Teaching and Learning

# A Productive Math Struggle: Questions and Assessment

Jenni Harding, University of Northern Colorado, Greeley, CO

Students need time to rationalize mathematics, play with it, and make it their own. Through a productive math struggle, teachers can create opportunities for students to conceptualize mathematics. Hiebert and Grows (2007) define a productive struggle this way: “students expend effort in order to make sense of mathematics, to figure something out that is not immediately apparent” (p. 387). Students can contemplate mathematical ideas, take risks, justify their own thinking, and evaluate other student ideas. Creating this environment in the K-12 classroom takes time and practice to establish. The purpose of this article is to focus on productive math struggle questions to enrich learning and guide assessment.

A productive math struggle can take place when students are answering problem solving questions within groups. In a productive math struggle, teachers focus students’ attention on ideas and sense-making and develop their math confidence. Also, teachers allow entry and exit points for a wide range of students, provide extensions and elaborations, and make connections between ideas, concepts, strategies, and procedures. Teaching responsibilities specific to a productive struggle include selecting high-quality tasks (some examples: Illuminations https://illuminations.nctm.org, YouCubed https://www.youcubed.org/tasks/, or Exemplars https://www.exemplars.com), asking high-level math questions, allowing students to verify and relate their strategies, listening to student responses, examining their work to extend and formalize their thinking, and providing targeted feedback. This structure of learning puts several of the Standards for Mathematical Practice into action, including: make sense of problems and persevere in solving them, construct viable arguments and critique the reasoning of others, use appropriate tools strategically, and attend to precision (Common Core State Standards Initiative, 2010).

I organized this article with questions and a rubric to support the structure of a productive math struggle. Teachers may use the first set of questions to support and challenge students’ understanding of mathematics content as they move between groups of students. The second set of questions guide student discussion, giving concrete stems to help students share their mathematics ideas and critique the ideas of others. The third set of questions model how to have students reflect upon their learning. Finally, the rubric demonstrates a specific way to evaluate students during a productive math struggle.

The teacher can extend learning by deciding what type of guidance is needed for students and then using their questions to scaffold mathematics thinking. Teachers can ask students questions to clarify students’ ideas, emphasize reasoning, and encourage student-to-student dialogue.

Teacher Questions: For a Productive Math Struggle

 Clarify Students' Ideas Did you use the red trapezoid as your whole? What parts of your drawing/diagram/web relate to the problem? Who could share what Julia just said, using your own words? Emphasize Reasoning Why does it make sense to start with these particular numbers? Can you give me an example? What connections do you see between Sara’s idea and Sam’s idea? Encourage Student-Student Dialogue Who has a question for Juan? Turn to your partner and explain why you agree or disagree with Shelly. Talk with Scott about how your strategy relates to his.

During a productive math struggle, teachers can give question stems to guide student discussion during math conversations. These question stems give structure to promote students’ participation. Because the stems help everyone to share their ideas, this can help more students to be heard, regardless of their background or status.

I suggest introducing one or two discussion stems each day that students participate in a productive struggle. Teachers can create a classroom anchor chart to hang in the room for student reference. Here are some sample math discussion stems:

• Explain why/how…

• What would happen if _____?

• How could _____ be used to _____?

• Why is _____ important?

• Did anyone think of this in a different way?

• Describe _____ in your own words.

• What are you thinking now?

• I agree/disagree with _____ because…

• That is good thinking because…

• I got different results because…

• My strategy is like yours because…

• My strategy is different than yours because…

• What I hear you say was…

By posing reflection questions for summative assessment, teachers can focus on the learning happening during the productive math struggle. This can exist in the form of math journals, exit tickets out the door, student self-evaluation, or group discussions to have students evaluate their own learning. Here are some sample math reflection questions:

• What were the main concepts or ideas you learned today?

• What questions do you have about ____?  If you don’t have a question, write a similar problem and answer it.

• Describe a mistake that you or a classmate had in class today. What did you learn from this mistake?

• How did your group approach today’s question? Was your approach successful?

What teachers assess and grade in their classroom demonstrates to students what is valued. For example, if teachers just grade homework, quizzes, and tests, it tells your students you only value formal assessments. I recommend grading students during a productive struggle session with a rubric to demonstrate the value of mathematics conversations happening in the classroom. This rubric gives students explicit guidance about what is expected of them during the productive struggle math group time. A group grade can demonstrate that conversation and understanding of mathematics is more important than the math answer. Teachers may use the rubric below by placing a tally mark each time they observe a group exhibiting one of the behaviors.

Productive Struggle Evaluation Rubric

 Group A Group B Group C Group D Leaning in and working in the middle of the table Equal air time (everyone takes a turn talking) Sticking together discussing each problem before going to the next one Explaining how they solved a task with justification and/or reasoning Listening to each other when someone is talking Asking each other questions to clarify and understand Providing solutions using multiple strategies Students persevere, persist, and don’t give up Following group roles or jobs Students encourage each other Overall Rating:____________

After teachers have all of the tallies recorded, they can give an overall group grade or they may use it as an informal assessment for that day's productive math struggle.  I use a 0-3 scale as an overall rating for all of the items on the rubric:

0 = No evidence during the observation
1 = A few isolated instances of evidence being observed (only a few items attempted; 5 out of 10 with at least one tally mark)
2 = Some evidence observed but does not seem frequent (many items attempted; 6 or more with multiple tally marks)
3 = Strong and frequent evidence observed; is regularly present (multiple tally marks in each)

Through questions, discussion stems, reflective questions, and assessment rubrics, teachers can create conditions for a productive math struggle. Through group conversations, students can delve deeper into mathematics. As a result, classroom math discussions can become more vibrant.

## References

Common Core State Standards Initiative. (2010). Standards for mathematical practice. Retrieved from http://www.corestandards.org/Math/Practice

Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F.K. Lester (Ed.), Second handbook of research on mathematics teaching and learning. (pp. 371-404). Charlotte, NC: Information Age.

## Credits

Author: Jenni Harding is a Professor of Education at the University of Northern Colorado. She teaches courses in education research, mathematics methods, and teaching to graduate and undergraduate students. Her students learn through productive learning struggles where they grow intellectually through multiple viewpoints of content, ideas, and concepts.

# Adapting an Instructional Routine: Stronger & Clearer Each Time

Dylan Kane, High Mountain Institute, Leadville, CO

Classroom routines can form a foundation for everyday teaching and learning (Kelemanik, Lucenta, & Creighton, 2016). Routines help to get a class started, to distribute materials, or to transition between activities. With routines, students can think less about what to do next or how to participate in an activity, which shifts the focus for students and teachers to mathematical ideas. Stronger & Clearer Each Time (S&C) is an instructional routine designed to support both mathematical understanding and language development (Zwiers, Dieckmann, Rutherford-Quach, Daro, Skarin, Weiss, & Malamut, 2017).

As a classroom teacher, I have adapted the S&C routine to fit my goals and better support my students’ learning. I adapt S&C by asking students to write notes summarizing the ideas of a discussion, often with the prompt to write as if the notes are for a future forgetful self. I use this framing to build on the idea of “meaningful notes” (Liljedahl, 2018), where I structure note-taking as an opportunity to occasion student thinking and create useful resources rather than copying information without thinking.

After giving students a few minutes to write, I use a spreadsheet randomizer (example) to put them into pairs. Then, I ask students to take turns sharing their notes with their partner, adding ideas to their own notes. I then randomize students a second time, and they share again with a new partner to allow students a second chance to share their ideas and hear a different perspective. An alternative is to ask students to talk to a shoulder partner, and then a partner in front of or behind them. Finally, students take a minute to summarize and clarify any additional ideas they want to have in their notes. I often finish by inviting students to share key ideas with the full class.

In a lesson on finding sums of arithmetic series, I asked students to take notes for their future forgetful self and gave them several minutes to write. When I assigned students partners to share and revise their notes, I walked around listening to their conversations. I noticed a few pairs talking about how when there are $$n$$ terms in a sequence, there are $$n-1$$ “jumps.” That is, if there are $$5$$ terms, the distance from the first to the last term is $$4$$ times the distance between consecutive terms. For instance, in the series $$11 + 13 + 15 + 17 + 19$$, there are five terms and four jumps. The total distance from the first to the last term is $$8$$, and the “jump” is $$2$$. This can help students to quickly calculate either the number of terms in a sequence or to calculate the final term given the number of terms.

The opportunity to listen in to students’ self-explanations allowed me to address a potential source of confusion in the moment. I asked one group to share their thinking with the class before putting students into new groups to summarize what they heard with a new partner. I encouraged students to add a similar example to their notes when they made their final revision to solidify their understanding of a challenging idea.

The adaptation of the S&C routine affords several opportunities. First, students have to self-generate explanations around key ideas and how they might use them to solve a problem. Self-explanations can help students integrate what they are learning with what they already know and support the transfer of their knowledge in the future (Lombrozo, 2006). By asking students to write notes to their future forgetful self, I hope to prompt them to go beyond summarizing what they learned, and consider how those principles might transfer to new problems. Giving students multiple opportunities to discuss with partners and revise their work increases the quality of their explanations, exposes students to more models of others’ high-quality explanations, and allows them to clarify misconceptions in the moment.

A second benefit of S&C is to teach students that revision is an essential part of learning and that students’ ideas are always valued, especially when ideas are partially formed. Jansen, Cooper, Vascellaro, & Wandless (2016) write:

If rough-draft talk is valued, brainstormed ideas are welcomed. More students are likely to take risks rather than freeze during challenging tasks. Valuing a wider range of contributions invites greater involvement, in contrast to the same students who participate frequently or not at all (p. 304).

S&C enacts rough draft thinking in practice. The teacher goes beyond telling students that rough drafts are valued; the S&C routine shows students the role rough drafts play in their learning by creating a resource through revision that is more likely to be useful to them.

Finally, S&C allows me to elicit evidence of student thinking. By listening in to conversations or looking at notes with revision, I can learn how students are thinking about a new idea and respond to that thinking. Transitions between partners can serve as an opportunity to address common issues, direct student attention to important features of a concept, or prompt students to add something to their notes and conversations. As I listen in to conversations and read student notes, I learn to understand content from a student perspective and see the many ways that students think about mathematical ideas differently than I expect. These opportunities help me to become a more effective teacher.

The S&C routine embodies the practice of mathematicians; rarely are initial ideas polished, and mathematicians revise their work and improve their explanations over time. I love this routine because note-taking can be a low-energy part of class, but with S&C students create useful resources while they rehearse and consolidate their thinking as they share and revise their explanations. My classroom becomes more equitable as I build in time and space for all students to reflect on their learning.

## References

Jansen, A., Cooper, B., Vascellaro, S., & Wandless, P. (2016). Rough-draft talk in mathematics classrooms. Mathematics Teaching in the Middle School, 22(5) 304-307.

Kelemanik, G., Lucenta, A., & Creighton, S. J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann.

Liljedahl, P. (2018, November). Building thinking classrooms. Presentation at the Regional Conference of the National Council of Teachers of Mathematics, Seattle, WA. Retrieved from:  http://www.peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-NCTM-Seattle-no-video.pptx

Lombrozo, T. (2006). The structure and function of explanations. TRENDS in Cognitive Science, 10(10), 464-470. https://doi.org/10.1016/j.tics.2006.08.004

Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the design of mathematics curricula: Promoting language and content development. Retrieved from Stanford University, UL/SCALE website: http://ell.stanford.edu/content/mathematics-resources-additional-resources

## Credits

Author: Dylan Kane is a National Board Certified high school math teacher in Leadville, Colorado. He blogs often about the intersections of research and practice in math education, and he is on the leadership team of the Teacher Leadership Program at the Park City Mathematics Institute.

# From soliciting answers to eliciting reasoning: Questioning our questions in digital math tasks

Heather Lynn Johnson, Gary Olson, Amber Gardner, Amy Smith

Students can interact with digital math tasks in different locations, on different devices, and for different purposes. What kinds of questions do students encounter when interacting with digital math tasks? And why might the kinds of questions matter?

We designed digital math tasks to provide opportunities for students to engage in math reasoning. Questions are a key component of the tasks. With our questions, our goal was to do more than solicit students’ answers. We intended to elicit students’ reasoning.

We share a digital math task and a question from the task. Then we provide three design principles guiding our questions.

We developed the The Toy Car task in collaboration with Dan Meyer and the Desmos team. The task begins with a video of a toy car moving along a curved path (Figure 1). Then students investigate and graph relationships between a toy car’s distance from a shrub, and its total distance traveled.

Figure 1. The toy car and the shrub

The Toy Car task is part of a group of tasks that we call Techtivities. The Techtivities include video animations and dynamically linked, interactive graphs. Students have opportunities to sketch different graphs to represent the same relationship between attributes. Then students reflect on what those graphs represent. To learn more about the Techtivities, see Johnson (2018).

In the Toy Car task, students sketch, then reflect on two different graphs, shown in Figure 2. Each graph represents the toy car’s total distance traveled as a function of the toy car’s distance from the shrub.

Figure 2. Two different graphs in the Toy Car task

Students might wonder how it is possible for two different looking graphs to represent the same function relationship. Furthermore, students might notice that the graph shown at right in Figure 2 does not pass the vertical line test, meaning that a vertical line would intersect the graph at more than one point.

Students can apply the vertical line test based solely on the shape of a graph, and they may miss how graphs can represent relationships between attributes in a situation (Moore, Silverman, Paoletti, & LaForest, 2014). In the Toy Car task, and across the Techtivities, our goal was for students to focus on relationships between different attributes in the situations. We worked to design questions that could help us to achieve our goals.

## A Question

We posed this question in the Toy Car task: Val says that both of these graphs represent the toy car’s total distance traveled as a function of the toy car’s distance from the shrub. Do you agree or disagree? Why or why not? (Graphs are shown in Figure 2.)

We purposefully posed this question as person’s (Val’s) claim, rather than as a claim devoid of human connection. Furthermore, we used precise language to clarify Val’s claim. In particular, we used the phrase as a function of, rather than the more general term, function. We did this so that Val’s claim focused on the function relationship that the graphs represented. Overall, we aimed to position Val as a capable doer of mathematics, who made a claim worthy of consideration.

## Three Design Principles

1. Provide opportunities for students to consider other students’ claims. Mathematics is a human endeavor (Freudenthal, 1973). In our questions, we decided to have students respond to another student’s claim. We could have asked students: Do both graphs represent the toy car’s total distance traveled as a function of the toy car’s distance from the shrub? By framing our questions as a response to another student, we aimed to humanize students’ interactions with the digital math tasks.

2. Allow for gender ambiguity when incorporating student names into task questions. Students can think that gender identity plays a role in mathematical ability (Boaler, 2002; Leyva, 2017; Rubel, 2016). In our questions, we aimed to use gender ambiguous names, and names we selected were often informal. We could have used a pronoun to assign a gender identity to Val, or selected a more gendered name. Instead, we intended to open possibilities for students to use a variety of pronouns, or no pronouns at all, when responding to the student claims given in the tasks.

3. Elicit sense making, rather than soliciting judgments of correct/incorrect. To promote students’ reasoning, we posed questions to elicit sense making rather than solicit judgments. We could have asked students if Val was right or wrong. Instead of asking students to judge another student’s claim as correct/incorrect, we chose to ask students to explain why they agreed or disagreed. We intended to offer students opportunities to consider possibilities, rather than rushing to judgments.

## Closing Remarks

Doing mathematics is so much more than finding answers. With our questions, we can work to create spaces for students to engage in reasoning and sense making. In designing questions for our digital math tasks, we are aiming to do just that.

Acknowledgments. This work was supported by a grant from the National Science Foundation (DUE-1709903). Opinions, findings and conclusions are those of the authors. We thank Dan Meyer and the team at Desmos for their work with us. We are grateful to our colleagues who provided feedback to help us to grow.

## References

Boaler, J. (2002). Paying the price for “sugar and spice”: Shifting the analytical lens in equity research. Mathematical Thinking and Learning, 4(2-3), 127–144.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel Publishing.

Johnson, H. L. (2018). Helping students see how graphs work | Edutopia. Retrieved July 30, 2018, from https://www.edutopia.org/article/helping-students-see-how-graphs-work

Leyva, L. A. (2017). Unpacking the male superiority myth and masculinization of mathematics at the intersections: A review of research on gender in mathematics education. Journal for Research in Mathematics Education, 48(4), 397–452.

Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157.

Rubel, L. H. (2016). Speaking up and speaking out about gender in mathematics. The Mathematics Teacher, 109(6), 434–439.