1

Overview

• Comments, questions• Continue work with base ten blocks

(decimals –– ordering and computation)

• Overview of Part 3 of our course• Learning to remediate student

difficulties (computational algorithms)• Assignment and wrap up

2

Ordering Decimals

Put the following strings of decimals in order however you usually do it:

a) 5.3 5.03 0.53 0.8 0.08 0.4 0.40

b) 0.4 1.4 .55 .0098 15 .4 .04 .40

3

What Makes Ordering Decimals

Difficult• The length of a number no longer a clue• Some numbers “look” large• Multiple representations for same number: .4, .40,

and 0.4• Lack of understanding of what the numbers mean

(how we read decimals)• Money overly supports “correct” answers with

tenths and hundredths• They think some digits are larger (e.g., .09 would

look larger than .3 because 9 is larger than 3)• 1.03 might look smaller than .53 because student

gets focused on decimals

4

Modeling Computation of Decimals with Base Ten

Blocks

2.46

1.57+

31.9

18.47+

33.02

19.71-

10

4.63-

5

Issues to Attend To

• Choice of unit with base ten blocks• Language of decimals, materials,

and operations• Correspondence between model

and written algorithm

6

Question

Is it necessary to line up the decimal places when adding or subtracting decimal numbers?

7

From Quiz Last Week

8. Pins cost 3¢ and stickers cost 2¢. If you have 10¢ and you want to spend it all, what are all your choices of what you can buy? Prove that you have found all the choices.

9. What is the definition of an ODD number?

10. What is the definition of an EVEN number?

8

Analysis of student work

• Purpose of the question and core mathematical ideas addressed

• What are examples of “good” answers at the third grade level

• What do you make of the answers given by Mick, Shekira, & Bernadette?

9

Part 3 of Our Course

• Learning to design, teach, and improve lessons

• Attending actively to equity

10

Focus for Learning to Teach

• Designing lessons (structure of lessons)

• “Types” of lessons• Assessing students’ learning • Remediating• Extending• Out-of-school assignments

11

What Do We Mean by “Equity”?

Differences among students are a given These differences shape teaching and learning Some group differences associated with

inequitable access and outcomes Equity means that school outcomes are not

associated predictably by race, class, gender (RAND, 2003)

Concerns about changing these patterns concerns for “equity” in teaching practice

12

The Endemic Problem of Inequalityin School Mathematics

• Persistent achievement gaps: race, social class

• Gatekeeping role of mathematics• Unequal access to opportunity• Unequal participation and

retention

13

teachersstudentsstudentscontent

14

teachersstudentsstudentscontentenvironments

15

Where Teachers Can Try to Gain Some Leverage

(in the face of “defaults” that tend to create inequities in opportunity and achievement)

• Selection of mathematical tasks: consider assumptions, contexts, scaffolding

• Work on becoming more self-aware of how our identities and experiences as teachers shape our interactions with students

• Unpacking and scaffolding important mathematical practices

• Knowing and using more about students’ out-of-school experiences

16

Working to Create Equitable Practice

Inequality 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 the intersection of concerns for equity and the work of teaching.

17

Building a Professional Culture of Inquiry and Sensitivity

for Developing Equitable Practice

• Being able to “revise” one’s thinking• Finding “hearable” and respectful ways to

question and respond to others’ ideas• Considering what each of us brings in relation

to these issues, and how we can offer them best to the group

• Learning to consider limits of our experience

18

teachersstudentsstudentscontentenvironments

mathematical tasks

unpacking and scaffolding mathematical practices

becoming more conscious of self as cultural being

learning and using more about students’ out-of-school experiences

19

Common Student Errors

18

29

37

74

+

1

62

28-

46

905

147-

8 1

748

463 - 35 = 113

• What is the conceptual difficulty?• Design a scaffolded approach to remediating

the procedure and its meaning (5 minutes)• Present to one other student, others observe• Comment, feedback

20

Remediating What does NOT work?• Repeating the same things over

again, slower, more loudly• Re-teaching everything• Teaching that “coerces” or forces

students to get the “right answer” • Teacher does the problem for the

student• Teacher refers the student to

easier problems that they could handle and cuts off work returning student to work on basic concepts

• Teacher gives students more problems, thinking practice is the issue

What DOES work?• Identifying carefully where the problem(s)

lie(s) (Teacher works to locate)• Teacher helps student to initially “see” the

problem • Teacher draws upon/helps to develop

student estimation skill • Using manipulatives or other representations

to focus on the meaning and the procedure• Sharing the talk with the student, scaffolding• Offering a similar example to try• Teacher reduces complexity in someway and

then encourages connection of that work with original problem

• Teacher invites another student to explain how he/she understood the problem and mediates the interchange to highlight productive ideas or contrasting ones

21

Wrap Up

• Assignments• Project 2• Ways to extend or deepen your

work