Comparing Solution Strategies to Promote Algebra Learning and Flexibility Bethany Rittle-Johnson Vanderbilt University In Collaboration with Jon Star, Kelley Durkin & Abbey Loehr 8th ICMI - EARCOME
These slides will be posted: ¤ My Researchgate.net profile for this project: ¤ https://www.researchgate.net/project/LeveragingComparison-and-Explanation-of-Multiple-Strategies-CEMS-toImprove-Algebra-Learning
2
We Often Learn Through Comparison Learn categories Vs.
Learn words
3
“Comparison is one of the most integral components of human thought” (Goldstone, Day & Son, 2010, p. 103)
4
Comparison Aids Learning Across Variety of Tasks and Ages ¤ Comparing multiple examples supports learning (Alfieri, Nokes-Malach, & Schunn, 2013). For example, it promotes: ¤ Perceptual Learning in adults (Gibson & Gibson, 1955) ¤ Transfer of Negotiation Principles in adults (Gentner, Loewenstein & Thompson, 2003)
¤ Category Learning and Language in preschoolers ¤ Spatial Categories in infants (Oakes & Ribar, 2005)
(Namy & Gentner, 2002)
¤ However, at time, prior research not with school age children nor on math learning
5
Comparison a “Best Practice” in Mathematics Instruction ¤ Share and compare solution strategies featured in standards in many countries (Australian Education Ministers, 2006; Brophy, 1999; Kultusministerkonferenz, 2004; NCTM, 2014; Singapore Ministry of Education, 2006; Treffers, 1991)
¤ Expert teachers use this approach in US and Japan (Lampert, 1990; Richland, Zur & Holyoak, 2007; Shimizu, 1999)
¤ Comparison is one of many teaching technique used. Is comparison key to improving student learning? 6
Does comparison support math learning? ¤ Was no empirical evidence that comparison supported student’s math learning in school. ¤ Raised need for guidelines for using comparison effectively. ¤ Our research goal: How can comparison support learning of school mathematics within a classroom setting? ¤ Short-term researcher-led studies ¤ Year-long teacher-led studies 7
Focus on Algebra Learning ¤ Proficiency in algebra is critical to academic, economic, and life success ¤ E.g. Necessary for access to many job opportunities
¤ Students have difficulties with Algebra ¤ (National Assessment of Educational Progress, 2011).
8
Phase 1: Our Initial Research on Comparison ¤ 5 short-term researcher-led studies: ¤ Redesigned 2 - 3 math lessons on a particular topic (e.g., equation solving) ¤ Implemented during students’ mathematics classes by researchers, using experimental design. ¤ All students studied and discussed worked examples with a partner. Based on advantages of: ¤ Worked examples (e.g. Sweller, 1988) ¤ Generating explanations (e.g. Chi et al, 1989; Rittle-Johnson, 2006) ¤ Peer collaboration (e.g. Johnson & Johnson, 1994)
9
How We Supported Comparison 1. Present strategies at same time. 2. Use spatial cues and common language in worked examples. 3. Ask specific explanation prompts to focus attention on key aspects. 4. Provided some direct instruction to supplement learners’ comparisons. (e.g., Gentner, 1983; Gentner et al., 2003; Gick & Holyoak, 1983; Richland, Zur & Holyoak, 2007; Schwartz & Bransford, 1998).
10
Compare Condition
(Rittle-Johnson & Star, 2007) 11
Sequential Condition e ag p xt ne
e ag p t x ne
e ag p xt ne
(Rittle-Johnson & Star, 2007) 12
Student Outcomes: 3 Components 1. Procedural knowledge: ability to execute action sequences to solve problems, including the ability to adapt known procedures to unfamiliar problems (Rittle-Johnson, Siegler, & Alibali, 2001). 2. Conceptual knowledge: “an integrated and functional grasp of mathematical ideas” (Kilpatrick et al., 2001). 3. Procedural flexibility: knowledge of multiple strategies as well as the ability to choose the most appropriate method based on specific problem features (e.g., Star, 2005; Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). 1. Supports efficient problem solving, greater accuracy solving novel problems & greater conceptual knowledge (e.g., Blöte, Van der Burg, & Klein, 2001; Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Hiebert et al., 1996).
13
Outcome Measures ¤ Procedural knowledge ¤ Accuracy solving equations: ½ (x + 1) = 10
¤ Conceptual knowledge (e.g., equivalence, like terms, composite variables concepts) ¤ “Is 98 = 21x equivalent to 98 + 2(x + 1) = 21x + 2(x + 1)?” ¤ “Which of the following is a like term to (could be combined with) 7(j + 4)?”
¤ Procedural flexibility ¤ Flexible Use - Use of more efficient solution strategies on procedural knowledge assessment (i.e., fewer solution steps) ¤ Flexible Knowledge
14
Outcome Measure: Procedural Flexibility Knowledge Knowledge of multiple strategies
Ability to evaluate strategies
15
Example Study ¤ Participants ¤ 70 7th-grade students (12-13 years old) ¤ Some prior instruction on solving equations using one method
¤ Research design ¤ Pretest - Intervention – Posttest ¤ Random assignment to ¤ Compare condition: Students compare and contrast alternative correct strategies for equation solving ¤ Sequential condition: Students study same strategies one at a time
(Rittle-Johnson & Star, 2007, Journal of Educational Psychology) 16
Results: Procedural Knowledge
F(1, 31) =4.49, p < .05
Greater gains in procedural knowledge, esp. solving equations with novel features. (Rittle-Johnson & Star, 2007) Equal gains in conceptual knowledge.
17
Results: Procedural Flexibility Flexible Use of Procedures Proportion of Problems Solved Solution Method Most Efficient
F(1,31) = 7.73, p < .01
Comparison
Sequential
.17*
.10
* p < .05
(Rittle-Johnson & Star, 2007) 18
Why Comparing Correct Strategies Aided Learning ¤ Why aids learning? Analogical Learning Framework (D. Gentner) ¤ Students were familiar with one method. ¤ Compare new method to known method. Identify similarities and differences with known method. ¤ Evidence: Frequency of comparative explanations predicted learning
¤ Was familiarity with one method important?
19
Study 2: Importance of Prior Knowledge ¤ Research question: Do children with different levels of prior knowledge benefit equally from comparing solution strategies? ¤ Participants: 236 7th & 8th-grade students in classes with limited algebra instruction ¤ Identified whether students attempted algebra at pretest ¤ 40% did not attempt algebra (e.g., guess and test) ¤ 60% attempted algebra (wrote partially-solved equation, often incorrectly)
(Rittle-Johnson, Star, & Durkin, 2009) 20
Importance of Prior Knowledge Sample Result Procedural Knowledge Procedural Knowledge % Correct at Posttest
60
Sequential Compare Methods
50 40 30 20 10 0 No Algebra
Use Algebra
Use of Algebra at Pretest
Use of algebra, not general math ability, was important. 21
Results: Flexibility Compare Methods
Flexible Use 40
Flexibility Knowledge % Correct at Posttest
Shortcut Use % Correct at Posttest
Sequential
35 30 25 20 15 10 5 0 No Algebra
Use Algebra
Use of Algebra at Pretest
70
Flexible Knowledge
60 50 40 30 20 10 0
No Algebra
Use Algebra
Use of Algebra at Pretest
22
Importance of Prior Knowledge: Results Summary ¤ General math ability was not important. ¤ Rather, familiarity with one of the solution methods was what influenced learning from comparison
¤ For students without prior knowledge of algebraic strategies ¤ Sequential study produced fewer signs of confusion ¤ Sequential study of examples was best for procedural and conceptual knowledge and flexibility
¤ For students with prior knowledge of algebraic strategies, comparison was best.
(Rittle-Johnson, Star, & Durkin, 2009) 23
Should Comparing Strategies Be Delayed? ¤ Follow-up study found that even novices learn more from comparing strategies if given adequate instructional support. ¤ In this high-support context, delaying comparison of methods reduced procedural flexibility for all student (Rittle-Johnson, Star & Durkin, 2011).
¤ Conclusion: Comparing strategies can happen at multiple phases of instruction. ¤ Early in instruction, students need more support to compare effectively, such as a slower pace. ¤ Waiting to compare strategies for too long reduces students’ procedural flexibility. ¤ Later in instruction, students can compare more effectively on their own. 24
Implications
25
Phase 2: Helping teachers use comparison effectively ¤ Need for materials and training to help more math teachers use comparison effectively. ¤ Comparison used infrequently in textbook lessons on Algebra ¤ Percentage of worked examples with comparison
No Compare Compare Strategies Compare Problems+
US 46 20 35
Japan 82 1* 17
*Few examples of fictitious students suggesting 2 different ways to represent a problem. +Compare problems example: (1) x2 + 2xy (2) 3ax – 6ay (identifying common factors) 26
Phase 2: Our approach ¤ Created supplemental Algebra I curriculum and professional development for teachers to integrate Comparison and Explanation of Multiple Strategies (CEMS) in their classrooms. ¤ Evaluated effectiveness in 2 studies.
27
Comparison and Explanation of Multiple Strategies (CEMS) Curriculum Which is correct?
Alex and Morgan were asked to solve
¤ Use in addition to other Algebra I curriculum, covering topics such as ¤ Equation solving ¤ Polynomials (e.g., factorization) ¤ Linear functions (e.g., quadratics)
¤ Grade 8 or 9 in U.S.
Alex’s “combine like terms” way!
Morgan’s “combine like terms” way
I first combined like terms on the left side of the equation.
First I subtracted 45y on either side; 60y - 45y is 15y.
Then I subtracted both sides by 60y.
Then I divided both sides by 15 to get the answer.
.
.
Then I divided both sides by 75 to get the answer.
* How did Alex solve the equation? * How did Morgan solve the equation? * Why did Alex combine the terms on the left as a first step? * Why did Morgan subtract 45y as a first step? * Which way is correct, Alex's or Morgan's way? How do you know? * Can you state a general rule about combining like terms that describes what you have learned from comparing Alex's and Morgan's ways of solving this type of problem?
3.2.2
28
Supporting Comparison Which is better? Alex and Morgan were asked to graph the equation
1. Present examples side-by-side
Alex’s “choose typical x values” way
First I chose some x values. As I usually do when I make a table of values, I picked x to be 0, 1, 2, 3, and 4.
For each x value in the table, I plugged it into the equation to find the corresponding y value. Then I plotted each ordered pair and connected the dots to give a graph of this line.
3. Prompt for specific comparisons, tailored to learning goals
x
using a table of values.
Morgan’s “choose x values more carefully” way
y
x
0
0
1
3
2
6
3
9
4
12
y
First I chose some xvalues. I chose multiples of 3.
For each x value in the table, I plugged it into the equation to find the corresponding y value.
x
y
x
y
0
4
0
4
1
13/3
3
5
2
14/3
6
6
3
5
9
7
4
16/3
12
8
Then I plotted each ordered pair and connected the dots to give a graph of this line.
y
y
x
2. Visual alignment and use of common language to draw attention to similarities
x
* How did Alex graph the equation? How did Morgan graph the equation? * What are some similarities and differences between Alex’s and Morgan’s ways? Why did Morgan chose to use only multiples of 3 for x? * Whose way is easier, Alex’s or Morgan’s? Why? 4.1.2
4. Be sure that students, not just teachers, are comparing and explaining.
(Richland, Zur & Holyoak, 2007) 29
Which is correct?
5. Include a lesson summary, highlighting key points of the comparison.
2 3 2 Alex and Morgan were asked to simplify 3 ⋅ 3 ⋅ 3
Alex’s “multiply the exponents” way
Morgan’s “product of powers” way
32 ⋅ 33 ⋅ 32
I multiplied the exponents.
32 ⋅ 33 ⋅ 32
Hey Alex, what did
2+3+2 3 2×3×2 we learn 3from
comparing these right and wrong ways? I raised 3 to the power of 12.
I got 531,441.
312
37
When multiplying numbers with the same base, you can add the exponents. Be careful not 2,187 531, 441 to multiply the exponents by mistake.
* How did Alex simplify the expression? * How did Morgan simplify the expression? * Which answer is correct, Alex’s or Morgan’s? How do you know? * What are some similarities and differences between Alex’s and Morgan’s ways? * What if the problem were changed to x 2 ⋅x 3 ⋅ x 2 ? What would the answer be?
I used the product of powers property and added the exponents.
I raised 3 to the power of 7.
I got 2,187..
9.2.1 30
Which is better?
Alex and Morgan were asked to solve Alex’s “distribute first” way
Compare Correct Strategies Which is better?
3( x + 2) = 15
Morgan’s “divide first” way
3(x + 2) = 15
3(x + 2) = 15
First I distributed across the parentheses.
3x + 6 = 15
3(x + 2) = 15
Then I subtracted on both sides.
3x + 6 = 15 −6 −6
3
First I divided on both sides.
3
x+2=5 .
3x = 9
I divided on both sides. Here is my answer.
3x = 9 3 3 x=3
x+2=5 −2−2
Then I subtracted on both sides. Here is my answer.
x=3
* How did Alex solve the equation? * How did Morgan solve the equation? * What are some similarities and differences between Alex’s and Morgan’s ways? * On a timed test, would you rather use Alex’s way or Morgan’s way? Why? * If the problem were changed to 3(x + 2) = 17, would Alex’s way or Morgan’s way be better? Why?
3.1.6 31
Why does it work?
Compare Correct Strategies 2 Why does it work?
Alex and Morgan were asked to solve 3x = 16 4 11
Alex’s “multiplication” way!
Morgan’s “cross-multiplication” way
3x 16 = 4 11
First I multiplied both sides of the equation by 4.
(4)
3x 16 = (4) 4 11
(11)3x =
33x = 64
64 (11) 11
33x = 64 33 33
33x = 64 Then I divided on both sides of the equation by 33. Here is my answer.
3x(11) = 16(4)
64 11
3x =
Then I multiplied both sides of the equation by 11.
3x 16 = 4 11
x=
64 33
First I cross-multiplied.
Then I divided both sides by 33.
Here is my answer.
33x = 64 33 33
x=
64 33
* How did Alex solve the equation? * How did Morgan solve the equation? * What are some similarities and differences between Alex's and Morgan's ways? * Even though Alex and Morgan did different first steps, why did they both get the same answer?
3.4.1 32
Compare to Incorrect Method Which is correct?
33
Study 1 Participants ¤ Algebra I teachers randomly assigned to condition ¤ 39 teachers (and their 781 students) used our comparison materials ¤ 29 teachers (and their 586 students) used their existing approach
34
Student Outcomes ¤ Standardized commercial algebra test ¤ Our own researcher-designed assessment of algebra knowledge ¤ Assessed conceptual knowledge, procedural knowledge, and procedural flexibility
¤ Teachers administered both assessments at beginning and end of school year
35
Results: Study 1 •
At end of school year, no effect of condition on any outcome measure. Students in two groups performed the same.
•
Perhaps due to •
Large variation in how frequently teachers actually used our materials. Used much less often than requested. •
Dosage: how often they used our materials x average length of time used
•
Average dosage was 140 minutes in entire school year; Less than 4% of math instructional time Range 0-864 minutes
•
36
Dose-Response Relationship All comparison types focused attention on procedures Attention to flexibility and concepts varied with type
(total minutes)
Increased dosage associated with increased procedural knowledge No other differences were reliable.
37
Reflections on Study 1 + Implemented infrequently ¤ Teachers needed to find time to plan which of our materials to use and when
+ When used, teachers’ implemented curriculum materials pretty well ¤ Prompting comparison, asking students questions ¤ But teachers struggled with: “Be sure that students, not just teachers, are comparing and explaining.” ¤ Teachers were doing most of the talking; little discussion
38
Revised Comparison and Explanation of Multiple Strategies (CEMS) Curriculum 1. Specify what materials to use when to increase dosage. 2. Increase support for having high quality class discussions.
39
Revised CEMS Curriculum 1. Specify what materials to use when. Focus on 5 units. To be used in most daily lessons during those units. Lesson
WEP Type
Suggested Use
3.1
Which is correct?
Mid-lesson
3.2
Why does it work?
Beginning of lesson
3.3
Which is better?
Mid-lesson
3.4
Why does it work?
Mid-lesson
3.5
Which is better?
End of lesson
3.6
Why does it work?
Beginning of lesson
40
Revised CEMS Curriculum ¤ 2. Increase support for explanation and class discussions. ¤ Provide routines and materials Compare ? Prepare to Compare • • •
What is the problem asking? What is happening in the first method? What is happening in the second method?
Make Comparisons •
What are the similarities and differences between the two methods? o Which method is better? o Which method is correct? o Why do both methods work? o How do the problems differ?
Discuss Prepare to Discuss (think, pair) • •
How does this comparison help you understand this problem? How might you apply these methods to a similar problem?
Discuss Connections (share) •
What ideas would you like to share with the class?
Identify the Big Idea •
Can you summarize the Big Idea in your own words?
41
Increased Professional Development ¤ One week (35 hours) during the summer ¤ After each unit, individual teacher meetings with a researcher ¤ Provide personal feedback on videotaped lesson ¤ Plan for next unit
Instructional Goals
Concrete Suggestions
+ Call on different students throughout the lesson.
(The following instructional goals are building on this strength)
Hear from at least two students whenever you ask a discussion question.
Aim for at least 2 responses per open ended question. Stay on a question longer by asking another student to summarize what they just heard, or if they agree/disagree with another student’s response. Students feel comfortable answering questions in your class, but their responses are brief. Use stems like Tell me more, or Why Ensure students compare the methods before moving on to the Discuss phase. Provide students with an opportunity to think independently before they pair. Make sure students still have their Discuss Connections worksheet in hand as they are discussing the Big Idea as a whole group.
Ask follow-up questions in response to student thinking. Attend to the sequence of the Implementation Model.
42
Study 2 Overview ¤ Teachers: ¤ Asked to use our materials several times a week, during 5 units of instruction. ¤ No researcher present during instruction. ¤ After each unit, met individually with a researcher for feedback and to plan
43
Study 2 Design ¤ Participants ¤ 9 treatment teachers used our CEMS approach with about 340 students ¤ 10 control teachers from different schools used their existing approach with about 230 students
¤ Data being collected this year ¤ Unit assessment at beginning and end of 5 units (pre & post) ¤ Overall assessment at beginning and end of school year ¤ Videos of instruction (2-3 videos per unit) ¤ Code for quality of instruction, particularly discussion
44
Instructional Quality Improving 1. Level of teacher questioning increases when using our materials vs. when using other curriculum materials ¤ Teachers much more likely to ask “why” and open-ended questions when using our materials ¤ E.g., “Can you generate another problem where Riley’s method could not be used?” vs. “What is the answer?”
2. Level of student responses also improves ¤ E.g., Student response focused on understanding, such as why an answer was correct or why a particular solution method might have been a good choice vs. states the answer.
3. Discussion among students also improves ¤ Multiple students respond to the same question, sometimes building on each other’s thinking vs. single student responds. 45
Reflections on Study 2 (so far) ¤Successfully improving ¤ Dosage. Most teachers using materials at specified times. ¤ Classroom discussions. Higher-level teacher questioning, student responses and discussion among students when using our materials. ¤ Student learning. ?? Don’t know yet
46
Conclusion ¤ Comparison can support mathematics learning ¤ Consistent benefits when used intensively for short period of time, especially comparing correct solution strategies when have some prior domain knowledge. ¤ Potential benefits when used throughout school year by teachers – if used frequently! ¤ American teachers need more support in leading high-quality discussions and in when to use comparison. ¤ Providing routines and feedback is helping.
47
48
Acknowledgements ¤ E-mail:
[email protected] ¤ Visit our Contrasting Cases Website at scholar.harvard.edu/contrastingcases for study materials, articles and more ¤ See my Researchgate profile for most recent findings
¤ Thanks to the Children’s Learning Lab at Vanderbilt University ¤ Funded by grants from the Institute for Education Sciences and the National Science Foundation ¤ Opinions expressed are those of the authors only!
48
Select References ¤
Rittle-Johnson, B., & Star, J. R. (in press). The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. In J. P. Mestre & B.H. Ross (Eds.), Psychology of Learning and Motivation: Cognition in Education (Vol. 55).
¤
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561574.
¤
Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529-544.
¤
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836-852.
¤
Rittle-Johnson, B., Star, J., & Durkin, K. (in press). Developing procedural flexibility: When do novices learn from comparing procedures? British Journal of Educational Psychology.
¤
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102, 408 – 426.
49
