# Presentation: 472726

## Research and Instructional Practices

- National Math Panel Meeting
- Stanford, CA     November 6, 2006
- James Hiebert, University of Delaware

## Two Points

- Teaching matters
- Teaching that helps students make sense of mathematics possesses a few key features

## Teaching Matters

- All educational innovations reach students through teaching
- By teaching, I mean the way teachers interact with students about content
- Unless teaching changes, students won’t know the difference
- Teach_ing_ is not the same as teach_ers_

## What Makes Teaching Effective?*

- Difficult to answer empirically
- Probably not *a best* way of teaching; some features might be better for some learning goals than others
- What about the goal of conceptual understanding, or making sense of mathematics?
- Two research-based features can be identified
- *References available upon request

## Features of Teaching That Help Students Make Sense of Mathematics

- Make conceptual relationships clear
- Attend explicitly, in some way, to relationships among facts, procedures, representations, ideas, etc.
- E.g., examine the similarities among patterns with constant rates of change, linear functions, and straight lines on a graph
- E.g., notice that a problem solved near the end of a lesson is a special case of a problem solved near the beginning

## Features of Teaching That Help Students Make Sense of Mathematics

- Allow students to do some of the important mathematical work
- Teachers pose challenging problems (just beyond the familiar) and allow time for students to work
- Major threat is teachers jumping in too quickly and giving students the answer

## International Data That Address the Two Features of Teaching

- TIMSS 1999 Video Study examined about 100 8th–grade math lessons in each of 7 countries
- At least 80% of lesson time in every country is spent working on math problems
- How problems are worked on with students gets at the heart of how teachers and students interact about content

## Types of Problems Presented to Students       

- Stating Concepts: recalling or applying definitions or conventions
- Using Procedures: applying learned procedures
- Making Connections: constructing relationships among ideas, facts, or procedures

## Examples: Making Connections

- Solve these two equations and describe what is different about their solutions
-  2x + 4 = x + 6         2x + 10 = 2( x + 5 )
- Find a pattern for the sum of the interior angles of polygons with varying numbers of sides

## Percentage of Problems Presented as Each Type 

**Notes:**
Making connections: JP>AU, CZ, HK, US
Using procedures: CZ>JP, NL; HK >AU, JP, NL, US; US>JP
NOTE: Analyses do not include answered-only problems (i.e., problems that were completed prior to the videotaped lesson and only their answers were shared). For each country, average percentage was calculated as the sum of the percentage within each lesson, divided by the number of lessons. English transcriptions of Swiss lessons were not available for mathematical processes analyses. Percentages may not sum to 100 due to rounding.
SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study-Repeat, Video Study, 1999.

## Types of Problems Presented AND Worked On During the Lesson

- Each problem was coded a second time based on how it was worked on and discussed during the lesson.
- This is where teaching really matters.
- Problems can be transformed

## Example: Transforming a Problem From Making Connections Using Procedures

- Problem: Find a pattern for the sum of the interior angles of polygons
- Worked on as Making Connections
- Measure the sum of the angles for 3- , 4-, and 5-sided polygons; predict for 6-sided polygons; for n-sided polygons; develop a general formula
- Worked on as Using Procedures
- Present formula [Sum = 180(n – 2)] and ask students to practice

## Again - Percentage of Problems Presented as Each Type 

**Notes:**
Making connections: JP>AU, CZ, HK, US
Using procedures: CZ>JP, NL; HK >AU, JP, NL, US; US>JP
NOTE: Analyses do not include answered-only problems (i.e., problems that were completed prior to the videotaped lesson and only their answers were shared). For each country, average percentage was calculated as the sum of the percentage within each lesson, divided by the number of lessons. English transcriptions of Swiss lessons were not available for mathematical processes analyses. Percentages may not sum to 100 due to rounding.
SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study-Repeat, Video Study, 1999.

## Percentage of Making Connections Problems Worked On in Each Way 

**Notes:**
This slide shows just the percentage of MC problems worked on as UP or MC. But these do not add to 100 because they might be worked on in other ways. Remember there was a category of Stating Concepts. And we had to add a fourth category to capture the cases when teachers gave only the answer; when they did no mathematics publicly with the students. 
Making connections: CZ, HK, JP, NL>AU, US
Using procedures: US>CZ, HK, JP, NL 
NOTE: Analyses only include problems with a publicly presented solution. Analyses do not include answered-only problems (i.e., problems that were completed prior to the videotaped lesson and only their answers were shared). For each country, average percentage was calculated as the sum of the percentage within each lesson, divided by the number of lessons. English transcriptions of Swiss lessons were not available for mathematical processes analyses. Percentages may not sum to 100 due to rounding. Lessons with no making connections problem statements were excluded from these analyses.
SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study-Repeat, Video Study, 1999.

## Percentage of Making Connections Problems Worked On in Each Way 

**Notes:**
Making connections: CZ, HK, JP, NL>AU, US
Stating concepts: JP, NL>US 
Using procedures: US>CZ, HK, JP, NL 
Giving results only: AU, US>CZ, HK, JP, NL
NOTE: Analyses only include problems with a publicly presented solution. Analyses do not include answered-only problems (i.e., problems that were completed prior to the videotaped lesson and only their answers were shared). For each country, average percentage was calculated as the sum of the percentage within each lesson, divided by the number of lessons. English transcriptions of Swiss lessons were not available for mathematical processes analyses. Percentages may not sum to 100 due to rounding. Lessons with no making connections problem statements were excluded from these analyses.
SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study-Repeat, Video Study, 1999.

## What Conclusions Can Be Drawn From the International Comparisons?

- Although high achieving countries displayed different styles of teaching, they shared a relative emphasis on implementing “making connections” problems as making connections
- This includes attending explicitly to key relationships and allowing students to do important mathematical work
- Good curricula is not enough; teaching transforms curricula

## Why Are the Two Features Absent From U.S. Math Teaching?

- Not a new finding
- Teaching is hard to change
- Ways of teaching are passed from generation to generation
- The cycle will not be broken just by selecting teachers with stronger academic records

## Changing Teaching

- Change will occur gradually as we
- Develop a consensus about the key learning goals for students
- Continue to refine research-based recommendations for features of effective teaching aligned with these goals
- Find levers for change that make sense to teachers and students
- Sustain support for teacher work aimed toward changing classroom teaching