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Fostering Algebraic Reasoning (K-12)

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Graham Fletcher

on 17 March 2015

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Transcript of Fostering Algebraic Reasoning (K-12)

In a particular condominium community 2/3 of all of the men are married to 3/5 of all of the women.
What fraction of the entire condominium community are married?
#1 Algebrafy existing Instructional Materials
not adding to the curriculum…we’re just changing the way we engage students in tasks
transform from arithmetic problems to opportunities for conjecturing, generalizing, and justifying mathematical relationships... make the purpose of task no longer just about finding answers
#2 Finding and supporting students’ algebraic thinking
Seamless and spontaneous integration of algebraic conversations whenever possible
#3 Creating a classroom culture that promotes algebraic reasoning
instrumental understanding
relational understanding
Fostering Algebraic Reasoning (K-12)
GRESA October 2, 2012
how we might see algebra
Algebra is an isolated strand of mathematics
Algebra starts in middle school
Arithmetic is required to perform algebra
What does algebraic thinking really mean?
How can we "algebrafy" our classrooms?
The art of “Algebrafication” involves three types of teacher-based classroom changes:

algebrafying instructional materials
finding and supporting students’ algebraic thinking
creating classroom culture and teaching practices that promote algebraic thinking
algebraic thinking is a process in which students build general mathematical relationships and express these relationships in increasingly sophisticated ways
A mile deep and an inch wide
Generalizing an activity
The Handshake Problem
Arithmetic task:
How many handshakes will there be if each person in your group shakes the hand of every person once?
Arithmetic task that fosters algebraic thinking:
How many handshakes would there be if each person in your group shakes the hand of every person one time? What if one more person joins your group? What if there were 20 people in the group? What if there were 100 people? How did you get your solution? How do you know it works?
The integration multiple algebraic ideas in a single mathematical task
Combine otherwise stand-alone algebraic tasks in ways that alter the complexity of the original task

Task 1:
Students use cubes to solve missing number sentences.
(E.g., ‘14 = 6 + n’)

Task 2:
Using pattern finding to solve missing number sentences.
Extended the sentence ‘14 = 6 + n’ into a family of number sentences while developing an understanding and reasoning in powers of 10:
140 = 60 + n
1400 = 600 + n
14,000 = 6,000 + n
Students look for patterns in their solutions to solve other problems in this family
Embedding algebraic task into other subject areas
S5L2. Students will recognize that offspring can resemble parents in inherited traits and learned behaviors.

b. Discuss what a gene is and the role genes play in the transfer of traits
Tell me what you were thinking
Did you solve this a different way?
How do you know this is true?
Does this always work?
We need to open our algebraic eyes and ears to find the opportunities that foster and support student’s algebraic reasoning (classroom discussions about “simple” tasks such as 6 + 8)
“Promoting algebraic reasoning in the classroom involves incorporating conjecture, argumentation, and generalization in purposeful ways so that students consider arguments as ways to build reliable knowledge. It requires respecting and encouraging these activities as standard daily practice, not as occasional “enrichment” separate from the regular work of learning and practicing arithmetic.”

(Kaput, 2003 in Teaching Children’s Mathematics)
Number Talks
Ways to help create a culture that fosters algebraic thinking
Never say anything a student can say
Process vs. product questions
Explain to students the value of me asking questions
Be patient and embrace the awkward silence
Don't have enough information???
What do you need to know?
Just in case you forgot your scale....
a single penny weighs 2.5 grams
your weight?
approximately 28 grams/ounce
Myth: while falling off a cliff a girl used her hair and wrapped it around a branch to stop herself from falling
Algebraic thinking is not about getting the answers!
Graham Fletcher
follow me on twitter @gfletchy
12/19 of the entire condominium community is married:-)




6/9 = 2/3

6/10 = 3/5
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