textbook suhmesablakewhittmore
TRANSCRIPT
An analysis of examples in college algebra textbooks: Opportunities for student learning
Heejoo Suh ([email protected]) Vilma Mesa ([email protected]) Tyler Blake ([email protected])
Tim Whittemore ([email protected]) 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