Number Talks in High School--STEP Conference

When 5x12 is more than 60: Exploring Number Talks in High School

Melissa JohnsonTara PereaBeth Peters

Agenda

Number Talk

How are number talks supported by research?

Dilemmas and Strategies in High School

Common Core Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively3. Construct viable arguments and critique

the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated

reasoning.

Make sense of problems and persevere in solving them “Mathematically proficient students…

consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution.”

Make sense of problems and persevere in solving them James Heibert: “Making Sense”

(1997)

“In order to take advantage of new opportunities and to meet the challenges of tomorrow, today’s students need flexible approaches for defining and solving problems.”

Reason abstractly and quantitatively “Quantitative reasoning entails…

attending to the meaning of quantities…and flexibly using different properties of operations and objects.”

Reason abstractly and quantitatively

Study by Eddie Gray and David Tall (1994)

Students ages 7-12 in two groups: above average ability & below average ability

Gave them three different types of simple addition problems

4 different ways of solving the problems: Count all, count on, known facts, derived facts

Reason abstractly and quantitatively Analysis:

The lower ability group was doing a different kind of math!!

Highlights the importance of exposing students to many different kinds of strategies for working with numbers.

Look for and make use of structure “Mathematically proficient students…

can step back for an overview and shift perspective.”

Look for and make use of structure Relational vs. Instrumental Understanding

in Math

Richard Skemp (1976) wrote about two different types of math understanding

Relational understanding = “knowing both what to do and why”

Instrumental understanding = “rules without reasons”

Look for and make use of structure Instrumental Understanding: the problem

with algorithms

Algorithm = “a process or set of rules to be followed in calculations or other problem-solving situations, esp. by a computer”

Students memorize algorithms, or “rules” without understanding them – and then often forget or mix up the many different rules

When the problem doesn’t exactly fit a rule, students do not know how to solve it

Look for and make use of structure Relational Understanding: creating

problem-solvers

Skemp writes, “the more complete a pupil’s schema, the greater his feeling of confidence in his own ability to find new ways of ‘getting there’ without outside help”

Students can create an overview of problems and develop creative ways to solve them

Attend to precision “Mathematically proficient students try

to communicate precisely to others.”

NCTM Principles and Standards. Heibert study

Look for and make use of structure James Heibert and Diana Wearne

Study (1993) Compared 6 2nd-grade classrooms

Two classrooms received alternate instruction (fewer problems, more discussion of methods and strategies)

Analyzed test scores before and after instruction

Before Instruction

Average Ability High Ability

Question Type# of

Questions

A B C D E F

Place Value 8 22% 25% 21% 27% 71% 69%

Instructed Computation 4 35% 31% 41% 32% 70% 80%

Story Problem 1 38% 56% 53% 43% 90% 97%

New Computation 8 3% 4% 6% 3% 25% 29%

After Instruction

Average Ability High Ability

Question Type# of

Questions

A B C D E F

Place Value 7 37% 24% 36% 53% 60% 86%

Instructed Computation 13 72% 71% 76% 83% 88% 88%

Story Problem 3 45% 44% 60% 79% 76% 93%

New Computation 5 38% 39% 39% 54% 59% 63%

Look for and make use of structure Analysis:

Allowing students to talk about their solution methods leads to better understanding and therefore better results

“The data reported in this study suggest that, in mathematics classrooms, certain kinds of instructional tasks and discourse encourage more productive ways of thinking.”

Construct viable arguments and critique the reasoning of others

“They justify their conclusions, communicate them to others, and respond to the arguments of others.”

Construct viable arguments and critique the reasoning of others

Noreen Webb Study (2009)

Looked at data from four 2nd and 3rd grade classrooms Focused on teacher questioning and

student explanations

Then compared test scores from the four different classrooms

Teacher probed students’ explanations to uncover details or further thinking about their problem-solving strategies

Classroom A B C D

Whole-Class 23 25 92 71

Small-Group 36 25 77 50

Student Explaining in Small Groups

Classroom A B C D

Group gave correct/com

plete explanation

16 33 72 56

Student Achievement

Classroom A B C D

Written Assessment 17 30 47 45

Interview 13 24 37 44

Construct viable arguments and critique the reasoning of others

Analysis:

Teacher practices of questioning led to more student explanation and greater percentage of correct answers

“By asking students to explain their methods for solving problems and refraining from evaluating students’ answers, teachers helped create expectations and obligations for students to publicly display their thinking…”

Possible Dilemmas in high school

Students afraid to share their thinking Maintaining number talks in high school – students think

it’s childish How much should a facilitator “push”? Traditional algorithm Receiving support from other faculty or parents – doing

number talks alone may be difficult Choosing a problem that is at the right level Supporting English Learners or students with special needs Timing Disjoint between number talk and lesson Status issues Introducing number talks in a way that is meaningful to

high schoolers

We will focus on… Students afraid to share their thinking Maintaining number talks in high school – students think

it’s childish How much should a facilitator “push”? Traditional algorithm Receiving support from other faculty or parents – doing

number talks alone may be difficult Choosing a problem at the right level Supporting English Learners or students with special

needs Timing Disjoint between number talk and lesson Status issues Introducing number talks in a way that is meaningful to

high school students

1. Students Afraid to Share

2. Traditional Algorithm

3. Choosing a Problem at the Right Level

4. Disjoint between Number Talk and Lesson Content

Questions?

Thank you for coming!