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

vs.

relational understanding

Fostering Algebraic Reasoning (K-12)

GRESA October 2, 2012

Algebra

Arithmetic

how we might see algebra

Misconceptions:

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

Mythbusters

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)

http://www.cleanvideosearch.com/media/action/yt/watch?videoId=eijmcqGe61o

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

Roadblock?

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

**Men**

**Women**

**Men**

**Women**

**Men**

**Women**

Algebraic thinking is not about getting the answers!

Graham Fletcher

gfletcher@henry.k12.ga.us

follow me on twitter @gfletchy

12/19 of the entire condominium community is married:-)

**2/3**

3/5

3/5

**4/6**

6/10

6/10

**6/9 = 2/3**

6/10 = 3/5

6/10 = 3/5