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) b)

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

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 Inequality in School Mathematics Persistent achievement gaps: race, social class Gatekeeping role of mathematics Unequal access to opportunity Unequal participation and retention

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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 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 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