The Focus

The Ventura County Mathematics Council is an affiliate of the California Mathematics Council (CMC) and the National Council of Teachers of Mathematics (NCTM). We serve educators in Ventura County and surrounding areas by conducting meetings, professional development and activities where ideas can be shared and networks formed.

Photo Credit: Felix Cortez

"An important message for students is the value of taking mathematical risks. Making mathematical errors and confusions public helps students make sense of them together, as a classroom of learners. A classroom that welcomes students’ unfinished thinking normalizes mathematical struggle as part of learning and positions all learners as belonging to the discipline of mathematics."

(2023 California Mathematics Framework, Chapter 2, page 32).

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Is it the flu? Is it an infection? When we are not feeling well, we go to the doctor for an examination. Using clues and evidence, the doctor diagnoses and prescribes a specific treatment to help us recover. We trust that this evaluation is thorough, because the wrong treatment won’t help. Eye drops probably won’t fix a broken bone, and cough syrup will most likely not cure a rash.

Our work as educators requires the same level of care and precision. It is our responsibility to investigate student thinking and use evidence to determine why a student is struggling. When a student has a knowledge gap, treating it as a simple error will not help them move forward. Effective instruction depends on our ability to accurately identify the type of incorrect response a student is making and respond in ways that meet their specific needs. Understanding the different types of student mistakes and how to address each one is essential for supporting meaningful learning and growth.

A knowledge gap occurs when a student lacks a key concept or procedure, preventing them from accessing the mathematics. These gaps often act as gatekeepers. Without a critical piece of understanding, students are unable to move forward. I often think of a knowledge gap as a missing bridge between two hills; without that bridge, the student cannot reach the other side, no matter how motivated they are.

In many ways, knowledge gaps can be the most challenging obstacles to address. How do we go back and fill in missing information for a classroom of 30 students, each with different needs? This is where Universal Design for Learning (UDL) becomes essential. Rather than attempting to remediate every gap in the same way, we must first identify the precise barrier and then decide how to respond. Do we need to pause and directly address the missing understanding, or can we provide a tool or support that allows students to move forward without being blocked by that gap?

Consider a student who lacks fluency with multiplication facts. This gap will likely interfere with their ability to perform long division. However, that missing knowledge should not prevent the student from engaging in grade-level content. At this point, teachers must clearly identify the purpose of the task or assessment. Are we measuring a student’s recall of multiplication facts, or are we assessing their understanding of the long-division procedure? If the goal is procedural understanding, providing support, such as a multiplication chart, removes the barrier and allows the student to demonstrate their mathematical reasoning. In this way, the knowledge gap no longer limits access to learning.

Instructional response:
Provide scaffolds, tools, or targeted instruction to remove barriers to access.

An error is often best understood as a simple mistake. In these cases, the student possesses the necessary content knowledge and can typically demonstrate proficiency. Errors may include forgetting a negative sign, adding instead of subtracting, or misreading the directions. While these mistakes are often minor, they require immediate attention. If left unaddressed, errors can become habitual, leading to persistent misconceptions or inefficient strategies.

When responding to an error, teachers should focus on clarity rather than on avoiding discomfort. Addressing an error is not punitive; it is instructional. Teachers must explicitly identify the mistake, explain why it is incorrect, and provide the student with an opportunity to practice the skill again correctly. Equally important is asking students to verbalize their thinking. Having students explain their reasoning aloud allows the teacher to pinpoint exactly where the error occurred and reinforce the correct understanding.

Instructional response:
Correct explicitly, give immediate feedback, and provide short, focused practice.

A misconception is a false or incomplete belief about how mathematics works that leads a student to an incorrect solution. It is important for teachers to identify and address misconceptions early in order to prevent them from solidifying into ingrained habits. When a student is struggling with a misconception, the first instructional move is to surface their thinking. Asking students to explain what they are doing and why allows teachers to uncover the reasoning behind the incorrect belief.

Rather than simply telling students they are wrong, effective instruction gives students the opportunity to discover the flaw in their reasoning. Teachers can do this by presenting counterexamples, posing purposeful questions, and offering tasks that directly challenge the misconception.

This process creates a more meaningful learning experience than simply correcting students individually. Once students recognize the error in their thinking, they should be supported in rebuilding their understanding through discussion, visual models, and multiple representations. This helps students replace the misconception with a more accurate understanding of the content.

Instructional response:
Elicit student reasoning, confront the misconception with examples or counterexamples, and rebuild understanding through discussion, visual models, and multiple representations.

After my students completed the FIAB on Geometric Figures, my student teacher and I sat down to analyze the results. Overall, the students performed well and demonstrated a strong understanding of how to create scale copies using a scale factor. However, one particular problem stood out, as most students answered it incorrectly. This immediately caught our attention because we had expected this problem to be easier for the students to answer.

We returned to the student work to investigate what had gone wrong. What puzzled us most was the repeated appearance of the same incorrect answer: 5.2. As we examined the responses more closely, my student teacher made an important realization. Several students had interpreted the scale factor 1:3 as the decimal 1.3, and then multiplied it by 4, resulting in 5.2. That moment of clarity changed everything.

This error was not due to a misunderstanding of scale factors. Instead, students held a misconception about notation. They believed the colon in a ratio functioned like a decimal point. Recognizing this distinction mattered. Reteaching multiple scale-factor problems would have been an inefficient use of instructional time and would not have addressed the root issue. What students actually needed was explicit clarification about the meaning of the colon and exposure to the notation used in CAASPP-style directions, which was unfamiliar to many of them.

Once we identified the misconception, we addressed it quickly. By targeting the specific misunderstanding, we corrected student thinking without slowing progress or lowering expectations.

On the same assessment, there was another problem with a similar structure, yet only a small percentage of students answered it correctly. Our analysis determined that the difficulty stemmed from the fractional scale factor. Fractions and number sense more broadly remain significant challenges for many of my students, and this pattern clearly indicates a substantial knowledge gap.

Recognizing this, we intentionally chose to address the gap by both going through and going around the barrier. To strengthen students’ fluency with fractions, decimals, and percents, we incorporated the game Spoons as a low-stakes, engaging way to build number sense. At the same time, we addressed the barrier by intentionally embedding fraction-based problems in our next unit and consistently reinforcing the use of calculators to convert fractions to decimals.

We made a deliberate decision not to spend extensive instructional time reviewing long-division procedures for converting fractions to decimals. Given that 7 of the 9 CAASPP targets allow calculator use—and that many students struggle with long division and multiplication facts—we determined that this approach would not be the most effective use of instructional time. Instead, our focus remained on ensuring students could access and demonstrate their understanding of grade-level mathematics.

Accurately responding to incorrect student work requires more than identifying that an answer is wrong; it requires understanding why it is wrong. When teachers distinguish between knowledge gaps, errors, and misconceptions, they move from reacting to mistakes to making intentional instructional decisions. This precision allows educators to remove barriers, provide timely feedback, and challenge faulty reasoning without lowering expectations.

By diagnosing students' thinking and responding with purpose, we create classrooms where all students have access to rigorous, meaningful mathematics and the opportunity to grow into confident mathematical thinkers.

As our school district considers adopting a new math curriculum, an important first step is determining what we value and believe about teaching and learning mathematics. We have a document titled "Philosophy of Mathematics " that outlines what an ideal math classroom looks like. When we revisited our previous philosophy, we realized that one significant element was missing: mindsets.

What students believe about learning—and about themselves as mathematicians—has a powerful impact on their outcomes. As a result, we intentionally added a focus on mindset to our philosophy. Here is what we included:

Students who view themselves as mathematicians bring a powerful set of mental habits to their work. By intentionally nurturing a growth mindset, teachers can help students develop persistence, resilience, and a willingness to think deeply and flexibly about problems.
Research in brain science shows that a strong growth mindset leads to higher engagement and stronger performance, while a fixed mindset often leads to fear of failure, avoidance of challenge, and giving up when learning becomes difficult.

I have found that investing in the development of positive mindsets—where students believe that struggle is a natural and necessary part of learning, and that hard work and perseverance lead to success—is a critical component of an effective math program. This is important for students who have always found math easy, as they will eventually encounter challenges that require persistence. It is equally important for students who struggle every day to understand that struggle can lead to deep learning and strong outcomes.

More and more math curricula are including mindset messages and activities, reinforcing the importance of this work.

Two of my favorite resources for learning more about promoting productive math mindsets are:

• Youcubed.org

• Productive Math Struggle by J. SanGiovanni