2010-11Algebraic Thinking

SD Counts Cohort MeetingPlatte South DakotaSeptember 29, 2010

8:30-3:30

House Keeping

Break at 10:00 and 1:45

Lunch at 12:00

Pop machine in break room

Bathrooms

Group Norms

Honor private think time.

Be prepared for sessions, everyone should have something to contribute.

Listen carefully to take in another participant’s ideas.

Cell phones off or on vibrate.

Start and end each session on time.

Take care of your needs.

Items to attend too

Website www.mathcounts.org

October meeting dates

Sign-in sheet

Syllabus

Training for next year-visit with principals

-upcoming workshop for Grades preK-2

How is your year going?

Take a few minutes of quiet think time and think about how this year has already been better than last year and why?

What did your e-metric scores look like?

Be ready to share

Main ideas of Algebraic Thinking in the Elementary

Equality

Relational Thinking

Understanding Properties

Conjectures

Justification

Algebra Readiness

does not start with pre-algebra.Is the foundation which begins in the elementary grades and is supported by sound teaching methodsBegins in Elem→MS→HS

Algebra should build on skills learned at the pre-K levelMust have curriculum and resources to achieve and meet the standards

Must get kids actively involved so they better understand what they are doing.Must be engaging and relevantPATTERNS, PATTERNS, PATTERNS-STUDENTS MUST HAVE THE OPPORTUNITY TO WORK WITH NUMBERS AND SEE PATTERNS BEFORE THEY CAN WORK WITH AND UNDERSTAND AN EQUATION

The patterns and patterning activities in which young children readily engage serve to form the basis of their emergent algebraic reasoning.Whether they are snapping cubes, clapping their hands or stomping their feet in rhythm, or using input-output machines, young children are eager to experience and are capable of identifying and describing change.A lot of concrete, hands-on activity.

Linking cube questions what would colors be?Different problem types

What should Algebra look like at each school?Algebra at the K-2 level..

Algebra at the 3-5 level….

The focus at this level is on patterns, relations, and functions and the ability to analyze situations using a variety of representations.

They begin to see how they can use algebra in the real-world.

A lot of hands-on, concrete activity.

Algebra at the 6-8 level….

Students need multiple opportunities over time and across contexts to demonstrate fluency, flexibility, and a deep understanding of algebra concepts and skillsMuch time should be spent examining tables, charts, graphs, and equations.Still a lot of concrete moving to abstract toward 8th gradeAs students develop their algebraic reasoning, they should be engaged in the use of tables, charts, graphs, and symbolic expressions containing variable's

Outcomes for this session

Establish that students have some very common misconceptions about the meaning and use of the “=” sign.Explore strategies to help students develop an understanding of the meaning and use of the equal sign.Develop an understanding of the importance of the concept of equality in algebraic reasoningTalk about “relational thinking”Develop as a cohort the “Big Ideas” of algebraic thinking.Visit the properties of math that students use

Table Tent Activity

Write your name

Write a 1-digit number below your name

Problem Solving Activity

Number of Chirps per Minute

Extension activity

-Once you see a pattern, try to write a

formula.

Misunderstanding

Students, across the state have a very common misconception of what the “=”

sign means.We gave students these numbers and wanted them to solve 8 + 4= ? +5

The next slide will show the resultsDiscuss how your students answered before we look to see if they answered like the rest of the students in our state.

Grade 7 12 17 12 &17

1 & 211 Students

2

18%

5

45%

2

18%

2

18%

3 & 4181 Students

48

26%

79

43%

29

16%

25

14%

5 & 651 students

22

43%

14

27%

9

18%

6

12%

234 students were asked this question 8 + 4= ? + 5. The following results will show the common misconception of what does the “=“ sign represent.

Read ArticleGroup 1-Beginning to The Next YearGroup 2-Beginning of The Next Year-end

Skip-about the Mathematics

Children’s Understand of Equality: A Foundation for Algebra

Read silently 10 minutes, share small group, discuss large group

Be looking for an answer to the following questionsWhat strategies does the author use to help her students understand the equal sign?How should students be thinking about the equal sign?Are there just some things you “just” tell the students

Share articles

Key ideas from your section???

Huge outcomes for this article were..

This teacher does a good job of teaching equality to students

Revisit equality; over, and over, and over.

2 tools to develop equalityTrue and false number senses

Open-ended number sentences

Why know about equality as a relationship?

First-In order to think about relationships expressed by number sentences, children must understand that equality is a relationship that expresses the idea that

two expressions hold the same value.Second-Because students do not understand equality as a relationship, this becomes a major stumbling block for students when they move from arithmetic to algebra.Must deal with misconception-Children must understand that equality is a relationship rather than a signal to do something.

Article Reading

Balancing Act: The Truth behind the Equals Sign

Read individually and be ready to share

Equality Handout

Ask “purpose” questions before each section

Watch video clips1.1, 1.2, 1.3, 1.4, 1.6

Steps to Equality

What are the steps in teaching equality to our students?

Where do we start?

Where do we move to?

What do we revisit?

Video 1.4

Questions

-What misconceptions about the equal sign do children in this episode express?

-How do the children who understand the equal sign help the other children to develop their understanding?

Watch video clip 1.6

Questions to answerHow does each child solve this problem?

What do their strategies tell you about their understanding of equality?

“Big Ideas” of equality

In your groups create a list of “big ideas”

about “equality” and put on the wall

What are the words that you have read or we have talked about that you would like listed on your sheets of paper.

Let us compare and add if we need to add any to our list.

Developing a series of true and false number sentences

Read page 14-15-16-17 in book. “Developing Children’s Conceptions of Equality”

Groups to create a series of true and false number sentences (Grade level)

Where do we start?

Where do we go?

What do we come back to?

Article: Fostering Relational Thinking while Negotiating the Meaning of the Equals Sign

This article will again talk about the meaning of the equal sign, but also talk about how relational thinking can be use

to help solve problems.

So what is relational thinking?

Relational Thinking

Take out handout “Intro to Relational Thinking”

Solve independently

USE ANYTHING YOU KNOW ABOUT NUMBERS TO SOLVE these number sentences.

Find at least 2 ways, if you finish early 3 ways

Definition of “relational thinking”

Solving problems by looking at expressions/equations completely, rather than as procedures to be carried out step by step

Must have understanding of equality before you develop “relational thinking”

“solving problem”= instead of procedure“add on, subtracting”=relational thinking

Relational Thinking

Take out Emma’s Problem Handout

Video clip 2.1 Decide why each number sentence posed might be particularly useful in developing relational thinking?

If you were this child’s teacher, what problems would you have her work on next?

Video 2.3

Take out Relational Thinking handoutQuestion

The number sentences posed led to the number sentences 4 x 7 = ?. Design a similar sequence of number sentences to lead to the number sentence 6 x 8 = ? What implications might such series of number sentences have for learning multiplication and division facts?

Video 2.4

QuestionDescribe each strategy. What strategy might each of these children use to decide whether 84 + 27 = 74 + 37 and 56 + 38 = 59 + 36

Big Idea’s of Relational Thinking

See handout, are there any you would like to add?

Course Expectation

Read and reflect a professional book

(I have some you can choose from)

-At our last meeting we will take time

to reflect on the book you read.

-How/has this book impacted you as

a classroom teacher.

Next meeting date

Snacks for next month

Evaluation

Read Chapters 4-5-6 Jot down key ideas for each chapter

Next meeting-CHANGED TO OCTOBER 25TH