the role of comparison in the development of flexible knowledge of computational estimation jon r....

The Role of Comparison in the Development of Flexible Knowledge

of Computational Estimation

Jon R. Star (Harvard University)

Bethany Rittle-Johnson (Vanderbilt University)

AERA, New York City. Tuesday, March 25, 2008. Session 33.074 Developing Mathematical Understanding paper session, Crowne Plaza Times Square, Room 509/510, 4:05 – 5:35 pm

Thanks to...• Research supported by Institute of Education

Sciences (IES) Grant # R305H050179 • All participating teachers and schools in

Nashville, Tennessee and Hale, Michigan• Graduate and undergraduate research

assistants at Vanderbilt, Michigan State, and Harvard

March 25, 2008 2AERA CCS4

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

March 25, 2008 3AERA CCS4

(Gentner, Loewenstein, & Thompson, 2003; Kurtz, Miao, & Gentner, 2001; Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005; Schwartz & Bransford, 1998)

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

March 25, 2008 AERA CCS4 4

(Ball, 1993; Fraivillig, Murphy, & Fuson, 1999; Huffred-Ackles, Fuson, & Sherin Gamoran, 2004; Lampert, 1990; Silver et al., 2005; NCTM, 1989, 2000; Stigler & Hiebert, 1999)

“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– Algebra equation solving– Computational estimation


Rittle-Johnson & Star, 2007• Experimental study in algebra classrooms with

70 7th grade students on equation solving• Intervention (Comparison condition)

– Comparing and contrasting alternative solution methods

• Control (Sequential condition)– Reflecting on same solution methods one at a time


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.

Results (from 2007 study)• At posttest, students in comparison condition

made significantly greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge

• The intervention worked! But need to replicate and extend these findings...


Computational estimation• Widely studied in 80’s and early 90’s

– Less so in recent years• Process of mentally generating an approximate

answer for a given arithmetic problem (Rubenstein, 1985)

– (Distinct from “mental computation,” which means finding the exact answer)

• Estimates of 2-digit multiplication problems13 x 27


Strategies for estimating 13 x 27• Round both (to the nearest 10)

10 x 30 or 300• Round one (to the nearest 10)

10 x 27 or 270(Alternatively, 13 x 30, or 390)

• Trunc (truncate) (Sowder & Wheeler, 1989)

1 x 2, or 2 Then append 2 zeros, or 200


Why estimation for replication?• Estimation is different from algebra equation

solving in several ways that play a potentially important role whether comparison will help


Equation solving vs. estimation• Equation solving

– Problems have a single correct answer• Estimation

– Correctness or “goodness” of estimate depends on two sometimes competing goals

– Simplicity: how easy it is to compute– Proximity: how close the estimate is to the exact

answer


Complexity of “goodness”• Sometimes easy-to-compute estimate is not very

proximal to the exact value, and vice versa• Some strategies present false illusion of

consistent proximity• Is round one always more proximal than round

two?– Intuitively, yes? The less you round, the closer you get– Actually no! It depends on the problem– Try it for 39 x 41 versus 39 x 37


Solution efficiency• Algebra equation solving

– Easy and visually apparent to judge relative efficiency of two compared solutions

– For example, just count the number of steps!• Estimation

– Not at all clear how one would judge efficiency of two compared solutions

– Efficiency is more of an individual or subjective judgment


Summary of rationale• Evidence for effectiveness of comparison in

algebra, but replication needed• Estimation is a good domain in which to replicate

– Certain features of estimation raise legitimate questions about whether comparison of strategies can have the same positive impact


Method• Estimation: 158 5th-6th grade students• 69 5th graders in urban private school

– 4 classes, taught by the same teacher• 44 5th and 45 6th graders in rural public school

– Two 5th classes, taught by same teacher – Two 6th classes, taught by same teacher


Design• Pretest - Intervention – Posttest – Retention test

– 3-day intervention replaced lessons in textbook• Intervention occurred in partner work during

math classes– Random assignment of pairs to condition– Both conditions present in all classrooms

• Students studied worked examples with partner and also solved practice problems on own


Comparison materials


Sequential materials


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Results• Procedural knowledge

– Ability to compute accurate estimates• Flexibility

– Knowledge of multiple strategies and ability to select most appropriate strategies for a given problem and problem-solving goal

– Direct measure (from procedural knowledge items)– Independent measure

• Conceptual knowledge


Procedural knowledge• Mental

– Estimate 32 x 17 mentally and quickly• Familiar

– Estimate 12 x 24 and 113 x 27• Transfer

– Estimate 1.19 x 2.39 and 102 ÷ 27


Procedural knowledge results• No difference between conditions on:• Accuracy of estimates

– Assessed accuracy a number of ways, none of which showed a difference for intervention students

• Both conditions improved the accuracy of students’ estimates


Accuracy of estimates


Conceptual knowledge sample• What does “estimate” mean?


Conceptual knowledge results• Comparison improved conceptual knowledge for

students who began with some initial procedural knowledge, but not for those who began with very little procedural knowledge

• In other words, comparison was particularly beneficial for students who began the study with some initial ability to compute estimates


Conceptual knowledge


Flexibility sample items• Independent measure• Estimate 12 x 36 in three different ways


Flexibility results• Comparison led to greater flexibility• Independent measure: Comparison students...

– Were more likely to be able to produce estimates for the same problem in multiple ways

– Were better at making judgments about which strategy would led to an easier or a closer estimate for a given problem


Flexibility independent measure


Flexible strategy use• Direct measure

– Strategies on procedural knowledge assessment• Comparison students were:

– More likely to use trunc, optimizing their strategy use for ease more than sequential student


Flexibility• Recall that students in both conditions saw the

same strategies demonstrated on the same problems

• Yet comparison students were– Better at generating multiple ways to find estimates– Were more likely to use the easiest strategy for a

given problem– Were better at predicting which strategy would led to

a close or easy estimate for a given problem


In sum...• It pays to compare!• Comparison led to:

– Greater flexibility (2007 and present study)– Improved procedural knowledge (2007 study)– Improved conceptual knowledge (for students with

modest procedural knowledge at pre-test; present study)

– In two very different mathematical domains, algebra and computational estimation


National Math Panel (p. 27)• “Teachers should broaden instruction in

computational estimation beyond rounding. They should insure that students understand that the purpose of estimation is to approximate the exact value and that rounding is only one estimation strategy.”


National Math Panel (p. 27)• “Textbooks need to explicitly explain that the

purpose of estimation is to produce an appropriate approximations. Illustrating multiple useful estimation procedures for a single problem, and explaining how each procedure achieves the goal of accurate estimation, is a useful means for achieving this goal. Contrasting these procedures with others that produce less appropriate estimates is also likely to be helpful.”


Thanks!This presentation and other related papers and

presentations can be found at:

http://gseacademic.harvard.edu/~starjo/or

by contacting Jon Star ([email protected])

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