Inspire and Cultivate Algebraic Thinking

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Presented by Aubree Short & Kelly Urlacher

Today’s Outcomes● Learn strategies to create algebraic thinkers in

your classrooms● Explore ideas for student discourse● Discuss the integration of algebraic thinking

through daily routines

Warm UpPinch Cards

Primary:Jenny has 13 pennies in her pocket. Some of her pennies fell out. If Jenny had 22 pennies to begin with, how many pennies fell out?

Intermediate: Manny ate ½ of the pizza and Nate ate ⅓ of the pizza. How much of the pizza is left?

http://www.heinemann.com/putting-the-practices-into-action/

TCSD’s Journey

Algebraic Thinking vs. Algebra Class

The main purpose of algebra is to learn how to represent general relationships and procedures; for through these representations, a wide range of problems can be solved and new relationships can be developed from those known…

However, students tend to view algebra as little more than a set of arbitrary manipulative techniques that seem to have little, if any, purpose to them.

L.R. Booth. Difficulties in Algebra. 1986

Algebraic ThinkingAlgebraic thinking is not just about an Algebra class, but encompasses how to think throughout a student’s math journey.

Algebra is often being referred to as the “gateway” to higher education.

Elementary school is the necessary time to start thinking about symbolic representations, the explanations of the students, and what are the underlying issues, misconceptions, and general ideas.

Why Algebraic Thinking Integration is Necessary

Over 60 percent of all students entering community colleges must take what are called developmental math courses … [that] are algebra-based and focus on linear and quadratic equations.

– Ginia Bellafante, nytimes.com, 2014

Because Algebra has come to be regarded as a gatekeeper … the high failure rate in Algebra, especially among minority students, has rightfully become an issue of general social concern.

– H. Wu, math.berkeley.edu

Algebra I is the key — and the barrier — to students’ ability to complete a challenging mathematics curriculum in high school.

– Southern Regional Education Board, publications.sreb.org

Reason Abstractly and QuantitativelyMathematically proficient students are able to:

★represent quantities in a variety of ways

★remove the problem context to solve the problem in an abstract way (equation)

★refer back to the problem context, when needed, to understand and evaluate the answer

-O’Connell & SanGiovanni, Putting the Practices Into Action (2010)

Match ItWork with a partner to match each expression or equation to the corresponding problem. Justify each of your matches.

Can tech support algebraic thinking?Blended learning is no longer a distant idea.

Technology can be a symbiotic element in the classroom.

Build conceptual understanding using their own strategies.

Recognizing transfer between conceptual and abstract.

Manipulatives to Explore Abstract IdeasResearch supports the use of manipulatives.

Students gain deep understanding by using manipulatives in their math learning.

Exploration comes to life.

Manipulatives are a means, not an end.

Teacher Tools as Tech Manipulatives

www.dreambox.com/teachertools

Mathematical Concepts and Misconceptions

Rhett Allain, Associate Professor of Physics at Southeastern Louisiana University, rightly points out that confusion is the sweat of learning; deeper learning could be considered an invigorating mental workout.

Misconceptions★ In the organic process of learning, mistakes are natural and

beneficial.○ Positive vs negative

★ Providing insight for teachers, but can be used by the students too.○ How can we learn from this?

★ Misconceptions are a new learning opportunity. ○ How can we build an environment that supports growth?

★ Conversation:○ How do you address misconceptions in the classroom?

How can you integrate algebraic thinking?Exploring symbolic representations of thinking.

All students can draw or write this out: using notation when suitable.

Question It: Tell the answer. Ask students to write the question (in the form of a problem.)

Answer = 10 The problem is…

Answer = 3 ½ The problem is…

Answer = $4.50 The problem is…

Contextualize & DecontextualizeDiscuss appropriate operations to

solve problems

model building appropriate equations

use diagrams to model math situations

Ask, “What operation makes sense?”

Ask, “How should we build an equation to match this problem?”

Ask students to write word problems to go with a situation.

Consistently ask students to explain the equations or diagrams.

Ask students to label answers by referring back to the problem to determine what a quantity represents.

Ask students if the quantity makes sense.

X + 4 =10

How can you integrate algebraic thinking?Math Talks: How are students thinking about the algebraic idea?

What does it sound like when they talk about their thinking?

True or False

80÷4=(80÷2)+(80÷2)

https://www.teachingchannel.org/videos/common-core-teaching-division

80÷4=(80÷2)+(80÷2)

20 2020 20

40 40 4040= +

Conceptual Understanding for All!Setting students up to be active learners using algebraic thinking.

Because [low-achieving students] are less likely to have acquired the basics on the same schedule as more advanced learners, struggling learners are often confined to an educational regimen of low-level activities, rote memorization of discrete facts, and mind-numbing skill-drill worksheets… [They] have minimal opportunities to actually use what they are learning in a meaningful fashion.

- Wiggins & McTighe, Schooling by Design

Authentic Performance

Constructing Meaning to MasteryGrant Wiggins, Educational Leadership 2014

Mastery is effective transfer of learning in authentic and worthy performance. Students have mastered a subject when they are fluent, even creative, in using their knowledge, skills, and understanding in key performance challenges and contexts at the heart of that subject, as measured against valid and high standards.

Students learn from constructing meaning. This can not be given to them, they need to have the power to try different ideas, different methods, and have the chance to build their cognitive map.

Empowering Students to LearnAs teachers, we often teach algorithms too soon and assume understanding.

We should be providing examples of problems that help the students realize that their informal procedures are beneficial.

By giving a thought-provoking question in new and unfamiliar situations, encourage:

▪ Independent, Critical Thinking

▪ Problem Solving

▪ Design own Solutions

Teaching StrategiesBy presenting mathematical concepts through multiple examples and problems, we can foster understanding with several different methods.

Examples:▪ Recipe: [y = 2k + a] To make y, you need to double k and add a to it.

▪ Stories: [3(s+2) = 12] There is a number. You add 2 to the number, then multiply by 3, and the number becomes 12.

▪ Function machine: Inputs and Outputs, find the rule.

▪ Balance model: Two Expressions are seen to be equal.

Today’s Outcomes● Learn strategies to create algebraic thinkers in

your classrooms● Explore ideas for student discourse● Discuss the integration of algebraic thinking

through daily routines

ReflectionWhat did you learn?

How are you feeling?

What are your next steps?

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