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Contrasting Cases in Mathematics Lessons Support Procedural Flexibility and Conceptual Knowledge Jon R. Star Harvard University Bethany Rittle-Johnson Vanderbilt University EARLI Invited Symposium: Construction of (elementary) mathematical knowledge: New conceptual and methodological developments, Budapest, August 29, 2007

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2 Acknowledgements Funded by a grant from the United States Department of Education Thanks to research assistants at Michigan State University and Vanderbilt University: –Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis, Tharanga Wijetunge, Holly Harris, Jen Samson, Anna Krueger, Heena Ali, Sallie Baxter, Amy Goodman, Adam Porter, and John Murphy

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3 Comparison Is a fundamental learning mechanism Lots of evidence from cognitive science –Identifying similarities and differences in multiple examples appears to be a critical pathway to flexible, transferable knowledge Mostly laboratory studies Not done with school-age children or in mathematics (Gentner, Loewenstein, & Thompson, 2003; Kurtz, Miao, & Gentner, 2001; Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005; Schwartz & Bransford, 1998)

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4 Central tenet of math reforms Students benefit from sharing and comparing of solution methods “nearly axiomatic,” “with broad general endorsement” (Silver et al., 2005) Noted feature of ‘expert’ math instruction Present in high performing countries such as Japan and Hong Kong (Ball, 1993; Fraivillig, Murphy, & Fuson, 1999; Huffred-Ackles, Fuson, & Sherin Gamoran, 2004; Lampert, 1990; Silver et al., 2005; NCTM, 1989, 2000; Stigler & Hiebert, 1999)

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5 “Contrasting Cases” Project Experimental studies on comparison in academic domains and settings largely absent Goal of present work –Investigate whether comparison can support learning and transfer, flexibility, and conceptual knowledge –Experimental studies in real-life classrooms –Computational estimation (10-12 year olds) –Algebra equation solving (13-14 year olds)

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6 Why algebra? Area of weakness for US students; critical gatekeeper course Particular focus: Linear equation solving Multiple strategies for solving equations –Some are better than others –Students tend to memorize only one method Goal: Know multiple strategies and choose the most appropriate ones for a given problem or circumstance

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7 Solving 3(x + 1) = 15 Strategy #1: 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4 Strategy #2: 3(x + 1) = 15 x + 1 = 5 x = 4

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8 Similarly, 3(x + 1) + 2(x + 1) = 10 Strategy #1: 3(x + 1) + 2(x + 1) = 10 3x + 3 + 2x + 2 = 10 5x + 5 = 10 5x = 5 x = 1 Strategy #2: 3(x + 1) + 2(x + 1) = 10 5(x + 1) = 10 x + 1 = 2 x = 1

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9 Why estimation? Widely studied in 1980’s and 1990’s; less so now Viewed as a critical part of mathematical proficiency Many ways to estimate Good estimators know multiple strategies and can choose the most appropriate ones for a given problem or circumstance

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10 Multi-digit multiplication Estimate 13 x 44 –“Round both” to the nearest 10: 10 * 40 –“Round one” to the nearest 10: 10 * 44 –“Truncate”: 1█ * 4█ and add 2 zeroes Choosing an optimal strategy requires balancing –Simplicity - ease of computing –Proximity - close “enough” to exact answer

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11 Flexibility is key in both domains Students need to know a variety of strategies and to be able to choose the most appropriate ones for a given problem or circumstance In other words, students need to be flexible problem solvers Does comparison help students to become more flexible?

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12 Intervention Comparison condition –compare and contrast alternative solution methods Sequential condition –study same solution methods sequentially

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13 Comparison condition

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14 Sequential condition next page

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15 Outcomes of interest Procedural knowledge Conceptual knowledge Flexibility

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16 Procedural knowledge Familiar: Ability to solve problems similar to those seen in intervention Transfer: Ability to solve problems that are somewhat different than those in intervention AlgebraEstimation -1/4(x - 3) = 10Estimate: 12 * 24 5(y - 12) = 3(y - 12) + 20Estimate: 37 * 17 AlgebraEstimation 0.25(t + 3) = 0.5Estimate: 1.92 * 5.08 -3(x + 5 + 3x) = 5(x + 5 + 3x) = 24Estimate: 148 ÷ 11\

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17 Conceptual knowledge Knowledge of concepts AlgebraEstimation If m is a positive number, which of these is equivalent to (the same as) m + m + m + m? (Responses are: 4m; m 4 ; 4(m + 1); m + 4) What does “estimate” mean? For the two equations: 213x + 476 = 984 213x + 476 + 4 = 984 + 4 Without solving either equation, what can you say about the answers to these equations? Explain your answer. Mark and Lakema were asked to estimate 9 * 24. Mark estimated by multiplying 10 * 20 = 200. Lakema estimated by multiplying 10 * 25 = 250. Did Mark use an OK way to estimate the answer? Did Lakema use an OK way to estimate the answer? (from Sowder & Wheeler, 1989)

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18 Flexibility Ability to generate, recognize, and evaluate multiple solution methods for the same problem “Independent” measure –Multiple choice and short answer assessment Direct measure –Strategies on procedural knowledge items (e.g., Beishuizen, van Putten, & van Mulken, 1997; Blöte, Klein, & Beishuizen, 2000; Blöte, Van der Burg, & Klein, 2001; Star & Seifert, 2006; Rittle-Johnson & Star, 2007)

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19 Flexibility items (independent measure) Algebra Solve 4(x + 2) = 12 in two different ways. For the equation 2(x + 1) + 4 = 12, identify all possible steps (among 4 given choices) that could be done next. A student’s first step for solving the equation 3(x + 2) = 12 was x + 2 = 4. What step did the student use to get from the first step to the second step? Do you think that this way of starting this problem is (a) a very good way; (b) OK to do, but not a very good way; (c) Not OK to do? Explain your reasoning.

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20 Flexibility items (independent measure) Estimation Estimate 12 * 36 in three different ways. Leo and Steven are estimating 31 * 73. Leo rounds both numbers and multiplies 30 * 70. Steven multiplies the tens digits, 3█ * 7█ and adds two zeros. Without finding the exact answer, which estimate is closer to the exact value? Luther and Riley are estimating 172 * 234. Luther rounds both numbers and multiplies 170 * 230. Riley multiplies the hundreds digits 1█ █ * 2█ █ and adds four zeros. Which way to estimate is easier?

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21 Method Algebra: 70 7th grade students (age 13-14)* Estimation: 158 5th-6th grade students (age 10-12) Pretest - Intervention (3 class periods) - Posttest –Replaced lessons in textbook Intervention occurred in partner work during math classes –Random assignment of pairs to condition Students studied worked examples with partner and also solved practice problems on own *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), 561-574.

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22 Results Procedural knowledge Flexibility –Independent measure –Strategy use Conceptual knowledge

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23 Procedural knowledge Students in the comparison condition made greater gains in procedural knowledge.

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24 Flexibility (independent measure) Students in the comparison condition made greater gains in flexibility.

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25 Flexibility in strategy use (algebra) Strategies used on procedural knowledge items:

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26 Conceptual knowledge Comparison and sequential students achieved similar and modest gains in conceptual knowledge.

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27 Overall Comparing alternative solution methods rather than studying them sequentially –Helped students move beyond rigid adherence to a single strategy to more adaptive and flexible use of multiple methods –Improved ability to solve problems correctly

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28 Next steps What kinds of comparison are most beneficial? –Comparing problem types –Comparing solution methods –Comparing isomorphs Improving measures of conceptual knowledge

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29 Thanks! You can download this presentation and other related papers and talks at http://gseacademic.harvard.edu/~starjo Jon Star Jon_Star@Harvard.edu Bethany Rittle-Johnson Bethany.Rittle-Johnson@vanderbilt.edu

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