1. Project 2: Preservice Teachers
Tasks for Teachers: Approaches to the Design of Tasks for Preservice and Inservice Learners
Project 2: Preservice Teachers
Dana Olanoff
Jennifer Tobias
Neet Priya Bajwa
Eva Thanheiser
Rachael Welder
Ziv Feldman

2.
Who are we? Six mathematics teacher educators of content courses for prospective elementary and middle school teachers throughout the US.
We need an updated picture…
Goal: To design and implement cognitively demanding mathematical tasks in mathematics content courses for prospective teachers.

3. First Task Sequence: Fraction Comparison
Task Design Cycle (Tobias et al., 2014)
Results from Implementing and Modifying Our Task Design Sequence with a Fraction Comparison Task (Thanheiser et al., 2016)
Results from Our Fraction Comparison Task Modification (Olanoff et al., 2016; Tobias et al., under review)

4. Task Design Cycle (Tobias et al., 2014)
1. Selecting a children’s task
2. Modifying the children’s task for teachers
3. Implementing the task and collecting data from the implementation
4. Analyzing the data from the implementation
5. Reflecting on the implementation
6. Re-designing the task based on the reflection
Repeat 3-6 as needed
We wanted to see if our task design cycle was valid across contexts, thus used this to develop a task based on whole number multiplication.

5. Second Task Sequence: Array/Area Model for Whole Number Multiplication
Why multiplication?
PSTs struggle to make connections across various representations of multiplication (Lo, Grant, & Flowers, 2008)
Representational fluency is the ability to make “connections among mathematical representations to deepen understanding of mathematics concepts and procedures” (NCTM, 2014).
From our proposal:
Research shows that K-8 prospective teachers (PSTs) struggle when making connections across various representations of multiplication and often over-rely on the standard multiplication algorithm without being able to clearly explain why it works (e.g., Lo, Grant, & Flowers, 2008; Eva’s TME chapter, 2014).
The ability to make “connections among mathematical representations to deepen understanding of mathematics concepts and procedures” is an important aspect of building problem-solving skills (National Council of Teachers of Mathematics, 2014, p.10).

6. Why Array/Area Models? Array models can be useful:
to portray the properties (associative, commutative, distributive)
this model can be extended from whole numbers to rational numbers.
as a connection between additive thinking (equal groups) to multiplicative thinking (area)
Goal of this slide is to motivate the array model:
Why do we care about the array model
What are some struggles students face when dealing with multiplication

7. Task Design Cycle (Tobias et al., 2014)
1. Selecting a children’s task
2. Modifying the children’s task for teachers
3. Implementing the task and collecting data from the implementation
4. Analyzing the data from the implementation
5. Reflecting on the implementation
6. Re-designing the task based on the reflection
Repeat 3-6 as needed
We wanted to see if our task design cycle was valid across contexts, thus used this to develop a task based on whole number multiplication.

8. Why Modify Children’s Tasks
Children’s tasks can be readily accessible for MTEs who may not have access to good tasks for PSTs.
At minimum, the content in the elementary curriculum is content that we want PSTs to know.
Showing PSTs that the tasks that they are working on are from tasks that they may teach in the future helps motivate them to want to learn the material.
Many PSTs believe that they know the math that they need to teach, so showing them children’s tasks and having them realize that they do not know as much mathematics as they think they do helps motivate them to learn.

9. Children’s Task (Bridges in Mathematics, Grade 3)
Because curriculum is the one resource one can go to.
Modified this task to larger numbers to elicit different strategies beyond 29 X 27

10. Mathematical learning goals for Multiplication
For PSTs to:
Conceptualize the array/area model of multiplication
Connect the basis of this model to an array of base-10 blocks
Use base-10 block models to develop the partial products multiplication algorithm
Identify the standard multiplication algorithm as a partial products algorithm
Recognize the distributive property as a driving force behind partial products algorithms
Our overall goal was to build our PSTs representational fluency in multiplication. Specifically, we set out to modify the task to help our PSTs in:
Recognizing the distributive property as a driving force behind partial products algorithms
Connecting the base 10 block models to the partial products and standard algorithms
Recognizing that the standard multiplication algorithm is a partial products algorithm
Using array models to explore the commutativity property of multiplication
As a result of engaging in our task design, we created 3 tasks to meet these goals.

11. Name
Amy has a flyer that she wants to distribute to everyone in her college. She walks into the campus mail room and sees the rows of mailboxes. She knows that she cannot count each mailbox because she needs to get to a class. She must figure out a way to determine how many mailboxes there are so she knows how many copies of her flyer she needs. Help Amy figure out how many mailboxes there are using the picture below. See if you can do this in at least 3 different ways. Use each picture to represent only one way.
Task 1 v.1

12. 2. Does this model represent the mailbox problem. If so, how
2. Does this model represent the mailbox problem? If so, how? Please provide your reasoning.
Reasoning:
Task 2 v.1

13. 3. Use the base ten block model in 3 different ways to find how many mailboxes there are total. Write a symbolic representation that matches what you did with the model.
Solution 1
Task 3 v.1
Symbolic Representation

14. Task Design Cycle (Tobias et al., 2014)
1. Selecting a children’s task
2. Modifying the children’s task for teachers
3. Implementing the task and collecting data from the implementation
4. Analyzing the data from the implementation
5. Reflecting on the implementation
6. Re-designing the task based on the reflection
Repeat 3-6 as needed
We developed a multiplication task and implemented the task in Spring We did steps 4-6 using anecdotal evidence from the instructors’ classroom observations and analysis of student work to reflect and modify the task and re-implement it in Spring 2019, Fall 2019, and (upcoming) Spring 2020.

15. Task 1
Name
Amy has a flyer that she wants to distribute to everyone in her college. She walks into the campus mail room and sees the rows of mailboxes. She knows that she cannot count each mailbox because she needs to get to a class. She must figure out a way to determine how many mailboxes there are so she knows how many copies of her flyer she needs. Help Amy figure out how many mailboxes there are using the picture below. See if you can do this in at least 3 different ways. Use each picture to represent only one way.
Task 1 v.1

16. Data Analysis from Task Implementation: Task 1
Almost all PSTs used methods that did not involve grouping methods for two of their three strategies:
Attempting to count all 667 mailboxes
Multiplying 23 by 29
Their third solution method was where we found preservice teachers used different grouping methods to find the total.
Also, the keyholes, which looked really cool, made it difficult to write on, when PSTs did try to use a grouping strategy.

17. 49

18. 414

19. Modifications from Task Implementation: Task 1
We modified the instructions to say “without counting one-by-one.”
We asked PSTs to create four solution methods instead of three.
We removed the keys from the mailboxes
Rather than tell them not to multiply the length by the width, because we wanted them to make connections between the area and set models of multiplication. i.e., recognize that one way to solve this problem was to multiply 23 x 29, but then realize that this was not something that most (all) of them could do in their heads, so how could they make the multiplication easier for themselves?

20. Task 1
Modified
Name
Amy has a flyer that she wants to distribute to everyone in her college. She walks into the campus mail room and sees the rows of mailboxes. She has to determine how many mailboxes there are so she knows how many copies of her flyer she needs to make. Help Amy figure out how many mailboxes there are using the picture below without counting each box one by one. See if you can do this in at least 4 different ways. Use each picture below to represent only one way. For each strategy, explain how you used the picture and why what you did makes mathematical sense.
Task 1 v. 7.2

21. Task 2
2. Does this model represent the mailbox problem? If so, how? Please provide your reasoning.
Reasoning:
Task 2 v.1
Some students said no this doesn’t match b/c it’s rotated.

22. Data Analysis from Task Implementation: Task 2
Our goal for this task was for PSTs to see the connection between the rectangle in Task 1 and the rectangle created by using Base 10 blocks. E.g., we wanted them to recognize how to create multiplicative arrays with Base 10 blocks.
Their answers to this task showed they many of them did not delve into how the Base 10 block model was related to the mailboxes or multiplying 23 x 29. They merely said that since both pictures showed 667, then the Base 10 representation was appropriate.
Some PSTs said no the representation doesn’t match because the representation is rotated.

23. Modifications from Task Implementation: Task 2
Completely changed the task to ask PSTs to try to create the Base 10 block model themselves
First with 6 flats, 6 rods, and 7 small cubes (just focusing on the 667)
Next with the smallest number of pieces to replicate 29 x 23

24. Task 2
Modified
Name
2. a) A student said that you can create a replica of the 29 x 23 grid with 6 flats, 6 rods, and 7 small cubes. Is this student correct? Why or why not? Support your reasoning by drawing the mailbox grid with these given pieces below. Use the given graph paper, if needed.
Drawing
Reasoning
Task 2 v. 7.2

25. Task 2 Modified Task 2 Modified
Name
b) What would be the smallest number of base ten pieces that you can use to replicate the 29x23 rectangular grid? Draw a picture of your representation below and provide your reasoning for how you know this is the smallest number of pieces. Use the given graph paper, if needed.
Drawing
Task 2
Modified
Reasoning

26. Task 3
3. Use the base ten block model in 3 different ways to find how many mailboxes there are total. Write a symbolic representation that matches what you did with the model.
Solution 1
Task 3 v.1
Symbolic Representation

27. Data Analysis from Task Implementation: Task 3
Our goal in this task was for PSTs to relate symbolic representations with pictures of arrays. Specifically we wanted them to break the arrays into useful pictures and relate their pictures to a symbolic representation of the distributive property.
This did not happen. Some PSTs did not know what we meant by the instructions, and even those who broke the array up in a useful way wrote symbolic representation that had no connection to 29 x 23.

28. Student 11
Task 1 v.1 is rotated. Jennifer gave the first version then we modified it for Neet’s class. This student work is from Neet which is why it’s rotated from what we present as Task 1 v.1

29. Modifications from Task Implementation: Task 3
We gave PSTs an already “broken” base ten grid and asked them to represent the broken grid symbolically.
Gave PSTs a symbolic representation and asked where it was represented in the base ten block representation.

30. Task 3 Modified Task 2 Modified Task 2 Modified Task 3 v.5
Name
3 a) Katia decided to break up the base ten block model in the following way:
Task 2
Modified
Task 3 v.5
Write a symbolic representation that shows the multiplication problem in the way that Katia broke up the picture.

31. Task 3
Modified
Name
3. b) Marcus solved the mailbox problem by multiplying 29  23 using the traditional algorithm: 2 9 x 3 8 7 + 5 6
He wants to use the picture below to show how the base ten block model relates to the standard algorithm. Indicate in the picture below where the different components of Marcus’s solution are in the model. (Hint: You may need to figure out how the standard algorithm actually works first.)
Task 3 v.5

32. Data Analysis from Task Modifications: Task 3
PSTs still focused on the discrete representation rather than continuous. This meant that they had trouble relating the symbols to what they had in their pictures.

34. Modifications from Modifications: Task 3
We switched the order of Marcus and Katia so that Katia was presented first.
Additionally, we changed Katia’s problem so that preservice teachers were given Katia’s numerical work and asked where that method was represented in the base ten block representation rather than vice versa.

35. Task 3 Modified Again Task 3 v. 7.2
Name
3. a) Katia solved the mailbox problem by multiplying 29  23 using the partial products multiplication algorithm 2 9 x 3 4 6 1 8 + 7
i). Figure out where the 400, 60, 180 and 27 are coming from.
ii). Show the 400, 60, 180 and 27 from the algorithm in the picture below.
Task 3 v. 7.2

36. Task 3 Modified Again Task 3 v. 7.2
Name
3. b) Marcus solved the mailbox problem by multiplying 29  23 using the traditional multiplication algorithm: 2 9 x 3 8 7 + 5 6
Figure out where 87 and 580 are coming from.
Show the 87 and 580 from the algorithm in the picture below.
Task 3 v. 7.2

37. PSTs’ Thinking
In our earlier task versions we found that PSTs largely relied on counting strategies and using discrete objects.
By modifying problems and directions we were able to shift their thinking from additive (discrete) strategies to multiplicative (continuous) strategies.