1. Presented by Dr. Linda K. Griffith March 14 and 26, 2012
Problem Situations for Addition and Subtraction and the Nature of “Equals”
Presented by Dr. Linda K. Griffith
March 14 and 26, 2012

2. Day 1 Session 1: Orientation
Session 2: Standards for Mathematical Practice Introduction
Session 3: The Nature of Equals & Both Addends Unknown Problems
Session 4: Situations The First Six Problem Situations

3. Session 1
Orientation

4. Introductions
Introduce yourself to the class. Tell your name, school/district/co-op, job position, and reason you are here.
In small groups, discuss the article Purposeful Pedagogy and Discourse Model:
Discuss the role of students’ current thinking in teaching them mathematics.

5. Standards for Mathematical Practice Introduction
Session 2
Standards for Mathematical Practice
Introduction

6. 8 Standards for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.

7. Small Group Discussion
Discuss briefly your understanding of each of the Standards for Mathematical Practice by considering these items –
What does this standard mean?
What does it look like when students are engaged in this standard?
What can a teacher do to model or encourage students to engage in this standard?
Why is this standard important?
Several content standards use the verb “understand.” What does the verb “understand” indicate in those standards?
Did you notice any themes that unite the Standards for Mathematical Practice?

8. Haley is solving the problem: Maria has 5 fewer pennies than Robert
Haley is solving the problem: Maria has 5 fewer pennies than Robert. Maria has 6 pennies. How many pennies does Robert have?

9. Haley Video PSAS-Sess2-Haley-mariasPennies.mp4

11. Class Discussion Consider the following prompts.
In what way(s) does Haley make sense of the problem, and in what way(s) is she stuck? What did you expect her to do that she didn’t? Why do you think she didn’t?
How did the teacher support her in learning to persevere in problem solving? Do you think Haley would have persevered anyway?
Do you think Haley is reasoning abstractly, quantitatively, or both? In what ways?
How are any of the other Standards for Mathematical Practice evident in Haley in this video?

12. Reflection
While reflecting on Haley’s work and your own experiences with students, examine this Wordle of SMP 1 Make sense of problems and persevere in solving them.
What words jump out at you from the Wordle of this standard?
Why are these items so important?
How do this graphic and the other information help you synthesize what this standard is all about?

14. The Nature of Equals & Both Addends Unknown Problems
Session 3
The Nature of Equals & Both Addends Unknown Problems

15. Try It Out
4 + 3 = ☐ What goes in the box to make this a true number sentence?
Are these number sentences true or false? Explain your thinking? Record your thoughts below.
2 = 2
2 = 2 + 0
Predict how you think students will respond to these problems.

16. Equality Video-First Grade PSAS-Sess3-Equality.mp4

17. Class Discussion Equality Video – Consider the following prompts.
How are these students thinking about equality? What misconceptions do they have?
In what ways did instruction impact or have the potential to impact their understanding?

18. Both Addends Unknown Video PSAS-Sess3-BothAddendsUnknown.mp4

19. Class Discussion
Both Addends Unknown – Consider the following prompts.
What is special about the “both addends unknown” problem type?
What did the students do to make sense of the problem?
What mathematical connections did the teacher make in her instructional moves as she facilitated children’s thinking about this problem? What is mathematically important about her connections?

20. Class Discussion
Reflect on the Standards for Mathematical Practice. What evidence of any particular SMP do you see reflected in these two videos?
What are the instructional implications of these two videos for you?

21. The First Six Problem Situations
Session 4
The First Six Problem Situations

22. Try It Out Analyze the problem situations in your handout.
Sort these six problems by operation. Which ones are Add To and which are Take From?
Reflect on your thinking. Justify your sort. What makes a problem an Add To or a Take From?

23. Small Group Discussion
Share and discuss your reasoning for your sort with your small group.
Do you agree with each other or not?
What are the areas of disagreement?
Why do you think these disagreements are occurring?

24. Jordan Video 1 PSAS-Sess4-Jordan1.mp4

25. Jordan Video 2 PSAS-Sess4-Jordan2.mp4

26. Jordan Video 3 PSAS-Sess4-Jordan3.mp4

27. Jordan Video 4 PSAS-Sess4-Jordan4.mp4

28. Jordan Video 5 PSAS-Sess4-Jordan5.mp4

29. Jordan Video 6 PSAS-Sess4-Jordan6.mp4

30. Class Discussion Consider the following prompts.
How did Jordan solve the problems? What were the number sentences that went with his solutions?
Why do you think he solved the problems the way he did? What was he doing?
What is the role of “action” in Jordan’ thinking about the problems?
Based on your previous discussions and the videos, would you reclassify your problem sort? In what ways? Share and justify your thinking.

31. Summary Video PSAS-Sess4-Summary-FirstSix.mp4

32. Assignment before Day 2
Read the article about Purposeful Pedagogy and Discourse Model for AR CCSSM PD.
Read the article Children’s Understanding of Equality: A Foundation for Algebra.
Pose some true-false equations to assess your students’ understanding of equality.  
Pose some addition and subtraction problems from the assignment sheet. Consider student thinking.