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Algebrafying the Elementary Mathematics Experience Part II: Transforming Practice on a District-wide Scale Maria L. BlantonJames J. Kaput University of Massachusetts Dartmouth USA The research reported here was supported in part by a grant from the U.S. Department of Education, Office of Educational Research and Improvement, to the National Center for Improving Student Learning and Achievement in Mathematics and Science (R305A60007-98). The opinions expressed herein do not necessarily reflect the position, policy, or endorsement of the supporting agencies.

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“Teacher training and not the abilities of children will, I believe, be the real challenge of early algebra.” Lesley Lee ICMI-Algebra Proceedings

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BIG QUESTIONS (with preliminary results) What are characteristics of an “algebrafied” practice that indicate generativity and self-sustainability? How can we quantify the robustness of such a practice? What can we say about possible trajectories in teachers’ development? What aspects of our professional development contribute to “algebrafying” teachers’ practice? (Likewise, what are the impediments to district change and how can they be addressed?) How is an “algebrafied” practice linked to student achievement? What counts as evidence?

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CHARACTERISTICS OF AN ‘ALGEBRAFIED’ PRACTICE (1) the seamless and spontaneous integration of algebraic conversations in the classroom; (2) the spiraling of algebraic themes over significant periods of time; (3) the integration of independently valid algebraic processes in a single mathematical task; (4) an ability to generalize an activity (what we describe as “activity engineering”); (5) embedding algebraic tasks in other subject areas; and (6) collegial sharing (in some leadership capacity) with peers in formal or informal settings; actively contributing to the development of a school community/culture of learning mathematics.

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Seamless and spontaneous integration of algebraic conversations requires the teacher to ‘algebrafy’ an arithmetic conversation as it occurs. Class Discussion about a Homework Problem. Teacher: Six plus eight, would your answer be odd or even? Student: Because 6 is even and 8 is even, so even and even is even. Teacher: Suppose you had 5+7. Would your answer be odd or even? Student: Even. Teacher: How did you get that? Student: I added 5 and 7 and then I looked over there (at the chart on the wall with containing even and odd numbers) and saw that it was even. Teacher: What about 45675 + 23675? Odd or even? Student: Even Teacher: 45678 + 85631? Odd or even? Student: Odd. Teacher: Why? Student: Because 8 and 1 is even and odd and even and odd is odd.

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The integration of independently valid algebraic processes in a single mathematical task. In this, the teacher combines otherwise stand-alone algebraic tasks in ways that alter the complexity of the original task. From earlier data (Year 2): Task 1: Students used Base-10 blocks to solve missing number sentences. (E.g., ‘14 = 6 + n’) Task 2: Using pattern finding to solve missing number sentences. Jan extended the sentence ‘14 = 6 + n’ into a family of number sentences: 140 = 60 + n 1400 = 600 + n 14,000 = 6,000 + n Students looked for patterns in their solutions to solve other problems in this family. NOTE: Jan spontaneously integrated these two algebraic processes.

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Generalizing an activity implies the teacher does not depend solely on activities used in our sessions, but has an explicit ability to cull other resources and adapt them to a particular grade level. Task: Handshake Activity (If the people in a group shake hands with each other once, how many handshakes will there be all together? What if another person joins the group? A second person? A third person? How many handshakes would there be if there were 20 people in the group?) Extended Tasks: Twelve Days of Christmas (How many gifts does the person get on the first day? Second day? Third day? Twelfth day?What if there were 26 days in the song? Big Lips and Hairy Arms - Investigating sound; Telephone Problem All tasks have a common solution that requires finding a pattern best solved by leaving the amount of handshakes, gifts, or phone calls as an indicated sum (i.e., ‘1+2+3+4+5+6+…+n’).

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Embedding algebraic tasks in other subject areas (e.g., science, literacy) Recent reflection from Jan (2001): We are studying sound and made telephones out of cups. I took advantage of reviewing the principles behind the telephone. I then read them a book called Big Lips and Hairy Arms. This book was about telephone calls. I then gave the class the telephone problem, which is exactly like the handshake problem. They had done the handshake problem a couple of weeks ago. I was hoping they remembered the similarity, but not many did. After talking with them, some of them started to use a T chart, others were trying to write names and whom they called. All in all, only a few were able to get it on their own and not very neatly done. I was disappointed. They, however, were able to add the numbers by grouping them. (That was good!) I gave this problem instead of the Handshake problem to the teachers. The way the problem was worded, one teacher thought that her class would think 1 phone call was made. The question asked, "What was the fewest number of telephone calls made?" She had a point. I am going to try and rewrite the problem.

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What does the progression of tasks used by Jan and how she uses them in instruction indicate about her (mathematical) development, in particular, or a possible trajectory for a teacher’s development in general? Is there an optimal way to sequence tasks in professional development?

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Original types of tasks focused on missing number problems - which possibly mirrors Jan’s conception of ‘algebra’ (and thus her initial conception of algebraic thinking) as solving equations for unknowns. (“Equality” problems were a good point of entry.) 1 x b = b5 + 8 = __ + 9 ∆ + ∆ = 6 9 + ∆ = 12 ∆ = ___ simple in/out charts. Later focused more on Generating patterns in a set of data, sometimes generated geometrically Looking for and describing symbolically relationships between data sets Does this suggest a way to sequence tasks in instruction?

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in/out chart 9/99 –I gave the class an In-Out chart. First we started by having them make their own charts. We played “Guess My Rule”. We discussed if the ‘in’ number was greater than the ‘out’ number. Then we had to subtract to get the answer. If the ‘in’ number was less than the ‘out’ number, we had to add. These charts went right along with finding the missing addends. It was interesting to hear how they arrived at the their answers. Stephanie said that she added 3 to 10 to get 13, then she thought she might have subtracted 3 from 13. Then she realized it didn’t matter. Ashley added that she counted from 10 to 13 and then said that that is really subtracting 10 from 13. The more these children are talking and explaining how they are getting their answers, the more they are willing to share their answers. This is good because it lets us know how they are thinking.

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in/out chart 10/01 (same grade level, same point in AY) Because the MCAS test had an in-out chart using letters, I decided that I needed to add a letter to the in and out and have the students write the equations using letters. I came across this idea to get them started. Flash cards. One side had x = 5, the other side had y = 7. I asked how I got y. They answered + 2. I tried another one (x = 7) and asked them to predict what was n the other side. They said y = 9. I asked them to write out the rule after doing several cards the same way. They wrote x + 2 = y. I then turned the cards over and we worked with the y = side. I was surprised that they picked it up right away. y - 2 = x. Then I had another set. x = 1, y = 3. They said + 2 again. The next card was x = 2 I asked them to predict, they said 4. WRONG. y = 5. Of course they said + 3, but that wouldn't work for all problems. As it ended up the rule was doubles + 1. Monnize figured it out after a few more problems. Now we had to find a way to write this. Monnize wanted to do x + x +1. Good deal. Then Candace said that we could just put that we need 2 x's then add 1. So, it came out 2x + 1 = y, I was impressed. Then for the fun of it, I tried turning it over to reverse the rule. That was a lot more difficult. We did end up with y-1 then 1/2 of the number. The class decided this themselves after much debate, but they wrote up the rule.

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Embedded the problem in a type of game using flash cards Created a classroom environment in which students naturally wrote rules for more complex relationships in increasingly formal ways (e.g., ‘2x+1’ vs “add 3 every time”) Explicitly focused on the relationship between two variables Increased the level of complexity by exploring inverse relationships between two variables Developed and used this activity based on a larger curriculum need (MCAS) Wrote her reflection in a more mathematically formalized way Focus is less on the organizational aspects of the problem (9/99: “First we started by having them make their own chart” ) The In/Out chart activity shows increased complexity in how Jan uses the task in instruction. In the later version (10/01) she

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How can Jan’s practice (and that of others like her) be replicated across the district? That is, what aspects of our professional development contribute to teachers’ development? Integrating our work with other district initiatives Professional development activities are not ‘add-ons’, but constitute a habit of mind whereby existing curricula can be transformed Teachers bring in and adapt their own resources Long-term sustained interaction with teachers Authentic math experiences that model what teachers can do in their classrooms Linking teacher professional development with professional development of principals and administrators in order to build school and district support for teachers

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