How Students Learn Mathematics in the Classroom June 18, 2009.

How Students LearnMathematics in the Classroom

June 18, 2009

Principle #1: Teachers Must Engage Students’ Preconceptions

Preconception #1 – Mathematics is about learning to compute.

What approximately is the sum of 8/9 and 12/13?

Conceptually Procedurally Sense making


Preconception #2 – Mathematics is about “following rules” to guarantee correct answers.

Systematic pattern finding and continuing invention Tower of Hanoi

Units used to quantify fuel efficiency of a vehicle Miles per gallon Miles per gallon per passenger

Compare different procedures for their advantages and disadvantages


Preconception #3 – Some people have the ability to “do math” and some don’t.

Amount of effort Some progress further than others Some have an easier time Effort is key variable for success

Informal vs. formal mathematical strategies

Brazilian street children could perform mathematics when making sales in the street, but were unable to answer similar problems when presented in a school context.

Carraher, 1986; Carraher et al., 1985

Informal vs. formal mathematical strategies

Men who successfully handicapped horse races could not apply the same skill to securities in the stock market.

Ceci and Liker, 1986; Ceci, 1996

Key Point to Remember

Without a conceptual understanding of the nature of the problems and strategies for solving them, failure to retrieve learned procedures can leave a student completely at a loss.

How to best engage students’ preconceptions and build on existing knowledgeAllow students to use their own

informal problem-solving strategies Wrong answer (usually partially correct) Find part that is wrong Understand why it is wrong Aids understanding Promotes metacognitive competencies

How to best engage students’ preconceptions and build on existing knowledge

Encourage math talk Actively discuss various approaches Learner focused Draw out and work with preconceptions Making student thinking visible Model the language

How to best engage students’ preconceptions and build on existing knowledge

Design instructional activities that can effectively bridge commonly held conceptions and targeted mathematical understandings

More proactive Research common preconceptions and points of

difficulty Carefully designed instructional activities

Key Point to Remember

Identifying real-world contexts whose features help direct students’ attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students’ informal experiences and the new formal mathematics they are learning.

Principle #2: Understanding Requires Factual Knowledge and Conceptual Frameworks

MDE HSCE for Mathematics p. 4 Conceptual Understanding Procedural Fluency Effective Organization of Knowledge Strategy Development Adaptive Reasoning

How Students Learn Mathematics in the Classroom

A Developmental Model for Learning Functions

A Developmental Model for Learning Functions.

Levels of Understanding 0 – Recognize and Extend Pattern 1 – Generalize the Pattern and Express it as

a function (y = 2x). 2 – Look at graph and decide if a particular

function could model it. 3 – Using Structures from Level 2 to create

and understand more complex functions.

Principle #3: A Metacognitive Approach Enables Student Self-Monitoring

Learning about oneself as a Learner Thinker Problem solver

Instruction That Supports Metacognition

An emphasis on debugging Finding where the error is Why it is an error Correcting it

Internal and external dialogue as support for metacognition

Help students learn to interact Model clear descriptions, supportive questioning,

helping techniques Seeking and giving help

Targets and Student Goal-Setting

Problem Learning Target Right? Wrong?Simple

mistake?More

study?

1 L2.1.2 translate roots to exponents



4 A1.1.2 understand the rules of exponents (integer)

5 A1.1.2 understand the rules of exponents (radicals)

6 A1.1.2 understand the rules of exponents (rational)

7 A1.1.2 understand the rules of exponents (integer)


9 A1.1.2 understand the rules of exponents (rational)



12 A2.1.1-3,A2.1.7 understand exponential equations

A2.1.1-3,A2.1.7 understand exponential table

A2.1.1-3,A2.1.7 understand exponential graph

A2.1.1-3,A2.1.7 understand exponential domain/range

A2.1.1-3,A2.1.7 understand exponential asymptote

Targets and Student Goal-Setting

I AM GOOD AT THESE!

Learning targets I got right:

I AM PRETTY GOOD AT THESE, BUT NEED TO DO A LITTLE REVIEW

Learning targets I got wrong because of a simple mistake:

I NEED TO KEEP LEARNING THESE

Learning targets I got wrong and I’m not sure what to do to correct them:

How Students Learn Mathematics in the Classroom June 18, 2009.

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