Connecting Arithmetic to Algebra

Connecting Arithmetic to Algebra
An On-line Course for Teachers Virginia Bastable Deborah Schifter TDG 2011 Susan Jo Russell intro—slides <10 to 15 minutes>

Foundations of Algebra project
A collaboration of the Education Research Collaborative at TERC, the Education Development Center (EDC), and SummerMath for Teachers (Mt. Holyoke College), and 25 teachers Funded in part by the National Science Foundation 5-year project First three years--working closely with teachers and classrooms to develop these ideas; 4th year developed and piloted course, now in 5th year; doing a field test. 5 sections with different pairs of facilitators—helping to guide the shape of the facilitator support materials

Connecting Arithmetic and Algebra
Today’s Session Goals and structure of the on-line course What do CAA participants do? What do CAA course facilitators do? What are we learning from this experience? In this session you will have the opportunity to experience the course first from a participant’s point of view and then from the point of view of a facilitator. We will conclude with comments about what we are learning from the experience.

Integrating algebraic thinking into arithmetic instruction involves:
Investigating, describing, and justifying general claims about how an operation behaves A shift in focus from solving individual problems to looking for regularities and patterns across problems Representations of the operations as the basis for proof The operations themselves become objects of study For example, when students are learning their single-digit addition combinations, they notice relationships that help them learn them, for example that they can solve sums involving 9, like 9 + 4, by thinking of an equivalent 10+ sum; = This work not only links arithmetic to the algebra students study later, but strengthens students’ grasp of their core work in number and operations –refer to Deborah’s plenary as additional examples You’ll get a glimpse of these key aspects in a warm-up activity that Virginia will now explain; then we’ll come back later to illustrate some of these aspects in more depth

What do teachers need to learn?
the mathematics content how to recognize opportunities instructional strategies how a range of students can engage with this content This area of content is new for most teachers. What is it they need to learn in order to engage their students in articulating, representing, and justifying general claims about the operations? Math content: where are the key generalizations that underlie core instruction in number and operations? What does it mean to prove a general claim? Recognize opportunities: notice what students are noticing and make noticing patterns and regularities a habit in the classroom. Learn instructional strategies: how to integrate this content into core math instruction on a regular basis. Engage a range of students: Develop experience and images of what students actually do and can achieve, including students who often struggle with math in relation to their peers and students who tend to excel in classroom math. These are the 4 things this course is designed to do.

Key Structures of CAA A year-long course in three 6-week parts
Every participant is part of a local team; each section includes approximately 20 participants Assignments are posted on a course web board; participants respond to one another’s postings Each 6-week part includes two 2-hour webinars to allow for “live” discussions and interactions Each 6-week part includes 3 Student Thinking assignments; participants receive personal responses from facilitators on these. In the pilot we have 65 people enrolled in 3 sections. They can be anywhere in the country. Scheduled different sections of webinars in both after-school times and evenings. There are no geographic boundaries. This year we have five sections and 90 participants. We work to create a sense of community in each section—through teams, web board response groups which cut across teams, and the webinar.

CAA Book Chapters and Course Sections
Part Two 5. Developing Mathematical Arguments Part I 6. Focus on the Range of Learners 7. Learning Algebraic notation Part One 1. Generalizing in arithmetic--noticing 2. Generalizing in arithmetic, getting started 3. Generalizing in Arithmetic with the range of learners 4. Articulating general claims Part Three 8. Developing Mathematical Arguments Part II 9. Looking ahead to middle grades 10. Building across the school year Highlight: focus on general claims; representation; articulating those claims; proof (2 chapters) Focus on the range of students in the class Connection to middle grades and How this work builds across a school year--we’re not talking about single lessons in which all this happens Each chapter is built around classroom episodes from the classrooms of teachers with whom we worked for the first 3 years.

What participants do: Read and respond to each chapter of the course text Respond to each other on the web board Do math activities with their team Carry out lessons with their class and write about them (“student thinking assignments”) Participate in 6 2-hour webinars <Previous should be 15 min> So we are going to give you a chance to experience some of the course content. Reading 1 Identify points in the reading that help you make connections to your own students. What teacher moves or approaches do you want to incorporate in your own practice?

Focus Questions Reading One
Identify passages that help you make connections to your own students. Identify passages that illustrate a specific question or action of the teacher that interests you. What was the impact of this teacher move on the students’ thinking? Read, marking points of interest and teacher moves. <10 minutes> Alert them to length of first reading-- Some time for small group discussion at your tables.—Make sure each person has a chance to share their points and teacher moves. <5 minutes> Whole group discussion <10 minutes>

What facilitators do: Read and take notes on web board responses
Adapt webinar plans to react to participants’ postings Co-teach the webinars Respond to participants’ student thinking assignments individually

Focus Questions Reading Two
What do you learn about the participants by reading their responses? What ideas might you want to bring out at the next webinar as a result? Read, take notes on questions. Small group <5 minutes> Whole group <5 minutes>

Is this number sentence true?
= Explain how you know it is true or not without calculating What ways might you represent this instance—story context, diagrams, using cubes Work on representations at tables < 5 minutes> Whole group < 5 minutes> 12

Is this number sentence true?
= The routine can provide a way to help students see the general claim is about an operation and not just about numbers,--by posing a similar question with a different operation. Will what we just found out work here? If not, what will work to create equivalent expressions in subtraction? and how can you prove it? <Work time <5 minutes> Whole group <5 minutes> 13

What general claim is suggested by this set of equations?
6 x 8 = 12 x x 64 = 300 x x 8 = 70 x 4 Develop at least two representations (story situations, diagrams, etc.) to illustrate this claim. Can you use your representations to talk about the claim without referring to specific numbers. We have looked at ways to make equivalent expressions in addition and also in subtraction. Now consider what happens for multiplication. Work time <5 minutes.> Whole group < 5 minutes> 14

Analyzing Web board Math Responses
Read the two responses to this assignment (Reading Three) What do you learn about the math understandings of the participants? What else do you want them to learn? . Read and table discussion <10 minutes> Whole group <5 minutes> 15

Examples of student arguments to justify this claim
1. What does each argument show that the student understands about proving the general claim? 2. What more would the student need to do to move towards proving this claim? An excerpt from a webinar. Some teams meet together to participate in the webinars. Some participants are alone.

Student argument #1 I figured out that 2 times 6 equals 4 times 3, and also 8 times 10 equals 4 times 20. So it works. Whole group <2 to 3 minutes>

Student arguments #2 and #3
I did a story context. I have 2 stacks of books, and each one has 6 books. That’s 12 books. Then I have 4 stacks of books, and each one only has 3 books. That’s 12, too. So they’re the same. Argument #3. I have 2 stacks of books, and each one has 6 books. But the stacks were too heavy to carry, so I put each stack in half. Now there are 4 stacks and each has 3 books. So when I doubled the number of stacks, there was only half of the books in a stack than there was before. Whole group <5 minutes>

Student arguments #4 and #5
Argument #4: Argument #5: I cut the 2 by 6 in half, and I put one piece underneath. It’s half across the top, but now it’s twice as long. It’s all the same stuff I started with, like if this was a carpet and I cut it and moved it around. Whole group < 5 minutes> See this is a 2 by 6, and this is a 4 by 3, and they both have 12.

Representation-based proof
The meaning of the operation(s) involved in the claim is represented in diagrams, manipulatives, or story contexts. The representation can accommodate a class of instances (for example, all whole numbers). The conclusion of the claim follows from the structure of the representation; that is, the representation shows why the statement must be true. <probably skip this slide>

Would any of these arguments hold if the numbers under consideration were not whole?
<no time for discussion—but pose this question as we do in the webinar something that will be coming up in future sessions of the course.>

What participants do: Read and respond to each chapter of the course text Respond to each other on the web board Do math activities with their team Carry out lessons with their class and write about them (“student thinking assignments”) Participate in 6 2-hour webinars In this session we have had experience with all of these activities excerpt the student thinking assignments- very significant part of the course work. Described in more detail in another TDG session. <10 min for summary—slides 22 to 29>

What facilitators do: Read and take notes on web board responses
Adapt webinar plans to react to participants’ postings Teach the webinars Respond to participants’ student thinking assignments individually Facilitator’s’ responses to the student thinking assignments provide means for challenging participants’ mathematics, helping them re-interpret the thinking of their own students and for offering suggestions for instructional practice.

Early algebra Notice a regularity about an operation
Articulate the generalization Prove why the claim is true In summary

What we are finding Elementary grade students are interested in examining generalizations about the behavior of the operations. Such discussions engage a range of students and support the development of computational fluency. Visual representations and story contexts provide a mechanism for proof accessible to elementary grade students.

Implications Elementary school: Studying the behavior of the operations supports the development of computational fluency. Middle school: “The kids who have the deepest trouble with middle school math are those without a clear and rich set of models for what multiplication is and how it is different from addition.”

Summary On-line/off line Asynchronous/synchronous interaction
Individual/team/webinar groupings Responses from individuals/teams/facilitators Alternate course work focused on their own math and implementation with students

What’s happened so far Participants are actively engaged in the mathematics content (in teams, on web board) Tape recording and analyzing class sessions is powerful (student thinking assignments) A focus on general claims is being integrated into instruction Students are engaged in significant mathematical thinking

Data we’re collecting:
Teacher assessments Teacher evaluations Student assessments Next year we will have more to report about what we’ve learned from the assessments.

email vbastabl@mtholyoke.edu Include CAAFall11 in the subject line
To receive information regarding CAA options for school year: Include CAAFall11 in the subject line

Working on math in teams
Adding 1 to a factor in a multiplication expression “I found that after writing the story and drawing the boxes of erasers, how helpful the story context could be for some of my students. The story context may make the statement easier to relate to for the children that struggle with relating to numbers.” This participant indicated that she didn’t usually use story contexts in her own math work, but after seeing others in the team use them she created one for this problem. 2 boxes with 3 erasers in each box, 3 boxes with 3 erasers in each box, 4 boxes with 3 erasers in each box. Each time you add 1 box, you add 3 erasers.

Adding 1 to a factor in a multiplication expression “Each time we solve a problem or examine a statement as a group, I am amazed at the different ways we all think about the problem. I always solve the problem and think about it before meeting with our group and then I always walk away with a new way of thinking about solving or representing the problem. It reminds me how important it is to have my students share their different strategies with their peers.” Expanding their own math, considering other participants as colleagues to learn from, and making connections between their own process of learning and how they want to organize their classroom instruction.

Adding 1 to a factor in a multiplication expression General claim #1: If you add n to factor b in a multiplication expression a x b, you add a x n to the result. I have 3 baskets with 4 apples in each basket. If I add an apple to each basket I added 3 more apples, one for each basket. General claim #2: If you add n to factor a in a multiplication expression a x b, you add b x n to the result. I have 3 baskets with 4 apples in each basket. If I fill another basket with the same number of apples, I have added 4 more apples. This participant shared that she and her partner had determined two general claims relating to this situation.

Adding 1 to a factor in a multiplication expression “What was so interesting was that my partner changed the first factor in the expression and I changed the second factor in the expression. This is how we discovered that we needed to have two general claims that could explain both situations.” Working in teams on the math provides the participants the opportunity to compare their thinking with that of others and to expand their mathematical ideas. In addition to the team work on the math assignments, they also post and respond to colleagues math work on the web board.

Grade 1 Is this number sentence true? 3 + 7 = 7 + 3

Grade 1 Are these the same amount?

Grade 1 My class and I continue to become more and more comfortable with asking “why”, using models to represent ideas, and pushing for articulation. These are connected processes that are hard to look at separately. It is helpful to analyze transcripts of classroom conversation to practice understanding students’ ideas and how to respond to them

Grade 1 As we move forward with the version of this routine, “Are These the Same Amount?”, I will be interested in hearing more about what it sounds like for first graders to articulate generalizations and what it sounds like if and when teachers press them to leave the specific numbers behind. Perhaps we can also revisit “Is this Number Sentence True?” and check for development in students’ ideas about the meaning of the equal sign.

Grade 3 6 x 2 = 3 x 4 An example from a third grade class. They were looking at doubling/halving. The teacher has asked students to create a representation or story context to show what is happening here--not just that these two expressions are equivalent by doing the multiplication, but why they are equivalent As students are working, she stops in to talk to Chad and Martin--

6 x 2 = 3 x 4 Thad: I have 6 apples and my mom gave me 2 more. My sister had 3 apples and my mom gave her 4 more. We have the same amount of apples. Teacher: So, Thad, I have 6 apples and I got 2 more, how many do I have? Thad: 8 Teacher: And if sister has 3 and gets 4 more how many does she have? Thad: 7. Oh I didn’t do it right. I was adding. Pause after Thad’s first utterance; what is going on here--what is Thad modeling? Thad is one of the “good math” students. Interesting to think about why he would make this kind of error. Students who are good at solving problems but aren’t used to noticing behaviors of the operations.

6 x 2 = 3 x 4 Teacher: And where is this story problem showing that doubling or halving that we really wanted to model? Thad: It doesn’t. Teacher: So let’s see if we can change your story to show both multiplication and the doubling and halving. We can start with the same 6 apples. But how can we show 6 x 2? Martin (Thad’s partner who is listening in): Each apple has two worms!

6 x 2 = 3 x 4 I have 6 apples with 2 worms in each apple.
If my sister only has 3 apples and the worms crawl over to her apples, they will have to double up, so there will be 4 worms in each apple. Now they have a story context that justifies the claim--for these particular values--that if you halve the number of apples from 6 to 3, but want to end up with the same product--in this case, the number of worms--and you want to maintain equal groups of worms, the worms have to “double up.”

The teacher reflects: I walked away from this class session knowing so much about what my students were thinking. I saw a common mistake when writing story problems for multiplication as Thad wrote an addition problem. I think this could be a great story to return to later when I want to compare the behavior of addition and multiplication.

Grade 4 Write a word problem, draw a picture, or another kind of representation to convince or prove to another class of fourth graders that = A technique used to make addition problems easier is based on an underlying general claim about addition. Students often learn tricks like this as rules, which they then often misapply because they don’t have a firm grasp of the reasoning that underlies the rule. Proving this particular case is a first step towards making and proving a general claim. Here is one story context the students come up with . . .

Prove that 237 + 195 = 232 + 200 Two cookie jars
237 cookies cookies

Prove that 237 + 195 = 232 + 200 move 5 cookies
The students argued that the total number of cookies hadn’t changed, but had only been rearranged. You can see how this context could be used to make an argument that this claim holds for any positive whole number. Starting with this image, the discussion continued as someone suggested that the whole, the sum, could be thought of as one thing, like a cake . . . 237 cookies cookies

Keith: It’s like x + y - y = x. Derrick: I don’t understand.
Sam: If you have a cake and you cut it into two pieces even if one of the pieces is bigger than the other and then you put the two pieces together you still have one cake. Teacher: Oh, what happens if one person gives some of their piece to the other person? Sam: Then you are just taking a little bit away from one and giving it to the other but you still have one whole cake. Keith: It’s like x + y - y = x. Derrick: I don’t understand. Teacher: Can you help Derrick understand by explaining what you said in numbers and words? Keith: If you have any number plus any number, to make an equivalent expression you add an amount to the first any number, then you need to take that same amount away from the second any number so they stay the same. These are just two examples of the kinds of conversations that we are saying happen in many of the classrooms of partcipants In the course. Like Megan (with her rep of hands holding objects), the class has moved from particular values to language and images that justify their claim for “any number.” (they are probably thinking about positive whole numbers--as they continue thinking about this claim, as fourth graders, they could also expand the class of numbers to include fractions. This is another topic covered in the course as are issues about notation--when it’s useful for elementary students, when it’s not useful, what issues come up.

Connecting Arithmetic to Algebra

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