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Rachel Rupnow

Virginia Tech

Previous research on teachers' and students' beliefs has focused on promoting availing beliefs, or beliefs that have been correlated with improved performance on achievement measures. Much of the literature on students' beliefs at the college level has shown that students hold non-availing beliefs. Thus, as teachers, one of our goals is to find ways to promote availing beliefs about math, such as beliefs about math as an ever-expanding field of knowledge or that math is about problem solving, not receiving knowledge. Information about availing and non-availing beliefs is considered, as well as ways to ascertain students’ existing beliefs.

Should we care about students’ beliefs about math? According to research, some beliefs about math are associated with better learning outcomes. Such beliefs are called availing beliefs (Muis, 2004). In this essay, I consider some previous research on availing beliefs, common beliefs held by students and teachers, and ways to consider our own students’ beliefs.



In her summary of the literature, Muis coined the term “availing beliefs” and defined them as  “beliefs associated with better learning outcomes” (Muis, 2004, p. 323). This was in contrast to nonavailing beliefs that have “no influence on learning outcomes or negatively influence learning outcomes” (Muis, 2004, p. 323). Muis’ classifications were based on statistically significant, large, positive correlations with learning and achievement outcomes. Based on her review of the literature, she identified some specific availing beliefs, including the idea that mathematical knowledge as a field changes and research is ongoing; that math as a discipline is focused on problem solving, not memorization; that problems may require extended time and multiple attempts to be solved; and that a growth mindset (Yeager & Dweck, 2012) focused on the ability to always improve on what we know is beneficial instead of having a fixed mindset focused on having or lacking gifts in an area.


While we would like students and teachers to have availing beliefs about math, teaching, and learning, many common beliefs seem to be nonavailing. Common student beliefs include that math has no meaning (Boaler, 2000); that any problem should be solvable in 5 minutes or less (Schoenfeld, 1988); that students’ role is to be passive in math class (Stodolsky, 1985); and that being able to reproduce a procedure indicates understanding (Schommer, 1998).

While it is important to consider students’ beliefs, as teachers, it is also important to reflect on our own beliefs about math, teaching, and learning and see how our beliefs and actions might align with students’ beliefs. For example, consider Figure 1 in which a set of beliefs a teacher might hold and questions that could be asked to aid reflection on how those beliefs are or are not being expressed in class are given.

Teacher’s belief

Reflective question

1.       Math is an interconnected structure.

1.   Do I teach disconnected procedures or show this interconnected structure?

2.       Math is about problem solving.

2.   Do I have students engage in problem solving regularly in class?

3.       To learn math requires hard work.

3.   Do I support students as they struggle?

4.       Anyone can learn math.

4.   Do students feel they can be successful?

5.       “Learning” math involves not only facility with procedures but also understanding why procedures work.

5.   Do my students know why they learn procedures?

6.       Teaching math requires allowing students to do math.

6.   Do I give students opportunities to do math meaningfully in class?

Figure 1: Potential teacher’s beliefs and questions for reflection based on beliefs


While reflective questions can be a place to ponder what students might believe, we want evidence to support the conclusions we draw. One way to gather this evidence is to directly ask students extra credit questions on homework. Some questions to consider include asking and explaining what they think math is or “What animal do you think math is like?” (Markovits & Forgasz, 2017, p. 56). For either of these questions, the explanation is the more interesting part, as it may reveal their beliefs about the structure or purpose of math or reveal positive or negative affective reactions to math. Other questions might relate more to students’ motivation in class. One way to examine students’ motivation in a class is with the MUSIC model of motivation, which is freely available for use (Jones, 2017). Other data collecting options include asking students when tutoring or in whole class discussion, and observing what students do when given time to work (e.g. persisting on a problem for a long time or giving up after a short time).


Based on previous research, we want students to have certain views about math, teaching, and learning because they are more likely to align with positive learning outcomes. Moving forward, we can consider our own beliefs and look to determine students’ beliefs in a variety of ways with a goal of influencing students’ beliefs through interventions or regular classroom practices so that students come to hold more availing beliefs.


Boaler, J. (2000). Mathematics from another world: Traditional Communities and the alienation of learners. Journal of Mathematical Behavior, 18(4), 379–397.

Dweck, C. S. (2007). Is math a gift? Beliefs that put females at risk. In S. J. Ceci & W. M. Williams (Eds.), Why aren’t more women in science?: Top researchers debate the evidence (pp. 47–55). doi: 10.1037/11546-004

Jones, B. D. (2017). User guide for assessing the components of the MUSIC model of academic motivation. Retrieved from

Markovits, Z., & Forgasz, H. (2017). “Mathematics is like a lion”: Elementary students’ beliefs about mathematics. Educational Studies in Mathematics, 96, 49–64. doi: 10.1007/s10649-017-9759-2

Muis, K. R. (2004). Personal epistemology and mathematics: A critical review and synthesis of research. Review of Educational Research, 74(3), 317–377.

Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of ‘well-taught’ mathematics courses. Educational Psychologist, 23(2), 145–166. doi: 10.1207/s15326985ep2302_5

Schommer, M. (1998). The role of adults’ beliefs about knowledge in school, work, and everyday life. In M. C. Smith & T. Pourchot (Eds.), Adult learning and development: Perspectives from educational psychology (pp. 127–143). Mahwah, NJ: Erlbaum.

Stodolsky, S. S. (1985). Telling math: Origins of math aversion and anxiety. Educational Psychologist, 20, 125-133.

Yeager, D. S., & Dweck, C. S. (2012). Mindsets that promote resilience: When students believe that personal characteristics can be developed. Educational Psychologist, 47(4), 302–314. doi: 10.1080/00461520.2012.722805

Virginia Council of Teachers of Mathematics
PO Box 73593
Richmond, VA 23235 

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