1

Overview

• Comments on notebooks & mini-problem

• Teaching equitably• Analyzing textbook lessons• End-of-class check (Brief discussion)

• Introduction to multiplication• Wrap up & assignments

2

Inequity in Mathematics Education

• Mathematics achievement is a problem in the U.S.

• Mathematics functions as a gate-keeper• Differences among students are a given• However: Persistent and dramatic differences in mathematics achievement, associated with race and class

• Many contributing factors • Equity goal: school outcomes not predicted by race and class (RAND, 2003)

• Instruction has effects on students’ opportunities to learn, and their achievement

3

Working to Create Equitable Practice

• Inequity is partly reproduced inside of instructional practice.

• Breaking this cycle depends on joining concerns for equity with the daily and minute-to-minute work of teaching.

• Teachers can have leverage at strategic points in their work.

4

Four Specific Practices to Attend to Equity

• TASKS: Select mathematical tasks with care– Consider implicit assumptions about contexts, other barriers

– Use students’ out-of-school knowledge and experience

• LANGUAGE: Notice and support language transitions that students face

• EXPLICITNESS: Make proficient mathematical practices explicit

• SUPPORT PARTICIPATION– Broaden concept of mathematical proficiency– Support students’ public classroom work– Attend to whose work is made public

5

Analyzing a Textbook Lesson

6

Getting the General Sense of a Lesson

• Determine the mathematical content and purpose– Review the objective as stated in the text– Do the tasks yourself– Relate the content to the strands of mathematical proficiency or mathematical processes

• Consider students as learners of this content– What knowledge or skills do students need to access the tasks?

– What methods are students likely to use? What solutions or responses are they likely to generate?

– What misconceptions are students likely to have? What errors might they make?

7

Examining and Revising the Specifics of a Lesson

• Tasks and their progression• Examples (e.g., specific numbers)• Context• Representations and tools• Language

With attention to mathematical integrity and the learning of all students

8

End-of-Class Check

• A short question or prompt to assess students at the end of a lesson

• Possible purposes:– to check on students’ understanding of content

– to help students reflect on their learning

– to see what students think about their work that day

– to reinforce skills or concepts

9

End-of-Class Check Assignment

1. Find out specifics of the lesson --- get copies of the teachers’ guide, student work pages, etc.

2. Design an end-of-class check about the specific mathematics content of the lesson. Consider your purpose, how long it will take students to answer it (keep it short), and how you plan to pose the prompt to your class.

3. If you would like feedback, please send an email at least three days before the lesson.

4. Carry out the end-of-class check that you designed. Collect student work, etc.

5. Review and analyze students’ responses.6. Complete the professional practice piece --- type your

responses to the questions listed in the assignment.7. Bring any artifacts and four copies of your

professional practice piece to class on Thursday, November 10.

10

Multiplication

Use blocks to show the meaning of2 x 4

in as many different ways as you can.

• Explain what makes them different from one another.

• In what ways are they also the same?

11

Two Different Interpretations of the Meaning of Multiplication

• Repeated addition: a x b is the result of adding b together a times

• Area: a x b gives the area (square units) of a rectangle of width a (linear units) and length b (linear units)

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2 x 4 = 4 + 4

Repeated Addition

• Counting• Usually used to introduce multiplication as a short cut for adding

• Convention in U.S.: in a x b, a represents “number of groups” and b represents “size of group”

13

aba x bab

2 x 4

Area

• Continuous model

• Convention in U.S.: in a x b, a represents width and b represents length

• Two kinds of measure:– a and b are lengths (linear units)

– a x b is an area (units of area)

14

A Third Interpretation:Cartesian Product

• Counting

• Each member from the first set is matched with each member from the second set; this mapping produces a set of ordered pairs

• Often taught, without naming it, in elementary school – (combination problems

such as “2 pairs of pants, 4 shirts, how many outfits”)

8 ordered pairsBACartesian product (of two sets)

A8 ordered pairsCartesian product (of two sets)B

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Modeling Multiplication Computation

• Calculate the answer

• Show as an area with base ten blocks

• Explain the correspondence between the area and the written procedure

3525x

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Modeling Multiplication as Area

35

25

17

Multiplication of Binomials

35 x 25 = (5 + 30) x (5 + 20)= 5x5 + 5x20 + 30x5 + 30x20= 25 + 100 + 150 + 600

(partial products)= 125 + 750 = 875

35 x 25 = 35 x (5 + 20)= 35 x 5 + 35 x 20= 175 + 700 (standard

algorithm)= 875

18

Three Common Student Errors

• What is the error?• What produces this error, and what does that show about the student’s understanding?

3525x175702453525x255801055211 2 3

3525x6258510560025

19

Reflection

Did you have any new insights about multiplication, or teaching and learning multiplication, from our work together today?

20

Assignments

• Student thinking interview– Conduct interview (if you haven’t already)

– Analyze data & prepare assertion sketch (see sample)

– Bring artifacts and assertion sketch next week

• Do your end-of-class check on the date scheduled

• Readings and other tasks