Routines that promote mathematical Reasoning

1. Routines that promote mathematical Reasoning Melissa Nelson Middle School Math Interventionist

2. Mission and Vision Mission: Dorchester School District Two leading the way, every student, every day, through relationships, rigor, and relevance. Vision: Dorchester School District Two desires to be recognized as a “World Class” school district, expecting each student to achieve at his/her optimum level in all areas, and providing all members of our district family with an environment that permits them to do their personal best.

3. Mel bought some doughnuts. She gave ½ her doughnuts and ½ a doughnut to her mom. Then she gave away ½ her remaining doughnuts and ½ a doughnut to her aunt. Then she gave ½ of her remaining doughnuts and ½ a doughnut to her sister, Michelle. This left her with ¼ of a dozen doughnuts. How many doughnuts had she bought?

4. What is mathematical reasoning? “Mathematical reasoning is essentially about the development, justification, and use of mathematical generalizations.” Susan Jo Russell (1999)

5. Mathematical Practices Related to Reasoning • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Look for and make use of structure • Look for and express regularity in repeated reasoning

6. The Mathematical Reasoning Process Model

7. Conjecturing • Conjecturing involves reasoning about mathematical relationships to develop statements that are thought to be true but are not known to be true. • Natural way to enter in to reasoning • Entry point to an activity or discussion • Emerge from a variety of sources (diagrams, an instance, an application) • Must be stated in the form of a statement

8. Conjectures • Opportunities needed to explore conjectures • Adjust textbook directions

9. Let’s make a conjecture • Write some questions that could be asked about a particular mathematical content area. • Example: What is the result of multiplying two fractions? • Re-write the questions in the form of conjectures (statements that can be shown as true or false). • Example: When two fractions less than one are multiplied, the product is less than either factor.

10. Generalizing • Generalizing is when students focus on a particular aspect of a problem or an idea and think about that aspect more broadly.

11. A Closer Look at Generalizing Identifying Common Elements Extending Reasoning Thinking about a relationship, idea, rule, pattern, or mathematical property and expanding it to be more general. Noticing what is the same across examples, problems, situations, and representations.

12. Making Generalizations Make as many generalizations as you can about the triangle paths in the figure. It is important for teachers to acknowledge generalizing as a PROCESS and not a product.

13. Generalizing Commonalities And Extending Reasoning If the denominator is 1 bigger than the numerator, the fraction with the bigger denominator will be bigger.

14. Creating tasks to generalize • Provide opportunities • Adjust textbook activities • Allow partner/group discussions

15. Consider the Relevance within the Domain “The quotient is smaller than the dividend.” Choose various real numbers a and b and determine a ÷ b. For what values of a and b does the generalization hold true? It is essential for students to revisit generalizations

16. Conjectures and Generalizations with “Math Language” Refining Mathematical Definitions Mathematicalvs Everyday Language Math terms and definitions essential for understanding and communicating concepts Potential for confusion arises when everyday language is allowed for mathematical terms Clarifying terms may arise during validating or refuting conjectures or generalizations Mathematical communication requires use of different symbols, terms, and representations Develop and clarify mathematical definitions to promote precision

17. Investigating Why • Essential pivot to go from conjecturing and generalizing to justifying and refuting • Attend to particular features to explain whether a generalization is true or false • Important to examine multiple factors to help understand with greater insight and promote deeper understanding • A process that leads to new generalizations or refinement of previous generalizations

18. Justifying There are 2 important characteristics for valid justifications • Language that states a general relationship and specifies a domain • Reasoning that supports the general relationship and shows that it holds for all instances in the domain • Initial attempts may include valid and invalid statements

19. Correct and incorrect reasoningExpanding to other examples • Read the students’ justifications. • Which, if any, of these justifications are valid, and which are invalid? • If a justification is invalid, how could it be improved so that it would be valid?

20. Refuting • Validation by refuting is important since conjectures can be true or false • A single counterexample is all that is needed to invalidate a conjecture • Demonstrating false conjectures can lead to development of new ideas and insights

21. Benefits of Justifying and Refuting Arguments • Creator defends truth in conjecture • Evaluators help creator to clarify • Opportunities arise for conjectures to make sense to all students • Opportunities arise where a conjecture must be revised or refined

22. Inappropriate Bases for Justifications • Authority of a teacher or parent • Textbook • Opinion of a peer • Class consensus • Support of many examples Understanding what does not make a valid justification is just as important as knowing what does make a valid justification.

23. Valid Mathematical Justifications • Must involve an argument that shows that a relationship holds for every possible case • May use a generic example since it considers how the idea applies to all cases

24. Mathematical Reasoning is the vehicle for engaging students.

25. Practice reasoning Instructional Practices to Promote Reasoning “Turn & Talk” Think – Pair – Share Non-routine Problems

26. Promoting Reasoning Four 4s Games to Promote Reasoning

27. Promoting Reasoning Four 4s Which Does Not Belong? Games to Promote Reasoning • 2, 6, 5, 10 • 2, 3, 15, 23 • , 2, 8, 16 • 9, 16, 25, 43 • ,

28. Promoting Reasoning Four 4s Which Does Not Belong? Tell Me All You Can Games to Promote Reasoning

29. Promoting Reasoning Four 4s Which Does Not Belong? Tell Me All You Can Four Strikes & You’re Out Games to Promote Reasoning

30. Contact Information Melissa Nelson Middle School Math Interventionist mnelson@dorchester2.k12.sc.us 843-323-7711

31. Session EvaluationParticipants are asked to complete a session evaluation for each session attended. Credit (attendance, renewal, and/or technology) will be added following evaluation completion. For each question, use 1=Strongly Disagree, 2=Disagree, 3=Neither Agree nor Disagree, 4=Agree, 5=Strongly Agree. Your responses will assist us in planning future professional development in Dorchester School District Two. • The instructor was well prepared for the workshop. • The materials for the workshop were appropriate. • The concepts presented were appropriate to my job. • I will benefit from attending this session. • I would recommend this training to others.