Educators and researchers have spent decades studying learners’ ideas, errors, and misconceptions, but the use of that knowledge has not yet permeated the classroom. Discovering learners’ conceptions and misconceptions can be a daunting activity for teachers, and today’s multiple-choice technology is not ready for analyzing networks of conceptual misconceptions.

How can we use all the research and knowledge on learners’ ideas, errors, and misconceptions to develop students’ mathematical thinking and increase math learning? How can the discovery of learners’ attempts to make sense of math lead us to better ways of thinking about and teaching math?

I have been passionate about the idea of misconceptions for ten years now. I used it to explain to learners the importance of reflecting in their process of learning and for them to see those misconceptions not as simple errors but as a natural part of building deeper understandings that need to be constantly thought of. I supported educators in using the MOZART project — a set of science tests to diagnose misconceptions — to learn how well students understand various concepts in science. Educators reported finding the MOZART project very useful, but I haven’t found a similar instrument for math.

This article describes a work in progress on math misconceptions. This first phase focuses on gathering the most common algebra-related misconceptions and errors in the existing literature. The reason is that passing 6th-grade math is one of the top predictors of high school graduation.

The Mapping Algebra Misconceptions (MAM) project is conducted by me, Nancy Otero. A special thanks to John Whitmer, who has supported this project from its beginning and provided valuable insights.

## Math Misconceptions--Background

Research suggests that learners construct more advanced knowledge from previous understanding; learners are not a black slate when they approach instruction. Learners’ previous conceptions can be a resource for cognitive growth (Smith et al. 1993). These conceptions (or misconceptions) are an opportunity to play, own, experience, and think about ideas as learners do in many other activities. This learner’s exercise in making sense of math also gives a window into how they construct their understanding. Making sense of and addressing learners’ misconceptions can improve math performance (Ashlock 2010) and support their ownership of doing math (Li 2019).

For these reasons I included a relation of the MaE to the Common Core, hoping it can be a tool for educators in their lesson planning. I included one example here:

## Errors vs Misconceptions

It’s worth understanding that errors and misconceptions are related, but different in important ways that matter.

Learners’ computational errors mean that they haven’t mastered how to use a rule. Overcoming computational errors is vital because the learner can carry on errors in fundamental procedures from which more complex math is built for years. For instance Bush 2011 found that 16.5% of her sample answers to algebra questions from middle school learners made basic computational errors with whole numbers.

As an example of a computational error, some learners consistently make the sum of two negatives a positive, -6 – 3 = 9 or -5 – 4= 9 (Ashlock 2010). The cause of computational errors can be multiple: because of lacking understanding of why that rule is that way, having a misconception on what the rule is that way, underdeveloped understanding of a particular underlying concept, an error while learning the rule, or a mistake applying the rule, etc. Researchers have found several common computational errors, and identifying them can be used to determine the cause.

Misconceptions are a mistake in conceptual understanding and they have relations with all the applications of those concepts. For example, a single misconception on the connections among proportional relationships (part/whole, part/part, whole/part) can cause problems in identifying those patterns in drawings and can be the cause of failing to realize all parts must be of equal size, therefore associating the denominator of the fraction with the total number of parts regardless their size.

Sometimes misconceptions about conceptual understandings are related to the form that teachers represent math. For example, Baroudi (2006) reported that misconceptions of equivalence can be attributed to the fact learners experience the equal sign always at the end of an equation and only one number comes after it. In a study, learners were asked to solve 8 + 4 = [ ] + 5, and all learners answered that either 12 or 17 should go in the box. Another cause of conceptual misconception is a teaching practice that focuses on procedures mostly instead of emphasizing concepts (Van de Walle et al. 2010).

One challenge to the distinctions between misconception and error is that there are other learning obstacles researchers and educators mention. For example, the natural learning bias (Van Hoof et al., 2014) is an inappropriate application of natural number’s features to rational numbers caused by the learners’ natural-numbers-based intuitive idea of what a number is (Vamvakoussi & Vosniadou, 2004). This bias has many consequences in learners’ practice. For now, I included expressions of this bias related to algebra, like ordering decimals incorrectly by basing the value of a decimal on its number of digits as if it was a natural number. For example: Thinking 0.25 is larger than 0.7, this bias will have implications for learners thinking about expressions such as “x/4 < x”. Including these learning obstacles might be a future extension of this project, given their importance and connection with misconceptions and errors.

An important difference between errors and misconceptions is that diagnosing conceptual misconceptions can be harder than computational errors in a typical classroom or by using multiple choice technology. Researchers have used interviews and paper-pencil work that show the learner’s thinking in order to discover them. This level of focus on each learner can be hard to achieve but today’s technology might be able to offer some help.

## Large Language Models and Tracking Misconceptions

Machine learning tools such as Large Language Models (LLM) can potentially be trained to identify math misconceptions in text and pictures (Wu et al. 2021). This is one of the reasons why I wanted to compile a list of the most common algebra and geometry misconceptions and the method used to diagnose them, as a first step in generating data that can be used to train LLMs.

Creating a relation between the MaEs and tests such as the NAEP is a step in identifying the misconceptions in the answers of those tests. Identifying MaEs in tests such as the NAEP would allow us to track those misconceptions and errors on existing data and follow their impact through time. The learning impact and importance of each misconception can be discovered by monitoring the effect of each misconception overtime on the existing NAEP data. Knowing which misconceptions have a higher effect helps prioritize which ones should be addressed first; e.g., knowing which misconceptions a student has can help to create targeted tutoring groups for example.

## Method and Data

For algebra, two central sources were used: Bush S (2011) and Karp K (2013). They included the analysis of 129 peer-reviewed journal manuscripts, proceedings, and papers from national or international conferences, technical reports, books, and dissertations.

Fifty-six different MaEs were found, a total of eighty papers were read to extract examples of the MaEs, and sixty sample questions to diagnose the MaES were transcribed.

The topics covered were based on Welder (2007), who has a broader framework and was used in both sources. Algebra-related MaEs developed during 4th or 5th grade impact middle school algebra; e.g., misunderstanding the connections among proportional relationships can later impact the understanding of slope. For that reason, 4th and 5th-grade MaEs that are algebra related are also included.

Topics:

- Number and numerical operations
- Ratios and proportions
- The order of operations
- Equality
- Patterning
- Algebraic symbolism and letter usage
- Algebraic equations
- Functions
- Graphing

The project includes further:

- A list of the 56 most common algebra-related Misconceptions and Errors (MaE) in middle school arranged by topic. (tab 1)
- An example of each MaE. (tab 3)
- An alignment between the 56 MaEs, the math Common Core (CC), and the National Assessment of Educational Progress (NAEP). (tab 3)
- List of NAEP’s topics and some sample questions.
- List of 6th to 8th-grade CC with link to an example worksheet
- Folder with all cited papers
- A deeper explanation of the MaE.

- A list of the 27 most common algebra-related Misconceptions and Errors (MaE) in middle school arranged by topic. (tab 1)
- An example of each MaE. (tab 2)
- An alignment between the 27 MaEs, the math Common Core (CC), and the National Assessment of Educational Progress (NAEP). (tab 2)
- List of NAEP’s topics and some sample questions (tab 3,4 & 5).
- List of 6th to 8th-grade CC with link to an example worksheet (tab 6)

## Ongoing Work to Deepen the MaE Data Set

Students can express a misconception in many different ways, e.g., it can be context dependent or be influenced by other misconceptions, so more than one single example per MaE might be needed to diagnose the MaE or understand its impact. Deeper explanations and more examples are already ready for some MaEs, but not all of them. MaE’s multiple examples and deeper explanations are a work in progress.

Misconception on geometry, measurement, and data analysis are also a work in progress. Finally, given that researchers and educators have found ways to address and change MaEs I hope to compile and match some of their methods to the current list one day. Feel free to contact me at nancy.otero.o@gmail.com if you have any questions or feedback.

-Nancy Otero