1. 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. 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. 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. 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. 5 Analyzing a Textbook Lesson

6. 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. 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. 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. 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. 10 Multiplication Use blocks to show the meaning of 2 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. 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)

12. 12 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. 13 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. 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”)

15. 15 Modeling Multiplication Computation Calculate the answer Show as an area with base ten blocks Explain the correspondence between the area and the written procedure

16. 16 Modeling Multiplication as Area

17. 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. 18 Three Common Student Errors What is the error? What produces this error, and what does that show about the student’s understanding? 123

19. 19 Reflection Did you have any new insights about multiplication, or teaching and learning multiplication, from our work together today?

20. 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