By:
**Laurie Cavey, Ph.D.**, * Professor, Department of Mathematics*, Boise State University

**Tatia Totorica, Ed.D**, * Clinical Assistant Professor and 7th Grade Mathematics Teacher*, Boise State University; Boise School District

**Jason Libberton, M.Ed.**, * High School Mathematics Teacher, District Math Coordinator*, American Falls High School

Equitable access to mathematics is a long-standing, yet unmet goal in mathematics education (Gutiérrez, 2018). There are many well-known obstacles to this goal, one of which is unexamined *biases about who does and does not do mathematics* (Walker, 2007). In this blog, we describe how our project data are teaching us that unexamined *biases about what is and what is not mathematics* can also be detrimental to establishing a classroom where each student has a fair shot at being a valued and contributing member.

**The VCAST Project**

The design-based VCAST research project, funded by the National Science Foundation (1726543), focuses on developing curricular materials that invite secondary teacher candidates to solve challenging mathematical tasks, watch videos, and examine written work of secondary students solving the same tasks, describe student reasoning, and evaluate and predict students’ problem-solving processes. The VCAST curriculum includes asynchronous components delivered through an online platform and synchronous components implemented by mathematics course instructors (Cavey et al., 2020).

The VCAST materials, while designed to be embedded in math courses for secondary teacher candidates, have also been successfully implemented in methods courses for teacher candidates and in a professional development setting for teachers. Over the last few years, the VCAST materials have been used at 14 different universities across the U.S. and with over 250 participants. Feedback from VCAST partner instructors is overwhelmingly positive, with instructors expressing appreciation for the way the modules are designed to engage candidates in doing mathematics that is truly relevant to their future work as teachers. As one partner instructor reported, “Grounding content knowledge in student thinking is a key component of an educator’s professional knowledge base. The modules enabled me to attend to this component throughout the semester, while at the same time making connections to course work.”

Here, we consider only one aspect of the VCAST materials, the Hexagon Task Module, where candidates encounter students’ problem-solving approaches to the Hexagon Task (See Figure 1). Candidates begin by solving the task themselves, describing what they paid attention to as they worked, and uploading an image of their full solution.

Figure 1. The Hexagon Task (adapted from Hendrickson, Honey, Juehl, Lemon & Sutorios, 2012)

Next, video-based evidence is presented featuring the work of Ashley and Maria (pseudonyms). Both high school students completed the task during a one-on-one interview. Ashley’s first language is English, whereas Maria’s is Spanish. All video clips include closed captioning in English, and a Spanish-English translator was present during Maria’s interview. We summarize both students’ approaches in Figure 2.

Figure 2. Summary of Ashley’s and Maria’s approaches

Candidates then respond to questions about Ashley’s and Maria’s work–some questions ask about the students’ mathematics and others focus more on pedagogy. Here, we highlight the mathematically-oriented questions, where candidates either use a 5-point scale to indicate their level of agreement with provided claims or compare the students’ approaches (See Table 1).

Table 1. Mathematically-oriented questions about Ashley’s and Maria’s work in the Hexagon Task Module

Repeated cycles of module development revealed that before candidates could meaningfully compare student approaches, they first needed to be able to recognize students’ productive reasoning. In addition, it is important to note that we didn’t fabricate the perspectives in our claim questions; they arose from earlier versions of the module where candidates responded to open-ended questions about the correctness and validity of student approaches.

**The Evidence, in Brief**

During the 2020-2021 academic year, 131 undergraduates completed the Hexagon Task Module at 13 different U.S. universities. Results yielded an unexpected number of candidates either indicated agreement with (45/131, ~34%) or were not sure about (27/131, ~21%), the Counts Claim for Maria (Counts-M). That is, nearly 55% of the candidates did not appear to notice Maria’s productive mathematical thinking, while only 27% of the candidates did not readily agree with the Counts Claim for Ashley (Counts-A).

To better understand this phenomenon, we first examined whether candidates’ own mathematical knowledge could explain the discrepancy. Take a look at Table 2 for a breakdown based upon the candidate’s own solution to the Hexagon Task. Notice that of the 57 candidates who submitted an incorrect solution, 32 (56%) did not recognize the productive nature of Ashley’s work and 43 (75%) incorrectly attributed Maria’s correct answer to luck. Perhaps more alarmingly, though, is that of the 74 candidates who submitted a correct solution, 4 (5%) answered Counts-A incorrectly and 29 (39%) answered Counts-M incorrectly.

Table 2. Correctness vs. Counts

This led us to wonder whether mathematical background mattered as much as conventional wisdom would suggest. We narrowed the data to those who appeared to be in the best mathematical position to recognize the validity of Maria’s approach. We identified 52 candidates who:

- uploaded a complete and correct solution to the Hexagon Task,
- correctly responded to Correct-A and Counts-A, and
- correctly responded to Correct-M.

Of the 52, 33 candidates responded to Counts-M correctly. Of the 19 who did not, 5 were unsure and 14 strongly agreed with the claim that “Maria’s approach won’t always work because it involves making a lucky guess […].” Why might those 19 teacher candidates who correctly solved the task and correctly responded to the other three claims questions not readily recognize the validity of Maria’s approach?

A potential answer to this question emerged when analyzing candidates’ comparisons of Ashley’s and Maria’s approaches (See Figure 3).

Figure 3. Teacher Candidate Responses to Compare What is Different

We call what you may notice in TC1’s response a *closed perspective*. This contrasts sharply against TC2’s *open perspective*, whose description is grounded in evidence and highlights the productive aspects of each student’s work. We wondered if this difference in perspective might be tied to biases about Maria’s cultural background—after all, there is significant data to suggest such biases can provide a plausible explanation (Botelho et al., 2015; Malouff & Thorsteinsson, 2016). To investigate this possibility, we examined the 19 candidates’ responses to subsequent mathematically-based questions in the module and found that 15 of the 19 expressed this closed perspective at least once. Further, more candidates expressed a closed perspective about the mathematical work of the white male student (Brandon) featured at the end of the module than with Maria. The number of closed perspective responses by module student are reported in Table 3.

Table 3. Closed Perspective Response Counts

Analyzing the closed responses across the three students, we concluded that many candidates devalued Maria’s approach because she didn’t represent her thinking with an equation—and the same is true for Brandon whose approach is summarized in Figure 4 alongside Ashley’s and Maria’s. Candidates took notice that neither Maria nor Brandon represented their thinking using a formal algebraic equation, like Ashley. From our perspective, all three students demonstrated productive thinking on the task. In fact, Brandon’s approach was very similar to Ashley’s (and the way Maria started). The difference occurs with Maria’s and Brandon’s use of intuitive methods that were not captured in a conventional way.

Figure 4. Ashley’s, Maria’s, and Brandon’s approaches to the Hexagon Task

Interestingly enough, candidates who appeared to learn mathematics during the module (evidenced by solving the initial task incorrectly and then correctly solving a related task after observing Ashley and Maria) exhibited overwhelming evidence of openness when comparing Ashley and Maria’s approaches (12/14; ~86%). In fact, later in the module, many of these candidates expressed appreciation for students’ creativity and their different (and valid) ways of approaching the task. (See Figure 5).

Figure 5. Candidate Evidence of Openness

**The Equitable Classroom and the Mathematical Preparation of Teachers**

Our evidence suggests candidates can assess Maria’s final answer as correct while simultaneously devaluing her way of doing mathematics. This is a problem! Students who engage with tasks using creative, sensible approaches need teachers who value and celebrate their thinking, especially when it deviates from traditional, conventional methods. Being able to recognize the valid mathematical reasoning of students, who often use informal ways of communicating their ideas, is essential for establishing a classroom community where each member is a valued mathematical contributor. And doing so is nearly impossible for someone who has a closed perspective on what it means to do mathematics–thinking for example, there is a right way to correctly solve and communicate that solution to the Hexagon Task.

We hypothesize that many candidates don’t recognize Maria’s and Brandon’s work as valid because they themselves have had few opportunities to engage in *doing* mathematics (Schoenfeld, 1994). Let’s face it; despite repeated recommendations that future teachers need opportunities to *do* mathematics (e.g. MAA Committee on the Mathematical Education of Teachers, 1991; CBMS, 2001, 2010), this is largely an unmet goal. As a result, our candidates’ experiences with mathematics are all-too-often limited to receipt and rehearsal of finalized mathematical products which adhere to conventional representations. Yet, data from our project suggests that a focus on content knowledge and fluency alone are not enough for developing teacher candidates’ capacity for creating an equitable classroom of their own. In short, this not only does a disservice to our future teachers but also to their future students.

**Implications**

Teacher candidates need opportunities to develop a better understanding of what it means to do mathematics–and it will not suffice to limit this to their experiences in one or two courses dedicated to future teachers. To that end, we urge those involved in the mathematical preparation of teachers to heed the following calls for action.

**University mathematics departments**

- Address candidate (and other student) perceptions of what counts as productive mathematical reasoning. This entails providing opportunities for candidates to use their own intuitive approaches to solving mathematical problems throughout their mathematics coursework.

**Mathematics educators**

- Broaden candidates’ exposure to the different ways students might productively approach problems. This is especially needed at the secondary level, where there are limited opportunities for candidates to engage in the mathematical work needed to connect across and build upon students’ ideas.

**Education researchers**

- Study secondary students’ intuitive approaches to key ideas in the middle and high school curriculum. This could increase mathematics educators’ knowledge base and potentially lead to further research and curriculum development focused on broadening candidate conceptions of what counts as doing mathematics.

**Teacher education standards and assessment writers**

- Emphasize the importance of doing and recognizing mathematics done from an intuitive perspective. This will necessarily involve a de-emphasis on the very narrow view of mathematical proficiency (consisting of quick recall and accurate application of conventional procedure) currently guiding teacher licensure assessment design.

We’ve known for years that providing opportunities for each and every student to engage in doing mathematics is essential to providing equitable access to mathematically-based career paths (Boaler & Staples, 2008). It is time to address the messages we all continue to send future teachers about what counts as mathematics.

**Acknowledgment**

This material is based upon work supported by the National Science Foundation under Grant Number 1726543. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

For more information about the VCAST project, watch this video or contact VCAST at vcast@boisestate.edu.