Algebra learning: Insights from Cognitive Psychology

People are bad at math

• Especially Community College students– 98% are put in remedial math at Chaffey college– Over 60% of students cannot pass the CC level

because they cannot pass math– Recidivism in Algebra is astonishing

• Average of 3X at SMC!

Problems with Algebra

• Lack of goal-directedness– Students do not know “the point” of concepts and

procedures in algebra• Students often think they don’t have one!• Results in disconnected procedures for solving

• Can we illuminate the goals so students know “the why”– Use the implicit process of causal induction

Using implicit learning processes• Insights from Learning research

Use error-correction (prediction error) to engage learning mechanisms

• Allow students to fail to solve a problem early in learning process

– Give a difficult problem, e.g. – If student gives wrong answer, allow them to discover the

difficulties that caused their failure/difficulty– This increases confidence and allows them to reason comfortably

• Give them contrasting problem(s)€

x 2 −13x − 30 = 0

Contrasts

• Contrasts: allow discovery of differences across two events/units

– causal learning: discover causal relationships» Implicit comparison across cause-present/ cause-

absent information» Implicit: happens naturally in presence of relevant

stimuli• Math stimuli

– Also: categorization, boundaries of categories» Compare whale v fish, differences matter

Summary: Use of contrasts in math

Student fails to solve: prediction error

Give contrast: implicit process of comparing/contrasting– Student discovers difficulty in original (goal 1)– Student discovers how to overcome difficulty (goal 2)– Student might learn “the point” of a certain concept of

procedure (goal 3)

Our study: Algebra, factorable quadratics

• Use of Zero-product Rule (ZPR) and Factoring to solve quadratic equations– If a student cannot solve :

– But can solve

– & knows ZPR

By comparing using contrasts, students discover • the goals of procedures in algebra• The causes of their failures• how to avoid those failures and reach their goals

€

(x + 4)(x + 2) = 0

€

x⋅ y = 0

€

x + 4 = 0

Our study: Design• Pretest > Intervention > Post-test

– Pretest• 5 algebra and alg. notation problems

– Intervention • Experimental “contrast” condition

– Script to use to prompt subjects

• Control “traditional” condition

– Post-test• 11 quadratics culled from Thai study

– We can compare our results to published data

Can’t solve

Can solve

Student can discoverthe difficulty of having x and x2

Goal: remove that difficulty

Goal: how to remove difficulty by using knowledge they already have Student can discover how factoring relates to

solving quadraticsGoal: factoring removes difficulty of x & x2

Student can discover why we use factoringGoal: learn why factoring allows us to go From 1d to something like 1a

Pilot data

Traditional

Contrast

0 1 2 3 4 5 6 7 8 9 10 11

# Correct

# Correct

Results are significant, t < .001 with an n=7

A closer look

Posttest: test for generalization of ZPR• Contrast: 4 out of 4• Traditional: 1 out of 3

• Contrast: 3 out of 4• Traditional: 0 out of 3€

x 2(2x +1)(x +1) = 0

€

(x −1)(x +1) = 3

Issues

• Is this the best control?

• What to do when Ss give the answers we arent looking for during the comparing and contrasting?