ACMS 2017: Cultivating Mathematical Affections through Engagement in Service-Learning

Here is some information on my talk at the 21st ACMS Conference (2017) at Charleston Southern University.

Abstract:

Why should students value mathematics? While extensive research exists on developing the cognitive ability of students, very little research has examined how to cultivate the affections of students for mathematics. The phrase “mathematical affections” is a play on the affective domain of learning as well as on the general notion of care towards something. Mathematical affections are more than a respect for the utility of the subject; the term is much broader and includes aesthetic features as well as habits of mind and attitude.

This paper will analyze the findings from a research project exploring the impact of service-learning on the cultivation of mathematical affections in students. This was a qualitative case study of high school students who recently completed a service-learning project in their mathematics course. Data was gathered from student interviews, reflection journals, and field observations. The framework for the analysis follows the definition of “productive disposition” offered by the National Research Council (2001) as well as the concept of formative “cultural liturgies” offered by the philosopher James K.A. Smith (2009).

The major themes that emerge from the data indicate that through service-learning students see math as sensible, useful, and worthwhile. This supports the potential of service-learning as a pedagogical tool that can be utilized to develop a productive disposition in students; addressing at a practical level how the affective objectives of national policy documents can be achieved.

PowerPoint:

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References:

Goldin, G.A. (2002). Affect, meta-affect, and mathematical belief structures. In G.C. Leder, E. Pehkonen, & G. Törner (Eds.),  Beliefs: a hidden variable in mathematics education? Netherlands: Kluwer Academic Publishers, pp. 59-72.

Hadlock, C. R. (2005). Mathematics in service to the community: Concepts and models for service-learning in the mathematical sciences (No. 66). Mathematical Association of America.

Krathwohl, D.R., Bloom, B.S., & Masia, B.B. (1964). Taxonomy of educational objectives: Handbook II. Affective Domain. New York: Longman.

National Research Council (2001). Adding it up: Helping children learn mathematics. Washington D.C.: National Academy Press.

Smith, J.K.A. (2009). Desiring the kingdom: Worship, worldview, and cultural formation. Grand Rapids, MI: Baker Academic.

Wilkerson, J. (2015). Cultivating Mathematical Affections: The Influence of Christian Faith on Mathematics Pedagogy. In Perspectives on Science and Christian Faith, 67(2), 111-123.

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The Importance of Playing Math

Math is fun.

It is amazing how many people today would simply scoff at that statement. In their minds, math is the complete opposite of fun. But I’m not stating an opinion, as in “I think math is fun” (though I do). This is a fact: Math. Is. Fun.

…at least it is when we first encounter it, as young children who simply have unending curiosity and interest in puzzles. The biggest flaw in our current math educational system is that it by in large removes that genuine curiosity and interest that students bring with them to the classroom. The result is that over time math becomes less fun and more of a rote chore.

Over the last few weeks a series of similar articles, all addressing the importance of recreational mathematics for children, came across my inbox:

“Bedtime problems boost kids’ math performance” (sciencemag.org)

“Mommy? Daddy? Read me a word problem,” is probably not a request that many parents hear. Yet if a school child’s parents replace a bedtime story with a math discussion even one night a week, the child’s math skills may improve markedly compared with peers who listen to nonmathematical stories, a new study shows.

“Where the Wild Fractions Are: The Power of a Bedtime (Math) Story” (npr.org)

…I talk about stress and performance, I mention how you don’t hear people walking around bragging that they’re not good at reading. But very intelligent people brag about not being good at math. And it turns out that that anxiety and social acceptability has implications for our nation’s success in math and science fields. And it’s really important that we as parents and teachers and adults try to convey to our kids that math is something that’s (a) enjoyable and (b) learned. You’re not born a math person or not; it’s something that’s acquired. And every time we talk about it and we integrate it into our daily lives, children may see the importance of it and that math is not something to be fearful of.

The Importance of Recreational Math (nytimes.com)

In his final article for Scientific American, in 1998, Mr. (Martin) Gardner lamented the “glacial” progress resulting from his efforts to have recreational math introduced into school curriculums “as a way to interest young students in the wonders of mathematics.” Indeed, a paper this year in the Journal of Humanistic Mathematics points out that recreational math can be used to awaken mathematics-related “joy,” “satisfaction,” “excitement” and “curiosity” in students, which the educational policies of several countries (including China, India, Finland, Sweden, England, Singapore and Japan) call for in writing. In contrast, the Common Core in the United States does not explicitly mention this emotional side of the subject, regarding mathematics only as a tool.

A colleague of mine, Scott Eberle, I know has a great interest in these issues of engaging children’s natural curiosity (particularly on the level of aesthetics), authoring an article on “The role of children’s mathematical aesthetics: The case of tessellations” for the Journal of Mathematical Behavior. I am still hoping to have Scott write a guest post for this site when he is able. For now, I’d like to share how I have tried to put this into practice at my school.

Our school recently started an after school recreational math club for kids in grades K-5. We use the materials from the first two articles cited above generated by Bedtime Math. The first activity actually had to do with tessellations (as referenced in Scott’s article). Below are some pictures of the kids playing math using glow sticks to make glow-in-the-dark tessellations:

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It has been exciting to hang out with the younger kids and play math. We have even involved our high school Mu Alpha Theta students as volunteers to further show the younger kids that even the older kids can still find this fun. Every activity also has follow up questions to help extend students’ curiosity to deeper levels of mathematical insight.

Everything we do in math club is meant to show how much fun math can be. Our hope is that all of our recreational math activities will instill in these kids an abiding affection for math.

Go play math and enjoy!

A Perspective on “Favorite Classes”

Slate.com recently ran a series on “Favorite Classes.” I thought that it would be worth sharing, and briefly commenting on, their perspective on mathematics and statistics.

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First, I love the title on their post about mathematics: You’re Not Actually Bad at Math. Here are some excellent quotes:

The idea that someone can be bad at math is wrong, and it hides several harmful assumptions. It’s an excuse to justify individual failure, rather than a real understanding of mental capabilities. Giving up on math means you don’t believe that careful study can change the way you think. No one is born knowing the axiom of completeness, and even the most accomplished mathematicians had to learn how to learn this stuff. Put another way: Writing is also not something that anyone is “good” at without a lot of practice, but it would be completely unacceptable to think that your composition skills could not improve.

I agree with the author that the phrase “I am not a math person” is used largely as an excuse to help the self esteem of a person who has had difficultly in mathematics. The reality of the situation is that difficulty and struggle are inherent in the process of doing mathematics well. The only reason a person would think that their initial failure to grasp a concept makes them ‘not a math person’ is that they have come to understand the practice of mathematics as solely about obtaining ‘correct’ answers. This leads to a second key quote:

It seems that the origin of math phobia is not the content of math itself; it cannot rest solely on someone’s inability to sit through logic puzzles, because people exercise careful abstract reasoning in every other field without the same sort of fear. Instead, I think the form is largely to blame. All of high school math is basically a one-way linear staircase that leads to calculus. If you fall off at any point, you’re doomed. Calculus prep has infiltrated the curriculum to such a degree that I think people conflate doing algebra with all of math. Students spend so much time memorizing computational tricks that they don’t get to see anything else—that those algorithms have a logical derivation, and that plenty of math isn’t even like that.

In short: the form of teaching matters, not just the content that is taught. I have discussed the form of teaching before. If math is taught purely as algorithmic thinking that leads to a single ‘correct’ solution then it should not be surprising to hear the people who don’t get the ‘correct’ answer to label themselves “not a math person.” As I have discussed before on the topic of the form of teaching, we as math educators need to be mindful of not just the content we teach but also how the manner in which we teach it is shaping student perspectives on mathematics. On this note I encourage you to search for articles on “Productive Struggle” in mathematics. Here is one such article from a colleague at Texas State University that was recently published in the Journal of Mathematics Teacher Education.

See also this article from the Atlantic on “The Myth of ‘I’m Bad at Math.'”

Finally, I believe this last quote speaks directly to the mindset we as math educators should be seeking to instill in our students:

Not every educated person needs to be a mathematician, but no educated person should be afraid of the steps it takes to get there.

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The second article is titled: “What are the Odds? To learn to think critically, take a statistics class.”

If there’s one skill almost everyone agrees schools should be teaching, it’s critical thinking, although what, exactly, critical thinking consists of is conveniently left undefined. For the longest time, I preferred to believe that it meant learning to be skeptical about words, specifically the arguments, exhortations, and beguilements foisted upon the public by politicians, advertisers, corporations, and the dodgier elements of the press. As a former English major, I figured I had this one nailed; if there was anything I mastered in college, it’s the ability to find the hidden and sometimes manipulative meanings in language.

What I, in my complacency, chose to ignore is just how much of the persuasion now aimed at the average citizen comes in the form of numbers, specifically numbers that tell us about the future, about how likely something is to happen (or not happen) based on how much it happened (or didn’t) in the past. These numbers sing to us the siren song of cause and effect, humanity’s favorite tune. Why do we like it so much? Because knowing what causes events and conditions is the first step toward controlling them, and we human beings are all about controlling our environments. That’s how we ended up ruling this planet, and it’s how some of us hope to save it.

…Statistics and the science of probability represent the ultimate in critical thinking, because they teach us how to criticize the ways we habitually think

I have written before on the Myth of Critical Thinking in Mathematics. To “think critically” is by definition “to be critical of thoughts” or in other words “to critique ideas” as they arise rather than accepting or rejecting them blindly. Critical thinking means thinking rationally and reasoning through arguments with care and consideration of the options on the table. This, of course, sounds all well and good until we as educators stop and actually consider the reality of the situation in the math classroom.

A critique can only be meaningful if you already have a standard by which to judge something. I can’t critique an argument as “true” unless I know what it means for something to be “true.” I can’t critique a painting as “beautiful” unless I know hat it means to be “beautiful.” I can’t critique an act of charity as “good” unless I know what it means to be “good.”

There is an underlying standard of judgment that is being instilled in students (whether knowingly or unknowingly) and this is the root of what is occurring in mathematics classrooms. The way in which students implement their “critical thinking skills” upon the completion of their mathematics courses is simply a symptom of a deeper reality that is being formed within them. That reality can be formed from a Christian perspective where it is God who sets the absolute standards for Truth, Beauty and Goodness, or it can be formed by very humanistic standards of relative truth, subjective beauty, and goodness defined by utility.

The humanistic standards that are pervasive in our educaitonal system is what I believe leads to the last part of the quote above: “We human beings are all about controlling our environments. That’s how we ended up ruling this planet, and it’s how some of us hope to save it.” A Christian perspective realizes that we as human beings are indeed all about controlling our environments, but that is as a result of sin and brokenness and not a trait to be admired. We wound up ruling this planet not by innate ability but by God’s granting to us the stewardship of His creation. And we can never save this planet – we can only rest in the promised hope of future redemption in Christ.

I strongly support the claims made by this article on the need for an understanding of statistics, especially in this age of digital media. However I believe we also must always be mindful of the worldview that we bring to the table in understanding statistics.