Formative Assessment Supports the Mathematical Practices

1. Formative Assessment Supports the Mathematical Practices Dr Anne M Collins Lesley University March 29, 2014

2. Persevere in Problem Solving • Inclusion of on-going formative assessments; • Inclusion of non routine tasks; • Facilitation of mathematical discourse; and • Opportunity to build new mathematical knowledge through problem solving. Dr. Anne M. Collins Lesley University collins2@lesley.edu

3. Pose Problems That Push Student Reasoning • Problems that do not require an exact answer. • Problems that peak a student’s interest in the possible solutions. • Problems that stimulate students to pose their own problems Dr. Anne M. Collins Lesley University collins2@lesley.edu

4. Include Essential Components of Formative Assessment • Assess prior knowledge • Use observational protocols • Ask open questions • Engaging • Clarifying • Refocusing • Collect exit cards/tickets to leave • Listen twice as much as you talk Dr. Anne M. Collins Lesley University collins2@lesley.edu

5. Village Houses and Roofs • In a small village houses are built in rows and either have a triangular roof or share a trapezoidal roof. • How many possible roof arrangements are there if 10 houses are joined in a row? Dr. Anne M. Collins Lesley University collins2@lesley.edu

6. Model with Mathematics • Use pattern blocks to model the situation • Solve a simpler problem Dr. Anne M. Collins Lesley University collins2@lesley.edu

7. Look for and make use of structure • What pattern might you see in this problem? • How do you know you have not miscounted? • How do you know you have counted all the different ways in which the houses might look? • Did you discover the Fibonacci Sequence? • Did you use combinations to solve it? Dr. Anne M. Collins Lesley University collins2@lesley.edu

8. Make Mathematical Arguments and Critique the Reasoning of Others • Conjecture Boards • Gallery Walks • Mathematical Congresses Dr. Anne M. Collins Lesley University collins2@lesley.edu

9. Incorporate Conjecture Boards • Pose a problem. • Record student’s responses as conjectures. • Test the conjectures, revise them, test them again…and repeat the process. • Conjectures remain on the board until a counter example is found or there is a negation. Dr. Anne M. Collins Lesley University collins2@lesley.edu

10. Consider the following: On a number line 7 is halfway between 6 and 8 is 1/7 halfway between 1/6 and 1/8? Dr. Anne M. Collins Lesley University collins2@lesley.edu

11. Construct viable arguments and critique the reasoning of others. • How might you support your reasoning and “prove” that 1/7 is, or is not, halfway between 1/6 and 1/8? • Given a list of three consecutive unit fractions will the middle fraction ever be halfway between the first and last? How do you know? Dr. Anne M. Collins Lesley University collins2@lesley.edu

12. Appropriate Responses to Student Ideas • React neutrally • Record all responses on a conjecture board • Listen for misconceptions • Ask if anyone has a different idea or did it differently Dr. Anne M. Collins Lesley University collins2@lesley.edu

13. Range Question • Tell me everything you know about a box-plot. Dr. Anne M. Collins Lesley University collins2@lesley.edu

14. Respond With a Clarifying ActivityUse appropriate tools strategically. • Distribute centimeter graph paper and envelopes with an odd number of either straws, licorice, or paper strips cut to various lengths. Next, instruct students to line the strips from shortest to longest on the graph paper. Marking on the y-axis, the minimum and maximum lengths. • Ask, what do you know about the data with a two-point summary? What DON’T you know? • Next, separate the strips into two equivalent parts…marking the odd strip as the median. • Ask, what do you know about the data with a three-point summary? What DON’T you know? Dr. Anne M. Collins Lesley University collins2@lesley.edu

15. Repeat with each half of the data • Separate each half of the data set and find the median of each half. Mark each median on the y-axis and label the first one as “quartile 1” and the second as “quartile 2.” • Ask, what do you know about the data with a five-point summary? What DON’T you know? • Connect the quartile markings to make the box. • Discuss the interquartile range. • Ask if they know the mean? Or Mean Absolute Deviation. Dr. Anne M. Collins Lesley University collins2@lesley.edu

16. Role of the teacher • Select and set up an interesting and rich mathematical task. • Support students’ exploration of the task by asking engaging, clarifying, or refocusing questions. • Oversee students as they share work and discuss the task. Dr. Anne M. Collins Lesley University collins2@lesley.edu

17. What are the differences between the following two problems? • 4/5 is closer to 1 than 5/4. Show why this is true on a number line. • What is closer to 1? • 5/4 • 4/5 • ¾ • 7/10

18. Responsive Teaching • Asks how you thought about the problem. • Asks if that strategy will always work. • Invites a student to explain why the strategy works or does not work.

19. Reasoning and Sense-Making 840 stickers were given to 42 students. 2/3 of the children were boys, and each of them received the same number of stickers. Each girl received twice as many stickers as each boy. How many stickers did each girl receive?

20. Misconceptions • b b b g g = 840 • 840 ÷ 5 = 168 • b = 3(168) or 504 • g = 2(168) or 336 Dr. Anne M. Collins Lesley University collins2@lesley.edu

21. Attend to Precision • If 2/3 of students are boys then 1/3 are girls. • 1/3 of 42 students are girls = 14 girls, 28 boys. • Girls received twice as many stickers as boys so the number of stickers shared by the boys and girls are the same. • 820 ÷ 2 = 420. • 420 ÷ 14 = 30 stickers per girl. Dr. Anne M. Collins Lesley University collins2@lesley.edu

22. Effective Teaching with Formative Assessment • Puts the student first and foremost when planning lessons • Asks questions that require more than a 1 sentence response • Assesses where the student is on each given day • Supports student learning, student discourse, student reasoning

23. Model with Mathematics Abby and Emma shared some beads in the ratio of 2 : 5. If Emma gave 30 beads to Abby, they would have the same number of beads. How many beads did Abby have in the beginning? Dr. Anne M. Collins Lesley University collins2@lesley.edu

24. Without solving the problem…answer the following questions. • How does the task build on students’ previous knowledge and experiences? • What definitions, concepts, or ideas do students need to know to begin to work on the task? • What questions will be asked to help students access their prior knowledge? • What are some the ways the task can be solved? • What particular challenges might the task present? Dr. Anne M. Collins Lesley University collins2@lesley.edu

25. What are some problem solving strategies? Dr. Anne M. Collins Lesley University collins2@lesley.edu

26. Starting Ratios Abby Emma Resulting Ratios +30 Abby -30 Emma Dr. Anne M. Collins Lesley University collins2@lesley.edu

27. Algebraic Solution • 2x + 30 = 5 x – 30 • 60 = 3x • X = 20 Emma had 100 beads to begin with. Abby had 40. Dr. Anne M. Collins Lesley University collins2@lesley.edu

28. Abby EMMA Dr. Anne M. Collins Lesley University collins2@lesley.edu

29. 100 90 Abby 80 70 60 50 40 30 20 10 50 100 EMMA Dr. Anne M. Collins Lesley University collins2@lesley.edu

30. Effective Teaching: Values Incorrect Answers Wrong answers: • Need to be part of the teaching/learning process • Give information about what the student is thinking • Might be an error of haste or an on-going misconception Dr. Anne M. Collins Lesley University collins2@lesley.edu

31. Answer Getting Versus Learning Math • United States • How can I teach my students to get the answer to the problem or computation? • Japanese • How can I use this problem to teach the mathematics of this unit? Dr. Anne M. Collins Lesley University collins2@lesley.edu

32. On a bulletin board in a high school AREA • A = lw • A = s2 • A = 2πr2

33. Model With Mathematics • Think about the mathematics content that is usually presented in an abstract manner. • Think about square roots. What do they look like? How might they be represented? • What models can you use to make those abstract concepts approachable for all students? Dr. Anne M. Collins Lesley University collins2@lesley.edu

34. Two Possible Models • Find a close approximation for the square root of 43 (without using a calculator) Dr. Anne M. Collins Lesley University collins2@lesley.edu

35. Geometric Model • On graph paper outline a 6 x 6 square. Dr. Anne M. Collins Lesley University collins2@lesley.edu

36. Number Line Model 34 35 36 37 38 39 40 41 42 43 44 45 46 44 48 49 Dr. Anne M. Collins Lesley University collins2@lesley.edu

37. Use Multiple Representations • A picture is worth a thousand words • Which of the two models gives a visual image of what a square root looks like? • Students learn best when they move from the concrete-pictorial-abstract. • We jump too quickly to the abstract! Dr. Anne M. Collins Lesley University collins2@lesley.edu

38. Fraction as Ratio • Discrete • Relationship between the numerator and the denominator • Either a part to part or part to whole comparison • Represent a Rate Dr. Anne M. Collins Lesley University collins2@lesley.edu

39. Geometric Representations • What representations might you use to compare ratios? • Which is greater 5:7 or 3:5? And why do you think I chose those two ratios? Dr. Anne M. Collins Lesley University collins2@lesley.edu

40. By Inspection, which ratio is greater? 5:7 3:5 Dr. Anne M. Collins Lesley University collins2@lesley.edu

41. Equivalent Ratios 5:7 3 : 5 Dr. Anne M. Collins Lesley University collins2@lesley.edu

42. Find the Least Common Denominator Least Common Denominator 5:7 3 : 5 Dr. Anne M. Collins Lesley University collins2@lesley.edu

43. Convert to Percents 5 : 7 3 : 5 Dr. Anne M. Collins Lesley University collins2@lesley.edu

44. I – We – You Teaching Strategy Is Ineffective! • I- Teacher presents a topic, shows how to do the procedural steps. • We- Students work through computations with the teacher • You- Students practice the procedures with worksheets Dr. Anne M. Collins Lesley University collins2@lesley.edu

45. You-We-I Supports Students • Students struggle with a problem situation. • Students collaborate with others to solve the problem situation • Students report out their strategies/solutions. Class discussion follows. Revisions are made. • Teachers pull together the important points made throughout the process and summarize that discussion. Dr. Anne M. Collins Lesley University collins2@lesley.edu

46. Underlying Tenet of CCSS-M • Teachers need to change the way in which they teach. • If we keep doing what we have always done the results will continue to be the same…and that is not good enough. • Coaches need to correct mathematical errors…even if it feels awkward. • Administrators must ensure student ideas, misconceptions, and engagement are at the forefront of every classroom. Dr. Anne M. Collins Lesley University collins2@lesley.edu