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Support Math Reasoning By Linking Arithmetic to Algebra
Virginia Bastable
GSDMC 2013
Developing Mathematical Ideas Professional Development for teachers
Investigations in Number, Data, and Space 2008-for students
Connecting Arithmetic to Algebra—book and on-line course
With: Deborah Schifter of EDC and Susan Jo Russell of TERC
Partially funded by the National Science Foundation
Early algebra
Generalized arithmetic
articulating, representing, and justifying general claims in the context of work on number and operations
Patterns, functions, and changerepeating patterns, number sequencesrepresenting and describing contexts of
covariationusing tables, graphs, symbolic notation
First Grade Ana
4 + 4 = 8
4 + 5 = 9
6 + 6 = 12
6 + 5 = 11
Grade 1 Video clip
Teacher: Who knows 9 + 9?
Class: Oh. My gosh.
Amalia: 18, because if it were 10+ 9 I would think it was 19… But it is 9 + 9.
Manuel: Its 18. If you add two more it would be 20. It would be two less off. .. It would be one less off and it would be 19 and then another 1 less off is 18.
Grade 1 Video Clip
Teacher 9 + 9 = 18
9 + 8 = ?
Coleman: 17.
I know that 9 + 9 is 18 and if you minus 1 from 18 you will be at 17.
Student: You’re right
Coleman: 9 is one more than 8. So this must be 1 less than 18. 17.
Identifying potential general claims
Key aspects of integrating early algebra into arithmetic instruction
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 are the basis for proof
The operations themselves become objects of study
Construct viable arguments and critique the reasoning of others.
They justify their conclusions, communicate them to others, and respond to the arguments of others.
Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.
A generalization is…
A claim you can make about the way numbers and operations work.
A claim is general if it applies to a range of numbers, for instance, all whole numbers or all positive numbers.
What behavior of addition is being revealed in these examples?
5 + 5 = 10 25 + 30 = 55
5 + 6 = 11 25 + 31 = 56
Use the first problem to help you solve the second:
15 + 15 = 30 75 + 25 = 100
15 + 17 = _____ 75 + 28 = _____
Grade 1: Adding 1 to an addend
1. On Saturday, there were 5 girls and 5 boys in the pool. How many children were in the pool?
2. On Sunday, there were 5 girls and 6 boys in the pool. Can you use the answer from the other story to help you figure out how many children are in the pool on Sunday?
Grade 1: Adding 1 to an addend
Saturday Sunday
Girls
Boys
boy
Grade 3: Adding 1 to an addend
Grade 3: Adding 1 to an addend
Megan: The picture could be used for ANY numbers, not just 3 and 4. I could have started with anything in one hand, and then anything else in the other hand, and put them together. If I got 1 more thing in either hand, the total would always only go up by 1.
Articulations using algebraic symbols
If a + b = c then a + ( b + 1) = c + 1
a + ( b + 1) = (a + b) + 1
Initial student reactions19 + 7 = 20 + 6
Initial student reactions19 + 7 = 20 + 6
The answer is to be placed here.
You can’t have more than one number on the right.
You can’t have that many plus signs
The teacher made an error. I see how she tried to fix it, but you can’t do that.
Explanations for why 19 + 6 = 20 + 5
Take 1 off the 6 and put it on the 19. So it is 20 + 5 = 20 + 5
Do the computation. Since both are 25 they are equal.
Use a story to show they are the same without needing to know the answer
Using a story situation as a tool for explaining why an equality holds---a third grade student
If I had some candy and I shared with my friend, but then I decided to share more with her, we would still have the same amount even though I’m sharing more with her. If I had 20 pieces and my friend had 5 pieces the sum would be 25. But then if I gave her another one of my pieces so she has 6 we would still have 25 together. So it doesn’t matter how we share the candy the total will always be the same. Unless we go get more or we eat some of it.
Symbolic Interpretations for19 + 6 = 20 + 5
a + b = ( a +1) + ( b – 1)
a + (1 + b) = (a + 1) + b
Early algebra
Notice a regularity about an operation
Articulate the generalization
Prove why the claim is true
Compare behavior of the operations
2 + 23 = 23 + 2
7 – 4 and 4 - 7
What about subtraction?
Will the same general claim be true or will it need to be modified?
Try some examples—make representations—articulate the claim.
Does it matter if you add to the subtrahend or the minuend?
Ways of Knowing
Accepting on authority
Trying examples
Applying mathematical reasoning based on a visual representation or story context
Proving using algebraic notation and the laws of arithmetic
What does it take to develop this practice?
Mathematics questions or tasks that are challenging enough so it is useful to take in more ideas than just one’s own.
Specific pedagogical moves on the part of the teacher to help students learn how to listen/critique the thinking of others.
Regular and consistent opportunities to develop and build this habit
Teacher Practices
Use of routines to provide a forum for students to notice, state, and test general claims.
Development of representations such as actions with cubes, number lines, arrays, and story contexts as tools for reasoning about operations.
Teacher Practices
Revisiting general claims and the arguments developed for them when the number system expands or as a different operation is explored.
Exploring the connection between the general claim and computational strategies
Teacher Moves to supportlistening
Teachers enact listening in their work with students
Teachers use techniques during discussions to help students learn to listen and to continue listening behaviors.
Turn and talk
Practice explaining in small groups
Paraphrasing
Index card to write responses first
Representations can make an idea tangible and visible
Looking across representations
Sentence Starters
I have a connection with what ________ just said…..
I think you are saying_________ (repeat in your own words)
I agree with ________ because …
I’d like to add to what ________ just said….
Conclusion
Expand student thinking beyond finding a pattern to seeing how the patterns works
Help your students to focus on the meaning of the operations
Use representations to make ideas visible
Support math discussion among students
Look for connections between computational strategies and larger principles of mathematics
Next steps
How can you modify math work you already do with students to incorporate the kind of thinking we have been describing in this presentation?
Speaker Evaluation1 2 3 4 5
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Text your message to this Phone Number: 37607
Strongly AgreeDisagree Neutral Agree
___ ___ ___ ___________ ”
Speaker was well-prepared
and knowledgeable
Speaker was engaging and an effective presenter
Session matched title
and description in program book
Other comments,
suggestions, or feedback
Example: 545 Great session! ”
“30995
“30995
For more information about the Mathematics Leadership Program
Contact Virginia Bastable at
Check out our website at
www.mathematics leadership/org
