Mind-Reading in Mathematics & the Undervalued Power of Discourse

By: Devan Smith

Although the invention of a mind reader sounds useful in theory, the creation of such a device poses several ethical concerns. A high-school educator might use the device to monitor students’ thought processes while actively working through exercises or to determine whether they are engaged during a lecture or simulation. However, a mind reader in the hands of a politician would give them an unfair advantage over their opponents. The implications of a mind-reading device possessed by a supervillain, or the real-life equivalent of such a character, also come to mind as having potential negative consequences. 

The likelihood of a device being invented is ambiguous in this day and age of massive technological improvements. Likely? No. Possible? Perhaps. However, it is unnecessary in the world of education, as math teachers (and arguably all teachers) have been effectively reading minds for years. This article will explore how mathematics educators, specifically, gain insight into student thinking through visible thinking strategies. In other words, how teachers of mathematics “mind read.” 

Mind-Reading Through Individual Written Work 

I recently designed a sticker for my teachers to incentivize engaging with district curriculum communications. It was instinctive to create “punny” graphics like the one shown in Figure 1, which has a coordinate plane overlaid with the phrase “Show Your Work.” Why? Well, because math teachers want students to show their work! 

Figure 1. 

Show Your Work 

Note. This image was created by Devan Smith using Canva.

Some may argue that math teachers, in particular, require work—as much as possible—because they are overly concerned about cheating. For a select few, this may truly be a concern. Students in the digital age have access to expansive, adaptive technology that allows them to provide answers to mathematical questions without a single thought beyond what to input into a program. However, most math teachers would argue that while they worry about cheating to some extent, they require students to show their work mostly because they want to see students’ thinking. After all, thinking is an “invisible endeavor” (Sliman, 2024, p. 502). Without some embodiment of thought shared between individuals, such as with discourse or written word, others cannot even begin to guess what someone is thinking, let alone ascertain if they can conceptualize topics in mathematics. 

By showing the steps taken to solve a problem, such as in a written format, teachers have insight into students’ mastery of mathematical concepts and procedural fluency that allows them to adapt their instruction and provide appropriate interventions for improvement. Given that mind reading is currently impossible, this is the solution teachers have at their disposal. It is effective given the learner provides enough information and can organize their thinking such that another person can interpret it.  

Why is it so important to “see” student thinking? For mathematics teachers, where learning prior skills is often imperative to master subsequent concepts, there may be a natural decay in procedural fluency, which the National Council for Teachers of Mathematics (NCTM) defines as the “ability to apply procedures efficiently, flexibly, and accurately, transfer procedures to different problems and contexts, build or modify procedures from other procedures; and recognize when one strategy or procedure is more appropriate to apply than another” (NCTM, n.d.b, para. 2). 

Procedural Fluency

Although procedural fluency is important, it must go hand in hand with conceptual understanding. Students of mathematics historically tend to struggle to conceptualize mathematics and it shows. The Program for International Student Assessment (PISA) measures 15-year-olds’ reading, mathematics, and science literacy (OECD, n.d.). There have been eight cycles since 2000 (every three years despite a one-year delay in 2020 due to the global COVID-19 pandemic), with the major domain (consisting of half the test) cycling between the three each cycle. Mathematics was featured in the most recent cycle in 2022. According to IES NCES, mathematics literacy is defined as “students' capacity to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts, and tools to describe, explain, and predict phenomena” (IES NCES, n.d., para. 3).  Prior to the 2022 cycle, the last time the major domain was mathematics was in 2003. Unfortunately, according to IES NCES (n.d.), the average score on the PISA for 15-year-olds in the U.S. in 2022, 465, was lower than in 2003, 483. The average score in 2018, when mathematics was a minor domain, was 478. In short, mathematics literacy is declining in the United States. 

This article will not delve into the why of this decline. However, a common reason for decay in procedural fluency over time, paired with the minimal conceptual understanding of mathematics as measured by PISA, leads to errors in student work on mathematical tasks that require mastery of prior skills, which, by the nature of how mathematics works, is most tasks (Brown & Skow., 2016). Could this decay occur because students were not given as many opportunities to master prior skills due to constraints of time or circumstance? Does it occur because they didn’t understand the concept before moving onto a skill that required the prior skill? Because they are experienced with such issues and often ask themselves these questions, math teachers who wish to pinpoint specific misconceptions may start by analyzing student steps to identify and categorize errors (Brown & Skow, 2016). 

Conceptual vs. Process Errors 

Teachers who wish to get to the root of the problem—pun intended—in student thinking often start with categorizing the errors made from student work samples to determine if students are displaying an error pattern (Brown & Skow, 2016). They will begin with identifying all errors made in a problem (or problem set), then either formally or informally decide whether the error is conceptual or procedural in nature. According to Brown and Skow (2016), there are four types of errors. These are factual errors, procedural (process) errors, conceptual errors, and careless errors. It is the opinion of this author that factual errors, which occur due to a lack of information, fall under the umbrella of conceptual errors. Process errors occur when the steps of a procedure are performed incorrectly. Conceptual errors occur when there is a “faulty understanding of the underlying principles and ideas connected to the mathematical problem” (Brown & Skow, 2016, p. 4). Careless errors are a breed of their own. As common as they are, typically pointing out the error is enough for a student to realize their mistake and correct it (Brown & Skow, 2016). 

Depending on the level of the course, a procedural error may have once been a conceptual error. For example, students in Algebra 2 should have mastered integer operations (i.e. adding, subtracting, multiplying, and dividing integers). Incorrectly combining like terms when adding and subtracting polynomials with integer coefficients because of a miscalculation (such as a student stating that -4xy + 5xy is equal to -9xy) would be considered a procedural error for these students. However, for a seventh-grade student exploring integer operations, stating -4 + 5 = -9 would be categorized as a conceptual error. 

The categorization of errors is not always clearly defined. When reviewing situations such as the one described with polynomial operations, teachers will typically probe students for more information, often through engaging in discourse or instigating written feedback (Brown & Skow, 2016). Teachers will engage in questioning strategies—aptly defined as probing questions—to surmise whether an error was indeed a process mistake or a misconception. Process errors can often point to incomplete mastery of foundational skills and this will arise from such probes (Brown & Skow, 2016).

A vast majority of the time, in my professional experience, learners make errors not in the skill they are actively practicing, but when applying skills assumed to have already been mastered. The ability to monitor student thought while working through problems would save math teachers a lot of time and cognitive energy when trying to identify misconceptions and correct them in the most efficient way. One way that I have saved time is through the utilization of technologies to provide instant feedback. However, while technology has allowed for teachers to quickly assess student mastery, there is no tool that this researcher is aware of that allows for teachers to instantly probe student thinking—save that of good old-fashioned visible thinking practices, which can be used hand-in-hand with platforms like Formative or Desmos Classroom, but are not exclusive to such programs. 

The Wonders of Visible Thinking

In her 2013 article “Visible Thinking in Mathematics”, Emily Silman shares two methods she employs to “elicit more critical thinking”—Chalk Talk and Claim-Support-Question (p. 502). Chalk Talk, developed through the School Reform Initiative, combines the power of silent collaboration with a shared working surface. Although spoken communication is prohibited, Silman found that she could learn a massive amount of information about how students were thinking because of the structures of the activity. Her students specifically recorded all discourse regarding a question, such as “What do you know about right triangles?”, through the use of markers on chart paper (Silman, 2013). She found that Chalk Talk was useful for showing how students form connections between their thinking and the thinking of their peers and acted as a springboard for classroom discussion, both during the activity and after (Silman, 2013). The process of communicating was more valuable than the products produced; however, the products, when left displayed in the classroom, allowed her to refer back to the learning experience and further anchor student comprehension after learning had occurred. 

Visible thinking can be formally defined as any active and observable representation of an individual or group’s thoughts, questions, reasons, and reflections, including such examples as mind maps, charts, lists, drawings, and diagrams (Tishman & Palmer, 2005). An important aspect of visible thinking routines is organization or structure (Tishman & Palmer, 2005). According to Tishman and Palmer (2005), visible thinking practices support student learning as they express a powerful view of knowledge, demonstrate the value of intellectual collaboration, and change the classroom culture. These routines help students “become more active, curious, engaged learners” (p. 3). In essence, visible thinking routines help students be… thinkers. 

A compendium of thinking routines was compiled by Project Zero (2022) called Project Zero’s Thinking Toolbox includes activities to promote visible thinking. Personal favorites of mine are: Headlines, where students consolidate their understanding of a topic using a news headline, 3-2-1 Bridge, where students formulate three words or ideas, two questions, and one metaphor or simile both before and after learning, and Same and Different (see Figure 2), where students compare and contrast the likes and differences of two opposing views or images that look different, but are grouped together. 

Figure 2

An Alike and Different Example Problem for Trigonometry Problems. 

Note. This figure was created by Devan Smith using Google Drawings, inspired by HCPS thinking routines.

In my own practice, I have seen students communicate through written mathematics, even beyond language barriers, using visible thinking structures or routines. During an activity where students worked together on a problem at a large whiteboard—similar to the posters used in Chalk Talk, but erasable—an English Language Learner in a class I was observing used written mathematics to “speak” to her classmates. It was, accidentally, a Chalk Talk task, because this student did not speak to her classmates, either due to inability or social anxiety. Although she could not or chose not to explain in English to them, she used her marker and body language to write her thoughts mathematically. Coincidentally, her thinking contradicted the thinking of her peers. The students were able to realize the error they made by analyzing her work compared to theirs, without talking aloud. 

It was fascinating to see this collaboration occur despite differences in spoken language—through visible thinking via writing on a shared collaborative surface. Without the collaborative structures, such as the shared workspace, this thinking may not have occurred, or it may have gone unnoticed. The mass misconception could have failed to have been addressed, which would have been detrimental for learning for this concept. 

Similarly, during an activity at an entirely different location, two students were arguing about the correct procedure to tackle the same problem, seen in Figure 3.

Figure 3

A Comparison of Students’ Visible Thinking in Algebra 1.

Note. This is a photograph of student work in a classroom, taken by Devan Smith 

The original problem provided to students was to fully simplify the square root of 45 divided by the square root of 5. One student (on the left) had written their answer, but their partner, who did not write his work because they only had one marker between them, disagreed with the procedure because they believed the problem needed to be started differently. When I arrived and could hear and see that the students were misunderstanding one another while speaking, I asked if the student who did not have the marker could write out how they would work out the problem, so that we could all see the differences he was explaining. Then, we could revisit the conversation and compare answers while discussing them. I emphasized that it would be easier to notice differences if we had something to refer to as we talked. 

Once that was done, I stepped back and allowed the pair to conduct an informal Notice and Wonder. A Notice and Wonder is a simple activity where learners are tasked with answering two questions: What do you Notice? What do you wonder? By asking these questions, students engage with sense-making through engaging curiosity and conceptualization through reflection and metacognition (NCTM, n.d.a). They immediately realized that they had arrived at the same answer, despite the different avenues taken. I asked both students to analyze their approaches and determine if anyone had made any mistakes. If neither had, what did that mean? Was one method more efficient than the other? Why? 

The act of comparing our thinking led to deeper learning for the students. We had moved out of procedural fluency into conceptual understanding simply by discussing our thinking and the work done as a group rather than independently. This brings me to my next point: that discourse is the most undervalued, yet ultimately very powerful visible thinking strategy to evaluate student thinking. 

The Undervalued Evaluative Power of Discourse

What does it mean to evaluate student thinking? It means to assess learners’ understanding of content and ability to perform skills. Think back to the students simplifying radicals at the whiteboards. As an observer, it was clear from just listening that both students were correct in their understanding of simplifying radical expressions. The student who had written the work on the left had prime factored the radical in the numerator, simplified, and then divided the like factors in the numerator and denominator. Simply by listening to the argument between them while filtering around the room, I knew that the student on the right was also able to arrive at the correct answer as he insisted that 45 was divisible by 5, which is a correct assertion. However, the students were fixated on the fact that there was only one possible way to complete the problem and so assumed one of them had to be wrong. Once encouraged to work out the problem his way, the students were able to see that that assumption was false. Both of them used appropriate algebra to arrive at the correct simplification. 

Without the structures in place and had I not been engaging in monitoring student discourse, this beautiful learning moment may have been overlooked. Visible thinking must and should include monitoring of student discourse, which makes the title misleading. However, according to the definition, hearing student thinking is an observable representation of thinking and is included under the visible thinking umbrella. It can be difficult, however, to formalize a structure to monitor student discourse. Or at least, it was in my experience, until the 5 Practices for Orchestrating Student Discourse in Mathematics. 

5 Practices for Orchestrating Student Discourse in Mathematics 

One resource or framework that helps to structure and organize discourse routines is 5 Practices for Orchestrating Productive Mathematics Discussions written by Smith & Stein (2011). The five practices are: Anticipating, Monitoring, Selecting, Sequencing, and Connecting (Knab et al., 2018). I like to categorize this further into three phases: before learning, during learning, and after learning. Anticipation occurs before learning. Monitoring and selecting occur during learning. Finally, sequencing and connecting occur after learning. 

Learning is continuous, and while units may end, true learning builds as each skill is developed. There will be some overlap, especially when you are teaching students multiple skills during these phases, or if you are following a model where you are engaging with a skill, consolidating, and then continuing. However, in this context, before learning refers to work a teacher has done before a learning experience, during learning occurs when students engage in the learning experience, and after learning is when students reflect on the learning experience to consolidate their understanding. 

Before learning, teachers wishing to orchestrate student discourse should anticipate student responses. They could ask themselves: what will students do? How will they do it? What misconceptions are likely (Knab et al., 2018)? For example, during a collaborative activity where students discuss the simplification of radicals, a teacher might anticipate that students will approach the problem of dividing the square root of 45 by the square root of 5 from different angles–such as with the student samples in Figure 2. Teachers might also brainstorm potential misconceptions students might have, such as failure to completely prime factor a radicand (the number beneath a radical) or confusing the procedure for square roots with the one for cube roots. When engaging students in a learning experience where they will need to use trigonometric ratios to solve for missing sides of a right triangle, a teacher might anticipate students will confuse the ratio for cosine with that of sine or fail to identify the sides of a right triangle correctly to determine the trigonometric ratio to use in the equation. Experienced teachers will often anticipate student responses informally. However, it is helpful for all teachers to formally engage with this process, especially if discourse is the focus. Through formally anticipating student discourse a teacher can prepare strategies to assess or advance student understanding through probing questions tailored to each anticipated response. 

During learning, teachers will launch a task or learning experience and then monitor student discourse using the anticipated responses. While a teacher is listening to students engage in collaborative work, they should be actively looking for student responses to include during the connection phase. They should identify what students are doing and what strategies are being used (Knab et al., 2018). This act of identifying student responses to share is called selecting. The idea is that teachers will select student thinking that captures important aspects of the experience. This could include a common misconception that needs to be addressed to avoid it from manifesting, or important themes that occur during a procedure. Teachers should ask while monitoring and selecting: Why should this group’s work be showcased? What strategies are being used? (Knab et al., 2018). 

After learning has occurred, a teacher will begin sequencing student responses. The selections made by the teacher should be organized and highlighted at the appropriate times, either formally or informally, by comparing simple approaches to more complex ones, or common approaches to unusual ones (Knab et al, 2018). This phase is where pacing can be ambiguous. It may be appropriate for a teacher to sequence student responses throughout an activity. For example, if students continuously incorrectly identify the adjacent sides of right triangles, it would be prudent to sequence a collection of responses that helps students to improve their understanding of what the adjacent side of an angle in a right triangle is. This could be through highlighting correct work, gathering around incorrect work and conducting an error analysis, or comparing a correct identification to an incorrect identification. However, it is also appropriate to sequence work after learning has occurred. 

The final phase of orchestrating student discourse is connecting. Student work should be showcased in a way that makes sense for the learning experience. Consider it like telling the story of the learning that has occurred in your classroom. What is it that you want that story to convey? Who are the characters? What is the theme? Are there any other ideas that should be discussed that were missed (Knab et al., 2018)? 

Conclusion

The decline in mathematical literacy among students in the United States highlights the urgent need to address both procedural fluency and conceptual understanding in mathematics education. By prioritizing a balanced approach that emphasizes a visible learning approach, including the orchestrating of student discourse, we can empower students to become proficient problem solvers and critical thinkers in an increasingly mathematical world. In doing so, we can, in fact, become “mind readers.” 

References

Brown J., & Skow K. (2016). Mathematics: Identifying and addressing student errors. Iris Center. https://iris.peabody.vanderbilt.edu/wp-content/uploads/pdf_case_studies/ics_matherr.pdf 

Project Zero. (2022). Project zero’s thinking routine toolbox. Harvard Graduate School of Education. https://pz.harvard.edu/thinking-routines 

IES NCES. (n.d.). Highlights of U.S. PISA 2022 results web report (NCES 2023-115). U.S. Department of Education. Institute of Education Sciences, National Center for Education Statistics. https://nces.ed.gov/surveys/pisa/pisa2022/mathematics/international-comparisons/ 

Knab, K., Hofacker, E. B., Ernie, K. T., & Ahrendt, S. (2018, March). Using the 5 practices in mathematics teaching: Selecting and sequencing student work with cognitively demanding tasks in a group environment can teach important mathematical skills. Mathematics Teacher, 111(5), 366-373. https://www.nctm.org/Publications/Mathematics-Teacher/2018/Vol111/Issue5/mt2018-03-366a/.   

NCTM. (n.d.a). Notice and wonder overview. NCTM.  https://www.nctm.org/noticeandwonder/ 

NCTM. (n.d.b). Procedural fluency in mathematics: A position of the National Council of Teachers of Mathematics. National Council for Teachers of Mathematics. https://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluency-in-Mathematics/

OECD. (n.d.). Programme for international student assessment. https://www.oecd.org/pisa/ 

Silman, E. (2013, March). Visible thinking in mathematics. The Mathematics Teacher, 106(7), 502-507. https://www.jstor.org/stable/10.5951/mathteacher.106.7.0502 

Smith, D. (2024). A comparison of students’ visible thinking in algebra 1 [Photograph].

Smith, D. (2024). An alike and different example problem for trigonometry problems [Infographic]. Google Drawings.  

Smith, D. (2024). Show your work [Image]. Canva.

Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. NCTM. https://www.mathedleadership.org/docs/coaching/5%20Practices.pdf 

Tishman, S. & Palmer, P. (2005). Visible thinking. Leadership Compass. http://pzartfulthinking.org/wp-content/uploads/2015/04/VT_LeadershipCompass1.pdf