. New York: Teachers College Press.

Sullivan, M. (2008). College algebra. Upper Saddle River, NJ: Pearson Prentice Hall.

Thanks

• The Teaching Mathematics in Community Colleges Research Group @ U-M

• Supported in part by NSF CAREER award DRL 0745474. Any opinions, findings, and conclusions or recommendations expressed in this presentation are those of the authors and do not necessarily reflect the views of the National Science Foundation