Students’ use of standard algorithms for solving linear equationsJon R. StarMichigan State University

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Acknowledgements

Thanks to graduate students at MSU: Kosze Lee, Beste Gucler, Howard Glasser, Mustafa Demir, and Kuo-Liang Chang

Thanks to Bethany Rittle-Johnson, Vanderbilt, for her collaboration in the design and implementation of this study.

Funds supporting this work provided by small grants from the Michigan State University College of Education.

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Starting definitions A procedure is a step-by-step plan of action

for accomplishing a task A strategy is a plan of action for

accomplishing a task I use these terms synonymously, as is the

norm among many psychologists who study strategy change (e.g., Siegler)

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More definitions A procedure/strategy can be either: A heuristic, which is a helpful procedure for

arriving at a solution; a rule of thumb An algorithm, which is a procedure that is

deterministic; when one follows the steps in a predetermined order, one is guaranteed to reach the solution

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Standard algorithms For some problems, a “standard algorithm”

(SA) exists Called “standard” because it is commonly

and often explicitly taught as THE way to solve problems within a problem class

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Strategies for 4(x + 5) = 80 A standard algorithm (SA)

4(x + 5) = 804x + 20 = 804x = 60x = 15

Alternative approach #14(x + 5) = 80x + 5 = 20x = 15

Alternative approach #24(x + 5) = 804x + 20 = 804x - 60 = 0x - 15 = 0x = 15

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SA, more generally

1. Distribute first, to “clear” parentheses2. Combine like variable and constant terms

on each side3. ‘Move’ variable terms to one side and

constant terms to the other side4. Divide both sides by the coefficient of the

variable term

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Pros and cons of SA Reasonably efficient Widely applicable Can be executed often

without attending to specifics of the problem

Are not always the best or most efficient strategy

Over reliance on SA may lead to difficulties on unfamiliar problems

Ability to use not always connected with why algorithm is effective; may lead to rote memorization; strategy may be easily forgotten

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Learning standard algorithms Learning and use of SAs has become a

flashpoint issue in US mathematics education

Should they be learned at all? explicitly taught? discovered?

Not a lot of research on students’ learning of SA to help resolve these issues

Particularly on algorithms other than arithmetic

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Not researched in high school? Key features of elementary school reform

instruction are less typical at high school level:

Sharing and comparing of multiple strategies for solving problems

Allowing students to discover their own algorithms, rather than providing direct instruction on a SA

Allowing students to use non-standard algorithms

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Not researched in high school? Discovery of SAs is presumed to be more

difficult, if not highly improbable, in high school “Are you saying you want my students to

‘discover’ the quadratic formula?!” As a result, many teachers feel that it is

necessary to provide direct instruction on strategies such as the SA “If I don’t teach students this algorithm, there is

no way that they would come up with it on their own.”

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Unquestioned assumptions Is direct instruction the only way that

students will learn the SA? Can students discover the SA largely on

their own? When some students discover a strategy

and others are shown it by direct instruction, is there a difference in how students use the strategy?

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Larger goal: Flexibility We want students to know the SA but also

to be flexible in their knowledge of problem solving strategies, meaning that they:

Know a variety of other strategies (SA and others) that can be used to solve similar problems

Are able to adaptively select the most appropriate strategy (SA and others) for solving a particular problem

(Star, 2001, 2002, 2004, 2005)

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Research questions Do students discover the SA for solving

linear equations when allowed to work largely on their own?

Do either of two instructional interventions affect the discovery and use of the SA among algebra equation solvers? Direct instruction Alternative ordering task (Star, 2001)

Goal was to see what strategies students develop and how they make sense of, use, and modify these strategies

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Method 130 6th graders (82 girls, 48 boys) 5 one-hour classes in one week (Mon - Fri) Class size 8 to 15 students; students

worked individually Pre-test (Mon), post-test (Fri); three

problem-solving sessions (Tues, Wed, Thurs)

Domain was linear equation solving 3(x + 1) = 12 2(x + 3) + 4(x + 3) = 24 9(x + 2) + 3(x + 2) + 3 = 18x + 9

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Prior knowledge & instruction Students had no prior knowledge of

symbolic approaches for solving equations Minimal instruction and feedback provided 30 minute benchmark lesson Combine like terms, add to both sides,

multiply to both sides, distribute How to use each step individually No strategic guidance provided during study No worked examples

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Alternative ordering task During problem solving, some students

were asked to re-solve a previously completed problem, but using a different ordering of steps (Star, 2001)

Random assignment to condition by class Control group solved new but isomorphic

problem 2(x + 1) = 10 3(x + 2) = 15

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Direct instruction At start of 2nd problem solving class (Wed),

3 worked examples presented to direct instruction classes

“This is the way I solve this equation.” Each problem solved with using a different

method; one was the SA Total time was 8 minutes of supplemental

instruction Random assignment to condition by class

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Analysis Students’ written work was analyzed for use

of SA Booklet problems (Tues, Wed, Thurs

sessions) - total of 31 equations attempted Post-test problems - total of 9 equations

attempted Three “markers” of SA:

Distribute first Combine like terms before moving Divide as a final step

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Results. About 2/3 of students did not discover SA Of those who did, a small number started

using SA very early

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Results..

25

66

9

0 20 40 60 80 100

Percent of students

Discovery of SA

Early UsersNo SASA

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Results... Those who discovered and used SA

performed better on the post-test than those who did not use SA (p < .01)

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Results...

94

76

59

0

10

20

30

4050

60

70

80

90

100Percent correct

Performance on post-test problems

Early Users SA No SA

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Results.... Direct instruction on SA did not increase

chances that a student would use SA on post-test

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Results.....

16 16

49 37

4 8

0 20 40 60 80Number of students

SA Users

Non-SA Users

Early Users

Learning SA from Discovery vs. Direct Instruction

Direct Instruction Discovery Only

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Results...... Stated somewhat differently (and not

including 12 Early Users): 25% (16 of 65) of students in the Direct

Instruction condition used SA on the post-test

30% (16 of 53) of students in the Discovery condition used SA on the post-test

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Results....... The alternative ordering task made it less

likely that a student would use the SA on the post-test (p < .05)

Alternative ordering task made it more likely that students would use other, more efficient or innovative strategies than the SA on the post-test

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Results........

12 50

20 36

0 20 40 60 80

Number of students

Alternativeordering

Control

Impact of Alternative Ordering Task on SA Discovery

SANo SA

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Summary of results. Do students discover the SA for solving

linear equations when allowed to work largely on their own?

Most did not Only about one-fourth of students learned

the SA on their own Is one-fourth high or low?

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Summary of results.. Do either of two instructional interventions

affect the discovery and use of the SA among algebra equation solvers?

There was no difference in the rate of SA use between the direct instruction and discovery conditions

The alternative ordering condition made it less likely that students used the SA on the post-test

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Implications for SA learning Neither a short period of direct instruction

(viewing of worked examples) nor pure discovery was particularly effective in promoting development of the SA

Is learning the SA a goal of algebra instruction? If so, how should it best be taught?

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Implications for flexibility Flexibility aided by activities such as the

alternative ordering task, where students generate and compare multiple strategies for solving procedural problems

Direct instruction did not improve chances of discovering the SA, so activities such as the alternative ordering task appear to be a win-win proposition

This presentation and other related papers can be downloaded at:

www.msu.edu/~jonstar

Jon R. Star Michigan State University jonstar@msu.edu