Math Exams

I gave exams this week, which means that both my students and I are in emotional turmoil. My calculus students struggled to complete the exam in time, which isn’t typically true of my exams, so I need to compare this year’s exam to last year’s. I did have a worry as I was writing it that it was a little long, and I should have listened to my gut! The exam I gave in abstract algebra didn’t seem too long for the students, but I know students struggled. Now I have to grade all of the accumulated exams and have the usual emotional baggage. I feel disappointed in myself and in the students. I doubt myself. I question my fitness for teaching. I get angry at the students for not trying harder, and I even get angry at them for making mistakes. I feel hopeless about the class and about the possibility of any forward progress at all. Dreary and gross stuff that I really don’t even want to admit.

Exams are a situation of artificial pressure. Exams are weak on authentic importance. These exams are only important because I will use it to write down grades for the students. The grades are important to me because they give me a way to assess my class and the students in it, determining whether individual students and the class as a whole met the objectives of the course. The grades are important to the students because they want good grades in order to stay in school, keep scholarships, look good to others, be attractive to employers, and meet requirements of a program or major. Note that none of those things involve student learning. What I want to do in my life is to help students understand and use mathematics, to be powerful with math. A test can only do that as an accidental outcome. A test might help me to assess if I have helped students, but the only way for a test to help with learning is if the pressure of the test helps students to put forth more effort, or if, when faced with a bad test outcome students make a change in their learning habits or approach.

But I have seen first hand what happens when students aren’t having tests — the majority of them don’t push themselves to work. Maybe students are addicted to tests, and thus we are all addicted to this unpleasant experience. Maybe I’m addicted to tests because I have developed too few other methods for helping students to motivate themselves. In any case, I don’t know what to do about it, so I keep giving tests.

Is Algebra Really Necessary?

Today I finally got around to reading Andrew Hacker’s opinion piece in the New York Times from July. Yes, I am behind.

Hacker starts the piece by describing how he used to believe in the “virtue” of learning algebra, but that now he sees the reasons for doing so as based on wishful thinking rather than evidence. He then talks about the problems with algebra taking resources away from other quantitative work, and causing students to drop out of high school. And universities often require algebra and other higher mathematics for admission, excluding students that may have real talents in other areas. Only 58% of entering college freshmen graduate college and math is often to blame. He also calls mathematics education to task for teaching math skills that have little relation to those needed in workplaces. He ends with a call for a focus on quantitative reasoning and a call to reduce the amount of abstract mathematics we require of all students.

Hacker also tries to rebut some of the likely counter arguments, noting that quantitative and statistical literacy are indeed vital in our world, but counters that there is no evidence that being able to do algebraic manipulation actually improves the kind of quantitative reasoning skills that we need to be citizens and have access to employment.

There is lots to love about this op-ed, starting with the fact that it is an editorial about math written by someone who is not a mathematician or mathematics educator. It doesn’t herald the coming of a great new reform that will fix all of our mathematical woes. It asks us to take a look at what we are doing and why, and that is something that we should be doing more of. Why is our mathematics curriculum structured in the way that it is? How did algebra and calculus get to be so important? There are answers to these questions, if we care to look. For instance, this history of math education by David Klein. And what are our real goals in our K-12 curriculum and in our college curriculum? Why those goals and not others? Who does our mathematics curriculum support well, and who is disenfranchised by it?

I’m thinking about these questions all of the time, so I’m curious to hear the answers of others and will be working on putting my own answers out there. I’m thankful to Hacker for opening the conversation, even if I was too wrapped up in my own thing to listen.


Math Anxiety and Critical Mathematics Education

I’ve been reading and writing a bit lately on two research and education movements that started in the late 70s and are still around in some form today — the math anxiety movement and critical mathematics education.

The term “mathematics anxiety” first appeared in the literature in the early 1970s (Suinn et. al., 1972; Nash, 1970), and was popularized by Sheila Tobias and others working in the late 1970s to address mathematical avoidance, particularly in women (Tobias, 1976, 1993). When Tobias published her article “Math Anxiety: Why is a Smart Girl Like You Counting on Your Fingers?” in Ms. Magazine in 1976, she was calling for an explicitly feminist and political response to a subject that was acting as a “critical filter” (Sells, 1978) that kept women out of science and engineering. When Tobias published her article on math anxiety in Ms. Magazine in 1976, she was calling for an explicitly feminist and political response to a subject that was acting as a “critical filter” (Sells, 1978) that kept women out of science and engineering. In response, a body of research blossomed to examine math anxiety and its correlates, including gender (Ashcraft & Ridley, 2004). However, the frame of anxiety pathologized the experiences of people struggling with mathematics and made those struggles individual rather than communal. The reality may be that what is termed “anxiety” is a normal response to mathematics education (Johnston, 1995). Anxiety is an individual problem, but we have a society-wide issue with mathematics in the United States (Ginsberg et. al., 2005), so we need context for mathematical struggles that can include community, culture, and politics (Lave, 1988; Apple, 1990).

Overlapping in time with the movement to create awareness of mathematics anxiety was another movement that impacted mathematical education — critical pedagogy. The critical pedagogy movement began with Paolo Friere’s Pedagogy of the Oppressed in 1970. In the 1980s the movement was continued with Shor, Giroux and others expanding the theory to colleges, K-12 schools, and producing work that allowed teachers to engage practically with the complex and theoretical issues Friere raised (Shor, 1987). There were several people working to bring critical pedagogy to mathematics, in what proponents termed “critical mathematics education” (see, for example Frankenstein, 1987).

As a movement, critical pedagogy was also concerned with mathematics as a critical filter (Skovomose, p. 2) as was mathematics anxiety, but the focus of the movement social justice rather than individual emotional reactions to mathematics. The push for critical mathematics education suffered arguably from too much politicization, as it was a highly leftist movement, and not every classroom is going to be able or willing to adopt and clearly left-wing agenda. Critical mathematics education and ethnomathematics show us that it is possible to empower people through mathematics. Critical mathematics education emphases the context in which mathematics education occurs, an aspect largely neglected by educational theory, which makes a variety of assumptions about the classroom — that the teacher has sufficient content knowledge, students are not struggling with poverty and hunger, that there is no violence, and so forth (Skovsmose, 2008). The problem with this is that it obscures study of how inclusion and exclusion operate within mathematics education (Skovsmose, 2008).

Mathematics eduction is assumed to be simply and universally a positive. This view is expressed for instance by Peter Hilton (1980), “Math avoidance has always existed; it is an urgent problem now because mathematics is so important today to the citizens of an industrialized nation-in their daily lives, in their job opportunities, in their understanding of the world around them. Math avoidance stultifies and impoverishes; mathophobia may be compared with the loss of one of the primary senses.” (Hilton, 1980) Indeed, there is a nearly universal view that mathematics is important, largely due to its usefulness even among people who hate mathematics and mathematics classes (Sam, 1999). People do not question the primacy of mathematics , because mathematics is seen as the bedrock of science and technology, which is the path of progress as a society, and a primary center of our positivist education (Frankenstein, 1987; for an exception, see Johnston). However, the apparent importance of mathematics does not mean that people really need to develop mathematical competence, for as Frankenstein says, “Both the (apparent) complexities of technology and the (superficially) wonderful concrete changes it has made…convince people that control over our high-tech society must be left to ‘experts’” (Frankenstein, 1987, p. 185). This produced a wonderful tension between a nearly universal belief that math is important and an equally universal avoidance of the subject!

I’m working right now looking and connections and disconnections between math anxiety and critical mathematics education right now (so I’ll write more later), but one of the things that I am consider is looking at all of this through an ethical frame. Kohlberg put framework of moral and ethical development in which a person moved from self-centered concerns to concerns that were oriented toward justice and universal rights (Nair, 2005). In the 1970s, Carol Gilligan proposed an alternative framework, called the ethics of care, in which development moves from care of the self, to care of others, to balancing ones own needs with others (Nair, 2005). This is an ethics based on care, responsibility and relationships. I see the mathematics anxiety movement as being very much situated within the ethics of care. Our wish to be of service to people who are anxious about mathematics would hopefully lead us to certain actions as a society and educational system that would be of service. The critical mathematics education movement fits more into an ethics of justice (although care is certainly still present). It is interesting to me that the math anxiety movement actually began with a justice concern, but became more about care. Of course justice and care are not mutually exclusive! I’m still trying to come up with coherent thoughts about this, so thoughts and suggestions are more than welcome.

References: I know I missed some here, so let me know if you want ones that aren’t here!

  • Frankenstein, M. (1987). Critical Mathematics Education: An Application of Paulo Freire’s Epistemology. In I. Shor (Ed.), Freire for the classroom: a sourcebook for liberatory teaching (pp. 180–210). Portsmouth: Boynton/Cook.
  • Hilton, P. (1980). Math Anxiety: Some Suggested Causes and Cures: Part 1. The Two-Year College Mathematics Journal, 11(3), 174–188. doi:10.2307/3026833
  • Nair, I. (2005). Ethics of Care. In C. Mitcham (Ed.), Encyclopedia of science, technology, and ethics. Macmillan Reference USA.
  • Sam, L. C. (1999). Public images of mathematics. Retrieved from
  • Skovsmose, O. (2008). Critical mathematics education for the future. ICME-10 Proceedings.
  • Suinn, R. M., Edie, C. A., Nicoletti, J., & Spinelli, P. R. (1972). The MARS, A Measure of Mathematics Anxiety: Psychometric Data. Journal of Clinical Psychology, 28(3), 373–375. doi:Article
  • Tobias, S. (1976). Math Anxiety: Why is a Smart Girl like You Counting on Your Fingers? Ms., (September 1976), 56–59.
  • Tobias, S. (1993). Overcoming Math Anxiety. W. W. Norton & Company.