Discussion-Worthy Tasks for Upper Elementary Math Groups

            In this Mathematics Teacher article, Nicola Hodkowski (Digital Promise) and Carolyn Carhart-Quezada (Cignition) say students learn more about mathematics when they discuss with classmates, connecting what they know with formal math content. To spark meaningful discussions, say the authors, teachers need to pose open tasks – that is, questions with multiple entry points, varied solution strategies, and more than one right answer. Well-chosen tasks allow students to “discuss, argue, represent, hear, and compare one another’s viewpoints.” 

            Hodkowski and Carhart-Quezada came up with a series of open tasks to help below-level fourth and fifth graders better understand fractional reasoning. Working on the tasks online in groups of four, facilitated by a tutor, students made significant progress. “By prioritizing conceptual understanding and promoting mathematical discourse,” say the authors, “we were able to both change traditional tutoring and empower our students to see themselves as doers and sense-makers of mathematics.” 

            Hodkowski and Carhart-Quezada developed five types of open tasks for their program and thought about where each type was most helpful: 

• Multiple strategies – More than one strategy can be used, but the solution or answer is the same – for example: The answer is ½. What is the question? 
• Multiple outcomes – Solution strategies may be similar, but solutions are different – for example: Add something to ¾ that sums to a number close to 1 but not exactly 1. Who is closest? How do we know? 
• Sorting and ordering – Students are asked to invent a sorting criterion, with different criteria leading to different sorting – for example: With the following fractions, quickly decide if they are bigger or smaller than ½. How did you decide? 3/11, ¼, 5/6, 7/8, 7/10, 6/12 
• Justification – Students need to explain and justify their answers to groupmates – for example: Ana says that 1/8 is bigger than 1/3 because 8 is bigger than 3. Margo thinks that Ana is not right. Who do you agree with? Explain your reasoning. 
• Group challenge tasks – Students work in pairs and discuss outcomes with another pair – for example: 6 people are going to share these 5 candy bars equally [show 5 rectangles]. Write a fraction that shows how much one person gets. 

            Students’ prior knowledge and readiness determine which type of problem will spark the best discussions. With students who are new to a concept, multiple-strategy tasks seem more engaging, allowing for more exploration as the group works toward a single solution. With students who had already been introduced to a concept and are working to deepen their understanding, a multiple-outcome task is more appropriate, with students discussing how they came up with different answers.                Hodkowski and Carhart-Quezada noticed that at first, students talked less and the tutors talked more – the opposite of what was intended. Why? Students weren’t used to having discussion in math classes and lacked confidence in their insights. “We quickly realized,” say the authors, “that rituals, routines, and collaborative norms were needed when using open tasks.” The tutors were prompted to establish these norms: 

• Explain why, not just the right answer. 
• Show active listening. 
• Compare and critique with groupmates. 
• Summarize what you’ve learned. 

Tutors supported these norms by saying, “Yes, and…” rather than immediately telling the answer and nudging students for “deeper noticings and wonderings about others’ work.” 

            But in some groups, setting norms was not enough. Over time, Hodkowski and Carhart-Quezada worked with tutors to develop several other ways to jumpstart discussions: 

• Individual think time before sharing in the group 
• Giving students 20-30 seconds to ponder the problem before beginning the group discussion.
• After presenting the problem, the tutor divides a slide into four sections, one for each student; students solve the problem on their section and then look at their groupmates’ solutions and discuss. 
• Students think about the task alone, then work together on solving it on one slide, then the teacher chooses one student to explain their solution, discuss, and decide on the correct solution. 
• Taco talk – Students are assigned to be tomato, lettuce, cheese, and taco shell, the first three present their ideas in that order, then the fourth student (taco shell) wraps up the discussion by explaining what their groupmates said. 

 “Let’s Give Them Something to Talk About” by Nicola Hodkowski and Carolyn Carhart-Quezada in Mathematics Teacher: Learning & Teaching PK-12, November 2023 (Vol. 116, #11, pp. 837-844); the authors can be reached at nhodkowski@digitalpromise.org and carhart-quezada@cignition.com.

Please Note: This summary is reprinted with permission from issue #1012 of The Marshall Memo, an excellent resource for educators.