In today’s classrooms, teachers are called upon to gather and use evidence of student thinking in a timely, formative way to implement and differentiate instruction that improves learning for all students. An important part of that process is “to identify and address potential learning gaps and misconceptions when it matters most to students, which is during instruction, before errors or faulty reasoning becomes consolidated and more difficult to remediate.” (National Council of Teachers of Mathematics, 2014, p.53).

Mathematics misconceptions are often considered overgeneralizations, that is, information extended or applied to another context in an inappropriate way. Since it is impossible to teach in a way that avoids creating misconceptions, we have to accept that students will sometimes make incorrect generalizations that remain hidden unless the teacher makes specific efforts to uncover them. A teacher’s role is to minimize the chances of students’ harboring misconceptions by knowing the potential difficulties students are likely to encounter, using formative assessments to elicit misconceptions and implementing instruction to help students build conceptual understanding of the mathematics. (Rose Tobey, Arline, Fagan, Minton 2009-2014, 2018; Hattie, Fisher, Frey, 2017)

**Defining Mathematics Misconceptions**

A misconception pertains to a flaw in *conceptual understanding* of a mathematical concept, which leads to an error in application. Mathematical errors can and do occur because of misconceptions but this is not true of all mathematical errors. For example, a copying mistake or simple calculation error within a larger multiple-step process are errors that are not caused from misconceptions.

The topic of comparing fractions can be used to demonstrate a number of conceptual flaws that students can harbor.

- Conceptual Flaw 1:
*Overgeneralizing from whole numbers*by comparing the given numerators or denominators and determining which is greater based on the “biggest” number.

*“** ^{5}/_{7} is larger than ^{3}/_{4 }*

*because 7 is greater than 4”*

- Conceptual Flaw 2:
*Overgeneralizing from unit fraction concepts*by associating the smaller numerator with the smallest fraction (i.e.^{1}/_{5}>^{1}/_{8}) regardless of the number in the numerator.

*“** ^{3}/_{8} is larger than ^{11}/_{12 }*

*because 8*

^{th}are bigger than 12^{th}”- Conceptual Flaw 3:
*Using gap reasoning*by determining the number of parts needed to make a whole, rather than reasoning about the size of the parts.

*“ ^{7}/_{8}*

*is equivalent to*

^{5}/_{6 }*because they both need 1 more to make a whole ”*

- Conceptual Flaw 4:
*Using different size wholes*when representing fractions by creating equivalent parts rather than equivalent wholes or by comparing parts of different size wholes.

*“** ^{2}/_{10 }is greater than ^{1}/_{65} *

*because more is shaded”*

**Moving Learning Forward**

Learning about common student difficulties and misconceptions and their root causes helps teachers to more effectively use the process of formative assessment (Creighton, Tobey, Karnowski, Fagan, 2015). This process includes:

- Developing learning targets for a series of lessons designed to close the gap in understanding.
- Engaging students in activities that develop conceptual understanding of the mathematics.
- Eliciting evidence by embedding additional formative questions and tasks throughout the series of activities.
- Building in structures and routines to help students learn to assess learning (their own and their peers) against the learning targets and to use feedback to discuss and revise thinking.
- Continuously making adjustments to the learning targets and associated instructional activities based on identified needs from the embedded assessment tasks and questions.

For the comparing fractions example described above, a teacher may need to develop a series of instructional activities designed to target the whole number overgeneralization misconception, focusing the learning target on using models to justify reasoning when comparing fractions. An example of an exploration in the series of activities follows.

Following additional explorations with different denominators, the teacher closes the lesson with an exit ticket in which students use models to describe the relationship between several pairs of fractions. Responses are analyzed to determine the extent to which students are meeting the learning target and to determine next steps.

When teachers are clear about what conceptual understandings they are working to build and know the potential root causes of misconceptions related to those concepts, they are well situated for the next step in the process: moving from diagnostic to formative assessment enhanced instruction by providing targeted learning activities and deeply involving the student in the learning process.

**References:**

Creighton, S.J., Rose Tobey, C., Karnowski, E., Fagan, E. (2015). *Bringing Students into the Formative Assessment Equation: Tools and Resources for Math in the Middle Grades. *Thousand Oaks, CA: Corwin.

Hattie, J., Fisher, D., Frey, N. (2017). *Visible Learning for Mathematics, Grades K-12*. Thousand Oaks, CA: Corwin.

National Council of Teachers of Mathematics. 2014. *Principles to Actions: Ensuring Mathematical Success for All. *Reston, VA: National Council of Teachers of Mathematics.

Rose Tobey, C., & Arline, C. (2014). *Uncovering student thinking about mathematics of the common core: Grades 9–12: 20 formative assessment probes. *Thousand Oaks, CA: Corwin.

Rose Tobey, C., & Arline, C. (2014). *Uncovering student thinking about mathematics of the common core: Grades 6–8: 20 formative assessment probes. *Thousand Oaks, CA: Corwin.

Rose Tobey, C., & Fagan, E. (2013). *Uncovering student thinking about mathematics of the common core: Grades K–2: 20 formative assessment probes. *Thousand Oaks, CA: Corwin.

Rose Tobey, C., & Fagan, E. (2014). *Uncovering student thinking about mathematics of the common core: Grades 3–5: 20 formative assessment probes. *Thousand Oaks, CA: Corwin.

Rose Tobey, C., & Minton, L. (2010). *Uncovering Student Thinking in Mathematics Grades K–5:25 Formative Assessment Probes for the Elementary Classroom. *Thousand Oaks, CA: Corwin Press.

Rose, C., & Arline, C. (2009). *Uncovering Student Thinking in Mathematics Grades 6–12: 30*

* Formative Assessment Probes for the Secondary Classroom. *Thousand Oaks, CA: Corwin Press.