1. USOE Professional Learning Series
Principles to Actions
Find a Half: I can determine what is half or not half and explain my thinking.
This module was developed by Carrie Ziegler, Nathan Auck, and Steve Jackson. They are the three principle designers of the course, Principles to Actions, in the UEN and Utah State Office of Education Professional Learning Series.

2. Overview of the Task Solve and discuss the Find a Half Task.
Connect the task to the Effective Mathematics Teaching Practices.
Watch a video clip of a third grade teacher facilitating small group work using a similar task.
Connect specific teacher actions seen in the video to the Effective Mathematics Teaching Practices.

3. Find a Half Task
Directions: Which of the following cards show a half? Sort them into two groups: those that show a half, and those that do not show a half. As you place the cards into each group record your thinking: Why is this card a half? Or: Why isn't this card showing a half?
Adapted from Treacy, K., & Cairnduff, J. (2009). Revealing what students think: Diagnostic tasks for fractional numbers (pp ). Ascot, W.A.: STEPS Professional Development.

4. Find a Half Task Independently sort and record your thinking.
Partner Talk: Compare your sorts. Are there any differences? What fraction concepts did you use to determine which is half or not half? What ideas did you grapple with during this task?
Make a list of concepts you used to determine half.

5. Find a Half Task Student Thinking
What understanding/knowledge is the student using to determine half?
What misconceptions/misunderstandings (unfinished learning) does the student have about the concept of a half?
What instruction does each need to support and extend their learning about the concept of a half?

6. Find a Half Task
What is Jak's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

7. Find a Half Task
Jak
What is Jak's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

8. Find a Half Task
What is Tia's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

9. Find a Half Task
Tia
What is Tia's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

10. What instruction does he need to support and extend his learning?
Find a Half Task
What is Khalia's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

11. Find a Half Task
Khaila
What is Khalia's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

12. What instruction does he need to support and extend his learning?
Find a Half Task
What is Newton's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

13. Find a Half Task
Newton
What is Newton's understanding and/or misunderstanding of the concept of a half?
What instruction does he need to support and extend his learning?

14. MTSS: Multi-Tiered Systems of Support
How would you address the learning needs of these students in a classroom? What support needs to be in place to see that their needs are met?

15. Ms. Brooks
Third Grade Classroom

16. The Half of a Whole Task
Identify all of the figures that have one half shaded. Be prepared to explain how you know that one half of the figure is shaded. Write a written description giving your reason why a figure is showing halves. If a figure does not show one half shaded explain why the figure is not showing halves.
Handouts
2-Task-HalfoftheWhole-ES-Brooks.pdf
4-Article-Watanabe-ES-Brooks.pdf (optional)
Facilitator Materials
3-SampleSolutions-ES-Brooks.pdf
This reference document provides descriptions of different strategies that participants and their students may use as they explore the task.
Facilitation Suggestion
Ask participants to individually work through the Half of a Whole Task.
Then in small groups, participants should compare their justifications for why a figure is or is not showing halves.
Optional Professional Learning Extension
Depending on your goals for the professional learning session, and if time permits, you might ask participants to read the article, “Ben’s Understanding of One-Half” by Tad Watanabe. The article describes a second-grader’s thinking about the Half of a Whole Task and several other tasks that are intended to uncover students’ conceptions of one-half.
(Adapted from Watanabe, 1996, p. 461)

17. Context of the Video Clip
Teacher: Millie Brooks Grade: 3 School: #26 District: Paterson Public Schools This lesson takes place in February. Students have just begun their study of fractions. After reading the instructions aloud, Ms. Brooks asked students to work on the task in small groups. The clip shows the interactions that Ms. Brooks had with one of the small groups, in which there was a disagreement about Figure (d).
Facilitation Suggestions
Summarize the context of the video. Then to establish a norm for the viewing and discussion of the video, remind participants that the teacher was willing to share her classroom instruction so that we could use the video to analyze and discuss the teaching of mathematics. It is important to be respectful of her work. We also need to remember that we are only seeing a “snapshot” of her interactions with the students.

18. Lens for Watching the Video
How does Ms. Brooks use and elicit student thinking?
How do the teacher and students use and connect mathematical representations?
What other mathematics teaching practices did you notice in the video?
What was the teacher doing that led you to identify the teaching practice?
Facilitation Suggestion
Summarize the lens for watching this initial viewing of the video.
Show the video.
Things that participants might notice:
Teacher asks students to prove their claim or to prove someone else’s claim wrong (lines 2-8).
Teacher revoices students’ ideas (lines 13-15, 35-37).
Teacher encourages students to explore using appropriate tools (scissors) (line 16).
Teacher holds students accountable. For example:
– Teacher leaves the small group with something to do or think about, and when she comes back to that group, she checks in with what they did (lines 17-18)
– Teacher makes sure that students are listening to each other, by asking students to explain what others in the group are thinking (line 21)
– Teacher holds the entire small group accountable for understanding – that is, it’s not ok if only some group members understand (lines 19, 41-42, 44, 55)
Some students in the group are not sure that Figure (d) shows one-half, because even though there are the same number of shaded pieces as unshaded pieces in Figure (d), there are 3 shaded pieces rather than 1 shaded piece.

19. Elicit and Use Evidence
of Student Thinking
Evidence should:
Provide a window into students’ thinking;
Help the teacher determine the extent to which students are reaching the math learning goals; and
Be used to make instructional decisions during the lesson and to prepare for subsequent lessons.
Formative assessment is an essentially interactive process, in which the teacher can find out whether what has been taught has been learned, and if not, to do something about it. Day-to-day formative assessment is one of the most powerful ways of improving learning in the mathematics classroom.
Wiliam, 2007, pp. 1054; 1091
Decisions about instruction should be based on what the students know and understand about mathematics. In order to determine what they know and understand, you have to make their thinking visible.
We could have called this formative assessment – that is what it is – but we think the use of the word assessment suggests a more formal process that obscures the focus on student thinking.

20. Mathematical Representations
Use and Connect
Mathematical Representations
Different Representations should:
Be introduced, discussed, and connected;
Focus students’ attention on the structure or essential features of mathematical ideas; and
Support students’ ability to justify and explain their reasoning.
Strengthening the ability to move between and among these representations improves the growth of children’s concepts.
Lesh, Post, Behr, 1987
The key here isn’t just being able to work in each representation but to be able to make connections between them with different starting points.

21. Mathematical Representations of a Half
The key here isn’t just being able to work in each representation but to be able to make connections between them with different starting points.

22. Effective Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.