1. Understanding Student Thinking in Algebra Session Equality 2008-2009

2. Sessions for Algebraic Thinking What does the equal sign mean? Relational Thinking Understanding the Properties of Arithmetic Conjectures Justification In order for students to have a good experience with Algebra, they must first understand what the equal sign means.

3. Outcomes Establish that students have some very common misconceptions about the meaning and use of the equal sign. Explore strategies to help students develop and understanding of the meaning and use of the equal sign. Develop an understanding of the importance of the concept of equality in algebraic reasoning. “Big Ideas” of Algebraic Thinking

4. 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

5. Problem-Solving Activity Number of Chirps/Minute Once you see a pattern, try to write a formula

6. Misunderstandings of the equal sign Students across the state have a very common misconception of what the equal sign We gave students these numbers and wanted them to solve 8 + 4= __ + 5 These are the results of the 234 students we polled

7. 234 students were asked this question 8 + 4=5+__. The following results will show the common misconception of what does the equal sign represent Grade71217 12 &17 1 & 2 11 Students 2 18% 5 45% 2 18% 2 18% 3 & 4 181 Students 48 26% 79 43% 29 16% 25 14% 5 & 6 51 students 22 43% 14 27% 9 18% 6 12%

8. Article Children’s Understanding of Equality: A Foundation for Algebra – What 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 your thoughts

9. Huge outcomes for this article Tell the students what the equal sign means Revisit equality over, over and over Two tools to develop – T/F number sentences – Open-ended number sentences

10. What about equality? 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. Third-We must deal with misconception-Children must understand that equality is a relationship rather than a signal to do something.

11. Big Ideas of the equal sign See handout

12. Watching students deal with equality Video Clip 1.1 (Title 1) 1’40” in length Kindergarten student Discussion after the video clip What does he understand about the equal sign? What strategies does he use to compute his answer?

13. Watch students use relationships to solve number sentences Video Clip 1.2 3’ 03” in length 1 st grader Listen to how this student explained how he decided what number went in the box. This is where we want our students to move. Then, watch how the teacher moved to bigger numbers to see if the student really understood the strategy he shared using smaller numbers -What strategy did the student use?

14. The Same but Different How did each student solve and what was the difference between the two students strategies from the video?

15. Developing Children’s Conceptions of Equality Video Clip 1.3-Teacher uses T and F number sentences 4 th grade classroom 10’ in length Listen to how this 4 th grade teacher took the misconception her students had and moved them through the process of T/F and open number sentences – Pay attention to questions asked by the teacher during discussion – Pay attention to how the equal sign appears in different places in the number sentences. – THIS VIDEO IS A GOOD EXAMPLE OF HOW TEACHERS BEGAN WITH A MISUNDERSTANDING AND MOVED STUDENTS TOWARD UNDERSTANDING

16. 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? In grade level pairs (K-1, 2-3, 3-4,5-6), generate a sequence of number sentences that might be used to help students develop the meaning of the equal sign. This could be a series of sequences or one-long series. Include the open number sentence that you hope the students can do at the end of that series of true and false statements. Record these on chart paper.

17. Developing Children’s Conceptions of Equality Over Time Video Clip 1.5 2 nd grade students Listen how the teacher carefully chooses problems to challenge children’s misconceptions about equality. Listen to how some of the children are beginning to use relational thinking. Pay attention to the sequence of problems Julie uses

18. Steps to Equality in your classroom First, find out who knows and understands the equal sign by using open-number sentences Then, move to true and false number sentences Next, go back to open-number sentences

19. Connecting equality to formal algebra Start with a simple example: 2x + 3 = 9 – Using what you know about numbers solve this – Solve as an elementary student would Solve this problem: 7x – 36 = 48 – 14x – Using what you know about numbers solve this – Solve as an elementary student would

20. Accountability Piece Assess your students on the “concept of equality” and get a schedule set up on how you will continue to deal with the equal sign Article-”Fostering Relational Thinking While Negotiating the Meaning of the Equals