Equality and Relational Thinking: Abstracting from Computation Part 1 Pathways to Teacher Leadership in Mathematics Tuesday, July 1, 2014

Homework Tasks Due Tuesday, July 1, 2014 “A well-developed conception of the equal sign applicable to elementary and middle school children is characterized by relational understanding…” (Matthews et al., 2012, p. 318) In your own words describe and provide an example of relational thinking. Reference the readings in your description. Use your readings to find 3-4 statements that exemplify and clarify the importance of relational thinking. Cite your readings for each example (author, year, page number). Summarize the four levels of indicators of mathematical equality in in Matthews et al. (2012). Draft only.

Synthesizing Our Learning How do the levels of understanding of the equal sign as defined by Matthews et al. (2012) provide insight into the nature of children’s knowledge of mathematical equality? What do you find yourself considering as you reflect on current instructional practices and “ah ha’s” you experienced as you read and reflected in preparation for today’s class?

Learning Intentions & Success Criteria We are learning to understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value. Success Criteria We will be successful when we can recognize the difference between computational and relational thinking. We will be successful when we can use relational thinking to build, express, and justify mathematical relationships.

Double Compare  Round #1 Directions: Mix the cards and deal them evenly. Each player turns over their top two cards.  The player with the larger total says “me” and takes the cards. If the totals are the same both players turn over 2 more cards. The game is over when there are no more cards to turn over.

Double Compare Debrief What strategies did you use to compare the two pairs of cards? Comment on how your strategies engaged you with the properties of the operations. Reflect upon the emerging use of MP7 and MP8 in this activity.

Possible Strategies for Double Compare Computational strategies - computing the total on each card separately. Such as: Counting up or counting on. “Just knowing” some number combinations Numerical reasoning about number combinations. Relational strategies Examining relationships between the quantities. (No computing to compare.)

Double-Digit Double Compare Work in groups of 4. Teams turn over the same color cards. Team with the larger total says “us” and takes the cards. Try to justify your reasoning without computing. In the case of a tie, draw again.

Thinking back on the strategies we surfaced from Round #1… Which type of strategy did you find yourself using in order to determine who won?

“Most elementary students and many older students as well, do not understand that the equal sign denotes the relation between two equal quantities.” Carpenter, T., Franke, M., & Levi, L. 2003. Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann. p. 9

True or False Work in teams of 4. Take out one card at a time. Study the equation on the card. Decide if the equation is true or false. Justify your reasoning without computing.

Supporting Relational Thinking Reflect on 583 – 529 = 83 – 29 10 + 0 = 100 – 100 + 10 82 – 28 = 86 – 29 What shifts in thinking did you have to make to view these true/false sentences through a relational lens? Chart the shifts in thinking that participants made to think relationally Structures can be: using place value, the properties of the operations, or generalizations about the behavior of the operations

In closing… When mathematical ways of thinking begin to become automatic – not just ways one can use, but ways one is likely to use – it is reasonable to call them habits: mathematical habits of mind. --EDC Transitions to Algebra

Learning Intentions & Success Criteria We are learning to understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value. Success Criteria We will be successful when we can recognize the difference between computational and relational thinking. We will be successful when we can use relational thinking to build, express, and justify mathematical relationships.

Disclaimer Pathways to Teacher Leadership in Mathematics Project University of Wisconsin-Milwaukee, 2014-2017   This material was developed for the Pathways to Teacher Leadership in Mathematics project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Any other use of this work—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors—without prior written permission is prohibited. This project was supported through a grant from the Wisconsin ESEA Title II Improving Teacher Quality Program.