By Shamira Underwood
We’ve all been there. You’ve checked all the boxes to prepare for the perfect math lesson. You’ve chosen an amazing high-level task. You’ve arranged your classroom space to encourage dialogue and collaboration, and you’ve done the work to strategically design questions that can challenge your students to engage in sense-making. Yet, a challenge remains.
In this piece, you’ll read a story from the field and learn about some ways a teacher encouraged one particular student to apply the confidence he displayed at recess to his approach to mathematical tasks.
Story from the Field
Darnell, a 10-year-old African American male, and student at Williams Elementary is generally kind and well-liked by his peers. During recess, he is often invited to join organized play opportunities like the game of kickball. While playing, Darnell can typically be overheard celebrating successes and lamenting perceived failures, proclaiming what he will do next time to get the opposing team members “out”, or make it to the base before being tagged. He regularly persists to have his voice heard and analyzes the progress of the game willingly and confidently. He interjects his thoughts and opinions (most times unsolicited) to contribute to the development of a group understanding of how the game should be played, who should be on what team, and strategies to prevent the ball from going down the hill or over the fence again. In the least, Darnell presents himself as someone who feels capable of engaging in a game of kickball despite small failures and setbacks along the way.
Darnell’s recess persona was in stark contrast to his disposition in Math class, where he seemed to procrastinate during problem solving lessons waiting for others to present answers so he could revise his strategy to be more like theirs. He could often be seen erasing his answers in response to hearing or seeing his peer’s answers without even weighing the mathematical accuracy of their claims. During whole group discussions, he typically seems quite comfortable allowing others to dominate the conversation about the problem. During a small group problem solving lesson, Darnell and five other students were given fractions divided into three columns and were asked to name a rule to describe each column. The first column contained fractions less than one. The second column contained fractions equal to one, and the third contained fractions greater than one. Darnell finished recording his observations quickly and avoided opportunities to share what he was thinking during the teacher’s presses for explanations. He listened intently to his peers and feverishly erased what he had written on his paper to match more closely with what he was hearing. Even though what he had written could add merit to the conversation, he displayed a hesitancy to challenge his peer’s ideas and contribute in the ways that he regularly employs during free play at recess.
Consider Darnell’s disposition and how the ways he views himself might contribute to how he initiates and engages in the pursuit of success at recess despite setbacks and failures. Consider also how teachers can support students in shaping the ways in which they view themselves in problem solving situations during Mathematics class.
Consider Darnell’s disposition and how the ways he views himself might contribute to how he initiates and engages in the pursuit of success at recess despite setbacks and failures. Consider also how teachers can support students in shaping the ways in which they view themselves in problem solving situations during Mathematics class.
- What might be contributing to Darnell’s perceived confidence in the pursuit of success during free play at recess?
- In what ways might you imagine that Darnell is able to rebound from perceived failures before, during, and after free play experiences at recess?
- How can teachers support students, like Darnell, to transfer the confidence to persist (despite setbacks) during Mathematics problem solving experiences in the same way that they engage in activities of their own interest?
Connecting Real-Life to Theory
We can look to education theory and scholarship to help us identify the type of struggle that should take place in effective teaching and learning environments. In Vygotsky’s theory, he talks about the “zone of proximal development” (ZPD) which is the sweet spot where productive struggle thrives. It’s the perfect space for teachers to step in as supportive guides, encouraging students along their sense making journey. As learners work to figure things out, the synapses of the brain are making strong and long-lasting connections that lead to students having a higher ability to apply their learning to future problem solving (Kapur, 2010).
In many ways, the idea of ZPD sits at the heart of NCTM’s effective teaching practice about productive struggle. It states, “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships” (NCTM, 2014). Because productive struggle has been characterized and described in many ways and because of the vital role it plays in supporting students to create new understandings, educator understanding is critical.
Preceding the opportunities to engage students in productive struggle, effective teachers plan for and provide students with cognitively demanding tasks that allow for meaningful discussion. Students should be expending effort to solve mathematical tasks, and the teacher’s role is to pay careful attention to the balance of support they will need and provide it through questioning and quality instructional materials.
Another important layer of ensuring a learning environment has opportunities for productive struggle is the consideration for the varying confidence levels that students may present. Students’ approach to mathematics learning is often discussed in tandem with how they have been supported in developing their mathematics identities. Understanding how math identities foster a growth mindset and students’ willingness to make and embrace mistakes can support a teacher’s efforts to ensure all students feel safe and supported to engage in the beautiful mess of learning from collaboration and risk-taking.
Four Types of Struggle
Kupar writes about four types of struggle related to learning.
- Productive Success: This type of struggle refers to achieving success after a process that involved challenges, effort, and growth. The journey might have been difficult, but it contributed to deeper understanding and skill development. The success is not just about reaching a goal; it’s about learning and improving along the way.
- Productive Failure: Productive failure is when an effort doesn’t achieve the intended result, but the process leads to valuable learning and growth. Even though the outcome wasn’t successful, the experience provides insights, reveals new knowledge, or builds skills that are beneficial in the long run.
- Unproductive Success: This occurs when success is achieved without much effort, challenge, or growth. It might happen due to external factors, luck, or a process that doesn’t require much engagement or skill development. While the outcome is positive, the journey does not contribute to personal growth or learning.
Unproductive Failure: Unproductive failure is when an effort ends in failure without any significant learning or growth. The process might have lacked direction, guidance, or meaningful challenge, resulting in an outcome that’s neither successful nor informative in terms of personal development.
Intention to Support Confident Math Identities as a Matter of Equity and a Means to Support Productive Struggle
While some students may readily jump into the dialogue, cognizant of the tools necessary to take risks and stay within the productive struggle, there may be other students who feel less confident or willing to engage. While everyone has the ability to be mathematical problem solvers, learning environments have not always sent the message to all students that they can be mathematicians, and schools have not always been equipped to support productive struggle for everyone. Consider Darnell, who seemed readily able to connect to his identity and abilities when engaging in a game of kickball at recess but didn’t exhibit the same level of confidence when working on challenging math tasks. Some theorists argue that children engage in play as a way to explore and learn from their experiences in a safe and supportive environment. Teachers can take a page from this book to help them design a safe and supportive space in the mathematics classroom—one that leverages student’s assets, encourages collaboration and free thought, and values mathematical depth over speed (Boaler, 2016).
Let’s look at three ways in which Darnell’s teacher supported him in transferring his already existing problem-solving abilities to the mathematics classroom.
1. Darnell’s teacher was aware that she needed to help Darnell feel safe and comfortable taking mathematical risk in front of his peers. She decided it would be a good idea to ask the small group to contribute to community agreements, posing that “challenging each other respectfully” be one agreement that was included. After relating to the group that everyone makes mistakes and the discussion of our thinking helps us fix them, the students seemed to understand why this agreement was important for everyone to be able to feel safe enough to contribute ideas. They went on to make more agreements.
2. Then, Darnell’s teacher positioned him as an expert of his own thinking by directly communicating her observations of him during recess and affirming him as a problem solver. She communicated the parallels of how what he does on the playground can be applied to his approach to the math work and the discussions within the small group. She asked him to be sure to share his “great ideas” even if he didn’t consider them perfect, and described his thoughts as the possible missing puzzle piece that the group needs. His response was one of joy and agreement.
3. Darnell’s teacher intentionally created opportunities for Darnell’s voice to be included in the discussion. He started off by accepting his teacher’s invitations to say back or add on to what his peers said, and eventually he began to ask for clarification when a more confident peer erroneously stated that “7/6 could also be called 7/12” because there were 12 equal parts between the two wholes. Although timid, he pushed back and said that he saw “one whole and a small piece on the other whole”. It became a beautiful transition to talking about fractions greater-than-one as mixed numbers.
A learning environment that supports students to engage in productive struggle sits at the nexus of equity, talk-based instruction, and cognitively demanding tasks. While cognitively demanding mathematics tasks and talk-based instruction are essential components to setting the stage for productive struggle, it’s also imperative to discuss how effective teachers also tend to the work of supporting the development of mathematics identities. An NCTM publication list affirming mathematics learners’ identities as one of its five equity-based practices (NCTM, 2013) and “validating and affirming” and “empowering” are two of the eight indicators of a culturally responsive learning environment (Gaye, 2010). Arguably, affirming the mathematics identities of diverse learners underscores the student’s willingness to engage in productive struggle.
In addition to equitable teaching strategies, such as collaboration and inquiry-based approaches, both girls and students of color-particularly—underrepresented minorities—need thoughtful and positive messages to be given to them, about their valued place in mathematics. They need this more than other students because of the prevailing stereotyped societal messages about math.
-Jo Boaler (2016)
Students of color and multilingual learners belong to demographic groups that disproportionately receive less experienced teachers, lower-quality teaching, and limited access to rigorous instruction (Goldhaber, 2018; Cherng, 2022; Umansky, 2016). Additionally, it’s important to recognize that girls are also an underrepresented sub-group that receives damaging messages about their abilities in mathematics (Beilock et. Al, 2009). Working to support students, specifically under-represented students, in building confident mathematics identities can empower them to approach cognitively demanding tasks and problem-solving opportunities with perseverance and confidence and transfer already existing assets and resiliency to the classroom.
Supporting all learners in seeing themselves as mathematicians includes encouraging students belonging to ignored and misrepresented subgroups to value the contributions they make to the learning community. It also means creating a safe and collaborative space for mathematics learning to occur. As students experience more opportunities for sense making and collaboration involving multiple perspectives, students refine what they think it means to “do” and “know” mathematics.
Here are some things to consider in pursuing equitable opportunities for productive struggle in the mathematics classroom:
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- Utilize careful planning in choosing cognitively demanding tasks that are within the zone of proximal development for the learners.
- Encourage and lay a foundation for students to regularly discuss and compare strategies for solving problems.
- Explicitly communicate your belief in your students’ abilities, especially students that belong to mathematically underrepresented subgroups of learners.
- Create a safe and collaborative environment and culture by normalizing mistakes and community discussion around them in the classroom.
- Intentionally debunk messages that being “good at math” always means solving with speed and accuracy. Encourage students to embrace the idea that good mathematicians reason and work to make sense of mathematics-even in the face of setbacks.
Tell us your story
We’d love to hear how you are working to support and provide more equitable opportunities for engaging in productive struggle in math class! Share you story with us here and we may reach out about including it in an upcoming article.
References:
Beilock, S.L., E.A. Gunderson, G. Ramirez, and S.C. Levine. 2010. Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, USA 1075: 1060 3.
Boaler, J., & Dweck, C. S. (2016). Mathematical mindsets: unleashing students’ potential through creative math, inspiring messages, and innovative teaching. (First edition). San Francisco, CA, Jossey-Bass; a Wiley Brand.
Cherng, H. Y. S., Halpin, P. F., & Rodriguez, L. A. (2022). Teaching bias? Relations between teaching quality and classroom demographic composition. American Journal of Education, 128(2), 171-201.
Gay. G, (2010) Culturally responsive teaching: Theory, research and practices. New York: Teachers College Press.
Goldhaber, D., Quince, V., & Theobald, R. (2018a). How did it get this way? Disentangling the sources of teacher quality gaps across two states. (Working Paper No. 209-1118-1). National Center for Analysis of Longitudinal Data in Education Research (CALDER).
Goldhaber, D., Theobald, R., & Fumia, D. (2018b). Teacher quality gaps and student outcomes: Assessing the association between teacher assignments and student math test scores and high school course taking. (Working Paper 185). National Center for Analysis of Longitudinal Data in Education Research (CALDER).
Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523–550.
National Council of Teachers of Mathematics Research Committee (Tarr, J., Walker, E., Hollebrands, K. F., Chval, K. B., Berry III, R. Q., Rasmussen, C. K., & King. K.), (2013). New assessments for new standards: The potential transformation of mathematics education and its research implications. Journal for Research in Mathematics Education, 44(2).
National Council of Teachers of Mathematics. (2014a). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: Author.
National Council of Teachers of Mathematics. (2014b). Access and equity in mathematics education: A position of the National Council of Teachers of Mathematics. Reston, VA: Author.
Umansky, I. (2016). To be or not to be EL: An examination of the impact of classifying students as English Learners. Educational Evaluation and Policy Analysis, 38(4), 714-737.