2 of 56

Project MathTalk

Videos used as examples

in these materials come from

www.mathtalk.org

3 of 56

Introduction of Theme: ��Analyzing Student Thinking

Project MathTalk PD Materials

4 of 56

Analyzing Student Thinking

Project MathTalk PD Materials

Students come to our classrooms with rich prior knowledge, ideas, and interpretations.

Our goal is to uncover these insights rather than just focusing on their ability to follow procedures.

5 of 56

Analyzing Student Thinking

Project MathTalk PD Materials

In order to see the full picture of student thinking, students need opportunities to share their interpretations of mathematical ideas, including their mental images and conceptions.

6 of 56

Analyzing Student Thinking

Project MathTalk PD Materials

Once students have shared their ideas, as teachers, it is important to then assess their thinking.

Formative assessment allows us to address misconceptions and guide students towards a deeper understanding of mathematical concepts in the moment.

It’s not just about assessing what they know or whether they can produce a correct answer, but about how they think and arrive at their conclusions.

7 of 56

Analyzing Student Thinking

Next, we will explore 3 examples of analyzing student thinking, by viewing and discussing videos from Project MathTalk.

Project MathTalk PD Materials

8 of 56

Example 1

Project MathTalk PD Materials

9 of 56

Grade 9 students

Video Participants

Video:

We will watch videos of two students as they reason about algebraic expressions and equations

Haleemah

Elijah (ET)

10 of 56

General Characteristics of MathTalk Videos

Project MathTalk PD Materials

MathTalk Videos feature pairs of students who

engage in dialogue
convey authentic confusion
persist to resolve mathematical struggles

11 of 56

Background for the Video Example

In the video we will watch, Haleemah and ET work on the Pool Task.
The task asked them to find the number of tiles in the border of a pool with a 10 tiles on a side.
This task is part of an instructional sequence in which students use the visual representation of the pool to create an arithmetic method and then generalize the method using algebra.

Project MathTalk PD Materials

How many tiles are there in the border of a pool with 10 tiles on a side?

12 of 56

Background for the Video Example

We will watch the video in two parts.
The first part will show Haleemah’s and ET’s initial approach to finding the number of tiles in the border.
For the second part, we will focus on the teaching actions that serve to shift their thinking.

Project MathTalk PD Materials

13 of 56

Before you watch the 2.5 minute video

Think about how you would find the number of tiles on the border of a pool with 10 tiles on a side.
Share various methods for finding the total number of tiles in the border with your group.
Then watch Haleemah and ET.

Project MathTalk PD Materials

14 of 56

Video, www.mathtalk.org, Algebraic Expressions Unit, Lesson 1, Episode 1, 0:58 – 1:30

15 of 56

Discussion

What’s their idea? Why do you think it’s reasonable to them?

Project MathTalk PD Materials

16 of 56

Discussion

What follow-up questions might you pose to Haleemah and ET?

Project MathTalk PD Materials

17 of 56

Teacher Actions that Support Student Thinking

We will watch the second part of the video.
As you watch the 2 minute video, notice the moves the teacher makes that guides the students to a more productive way of reasoning.

Project MathTalk PD Materials

18 of 56

Video, www.mathtalk.org, Algebraic Expressions Unit, Lesson 1, Episode 1, 1:38 – 3:37

19 of 56

Discussion

How did the teacher’s actions help promote a shift in Haleemah and ET’s thinking?

Project MathTalk PD Materials

20 of 56

Example 2

Project MathTalk PD Materials

21 of 56

Comparing a Variety of Mathematical Conceptions Can Lead to Deeper Meaning

When exploring complex mathematical ideas, there are often multiple ideas that come up that are valid.
Encouraging students to compare these ideas reinforces that there is rarely one “right” way to think about a problem.

Project MathTalk PD Materials

22 of 56

Comparing a Variety of Mathematical Conceptions Can Lead to Deeper Meaning

Students may also share these ideas without knowing whether they are all correct.
Discussing and comparing the variety of conceptions can be used as an opportunity to deepen classroom discussion.

Project MathTalk PD Materials

23 of 56

Background for Example 2

Prior to this video, Haleemah and ET have applied a method for finding the number of tiles in the border of a square pool of different sizes.

24 of 56

Background for Example 2

They generalized their method and wrote an algebraic equation that showed the relationship between the number of tiles on one side (x) and the total number of tiles in the border (B).

Project MathTalk PD Materials

25 of 56

Background for Example 2

In the following episode, they work to explain what x in their equation means in the pool context.

Project MathTalk PD Materials

26 of 56

As you watch the 3 minute video

Think about the following questions:

How is ET thinking about the meaning of x?

How is Haleemah thinking about the meaning of x?

Project MathTalk PD Materials

27 of 56

Video, www.mathtalk.org, Algebraic Expressions Unit, Lesson 1, Episode 4 (2:55 – 4:54)

28 of 56

Discussion

How was Haleemah thinking about x?

How was ET thinking about x?

Is there a way to see both as correct mathematically?

Project MathTalk PD Materials

29 of 56

Activity

What benefit do you see for your students in comparing these two conceptions of x?��How could you orchestrate a discussion around these ideas?

Project MathTalk PD Materials

30 of 56

Example 3

Project MathTalk PD Materials

31 of 56

Assessing Students’ Mathematical Thinking

Attending to student reasoning can provide insight into what is difficult for students.

Understanding their thinking can inform how much time to allot for a mathematical idea, designs for additional tasks, or decisions about elaborating on ideas as a whole class.

Project MathTalk PD Materials

32 of 56

Background for Example 3

One topic that can be difficult for students is interpreting expressions from multiple perspectives.

We want students to be able to view algebraic expressions in two ways

Both perspectives are important for developing deeper meaning such as labeling expressions on a drawing or making sense of more complex equations.

Project MathTalk PD Materials

Process:

Substitute letters and follow as a computational recipe

Product:

When I imagine substituting numbers, the result is a number itself, which I can, in turn, operate on.

33 of 56

Background for Example 3

In the videos we will watch Haleemah and ET encounter a challenge in shifting from a process to a product perspective for their algebra equation.

Project MathTalk PD Materials

34 of 56

Background for Example 3

Project MathTalk PD Materials

35 of 56

Background for Example 3

Project MathTalk PD Materials

x tiles

Take 2 corner tiles away

36 of 56

Project MathTalk PD Materials

Background for Example 3

37 of 56

Background for Example 3

Project MathTalk PD Materials

38 of 56

Background for the Video Example

Project MathTalk PD Materials

39 of 56

As you watch the 1 minute video

Project MathTalk PD Materials

40 of 56

Video, www.mathtalk.org, Algebraic Expressions Unit, Lesson 1, Episode 4 (5:47 – 6:55)

41 of 56

Discussion

Project MathTalk PD Materials

42 of 56

Background for Example 3 Part 2

To help make sense of the algebraic equation, the teacher had Haleemah and ET test their equation when x = 8.
In the episode we will watch, they will make sense of their arithmetic equation first.
Then, they will make sense of the algebra equation again.

Project MathTalk PD Materials

43 of 56

Teacher Actions that Address Student Challenges

Project MathTalk PD Materials

44 of 56

Video, www.mathtalk.org, Algebraic Expressions Unit, Lesson 1, Episode 5 (3:08-3:55; 5:48-6:58)

45 of 56

Discussion

What was Haleemah’s resolution?

How did the teacher’s actions address Haleemah’s challenge and support her eventual resolution?

Is there anything else you would do as a teacher?

Project MathTalk PD Materials

46 of 56

Summary

Project MathTalk PD Materials

47 of 56

Summary

Analyzing student thinking can be beneficial for informing teaching practice:

Recognize that students' mistakes often come from their own logical thinking. Use these mistakes to understand their interpretations and guide them toward more productive thinking.
Use students’ variety of conceptions to develop deeper meaning. When students express multiple ideas, encourage them to elaborate and discuss their ideas as a whole class.
Listen to and informally assess students' reasoning, especially during difficulties. These moments reveal specific challenges. By identifying these issues, you can tailor your teaching actions and curricular choices to address them directly.

Project MathTalk PD Materials

48 of 56

For More Information on This Topic

Additional reading on formative assessment:

McCloskey, A. V., & Norton, A. H. (2009). Using Steffe's advanced fraction schemes. Mathematics teaching in the middle school, 15(1), 44-50.
Norton, A. H., & McCloskey, A. V. (2008). Modeling students' mathematics using Steffe's fraction schemes. Teaching Children Mathematics, 15(1), 48-54.

Project MathTalk Algebraic Expressions Unit

50 videos starring Haleemah and ET available at www.mathtalk.org, showing their journey developing meaning for algebraic expressions and equations.

Project MathTalk PD Materials

49 of 56

Project MathTalk Resources for Teachers

Project MathTalk PD Materials

50 of 56

Project MathTalk Website�

Project MathTalk PD Materials

All MathTalk videos are available at www.mathtalk.org

51 of 56

Project MathTalk PD Materials

Website contains:

8 video units

Each unit has about 40 short videos

52 of 56

Project MathTalk Website

Has a section for STUDENTS

Project MathTalk PD Materials

a section for TEACHERS

53 of 56

Resources in the Teacher Section

For every video lesson, there is:

A description of the understandings that students can develop from the lesson
An alignment of the lesson with the Common Core Math Standards and Practices, with specific examples from the videos

Project MathTalk PD Materials

54 of 56

Resources in the Teacher Section

For every episode in every lesson, there is:

An identification of conceptual challenges that the students in the video, or your students, may face
A set of questions that you can pose to your students as they view the video
Ideas for supporting dialogue in your classroom as your students discuss the video

Project MathTalk PD Materials

55 of 56

Resources in the Teacher Section

Example of a Focus Question

Project MathTalk PD Materials

[Pause the video at 2:20] Haleemah and ET created a more efficient equation that relates the cost of each game app to the total amount spent by the 3 friends.

Ask your students, “Why do you think the new expression, 9c, represents the total amount spent by the three friends?”

56 of 56

The videos used in this presentation are supported by the National Science Foundation through Awards DRL-1416789 and 1907782. The views expressed here are not necessarily those of the NSF

Project MathTalk Team Members

Dr. Joanne Lobato, PI, San Diego State University (SDSU)

Dr. John Gruver, Co-PI, Michigan Technological University (MTU)

Dr. Michael McKean, Technical Specialist, SDSU

Dr. Alicia Gonzales, Postdoctoral Scholar, MTU

Isabel White, Doctoral Student Research Assistant, SDSU & UCSD

Dr. Michael Foster, Postdoctoral Research Associate, Rochester Institute of Technology

Dr. C. David Walters, Creator of Teacher Support Materials, Weber State University