math education

The Role of Mathematical Aesthetics in Christian Education

joshwilkerson1 Comment

by R. Scott Eberle

Scott Eberle has a Ph.D. in Math Education and currently serves as a missionary in Niger, working to spread the Gospel message through Christian education. Scott works to build up Christian leaders and educators in Niger who approach mathematics through a distinctly Christian perspective. You can follow Scott and his family at nigerministry.tumblr.com.

Josh Wilkerson invited me to contribute something on the aesthetics of mathematics from a Christian perspective. I’d especially like to discuss how such a seemingly abstract idea has application in Christian education.

Detail of the Mandelbrot Set in the plane of complex numbers
Detail of the Mandelbrot Set in the plane of complex numbers

Mathematical Aesthetics

Mathematics has been considered an aesthetic subject from antiquity. The Greeks considered mathematics to be the highest form of aesthetics because of its perfection. The Pythagoreans and Platonists considered mathematical concepts to have a real, mystical existence in some perfect realm.

Throughout history, mathematicians and philosophers have continued to claim that mathematics is beautiful for a variety of reasons. For example, whereas the Greeks saw beauty in the ontology of mathematics, the French mathematician Henri Poincaré saw beauty in its epistemology. Because of the way we teach mathematics, many students believe there is always one hard-and-fast method for cranking out the answer to any mathematical problem. But as mathematicians know, true mathematical problems require a great deal of creative intuition to solve. Poincaré pointed out that mathematicians rely on aesthetic-based intuition to distinguish fruitful paths of mathematical inquiry from dead ends. He wrote, “It is this special aesthetic sensibility which plays the rôle of the delicate sieve” (1908/2000, p. 92).

Today, nearly all mathematicians continue to recognize the aesthetic nature of mathematics (Burton, 1999). The British mathematician John Horton Conway went so far as to claim, “It’s a thing that non-mathematicians don’t realize. Mathematics is actually an aesthetic subject almost entirely” (Spencer, 2001, p. 165). The reason the general population doesn’t realize that mathematics is an aesthetic subject is probably due to the mechanical way in which we frequently teach mathematics. School exercises are often artificial, simplistic, and have only one right answer. There is nothing creative or aesthetic to see in the average math lesson.

Conway’s claim that mathematics is almost entirely aesthetic is a bold one. But actually, modern mathematics can be seen to be an aesthetic subject from its foundations to its methods to its end results. This is especially true since mathematics’ divorce from physics in the 19th century as it became a purely abstract study, inspired by, but independent of, the natural world.

  • Foundations: Mathematics rests on a foundation of axioms and definitions. But these are chosen, not deduced. Mathematicians choose definitions and axiomatic systems based on criteria of logic, relative completeness, consistency, mutual independence, simplicity, connectedness, and elegance. These criteria are partly aesthetic in nature.
  • Methods: As Poincaré pointed out, mathematicians rely on a certain aesthetic sense to guide their explorations. Paths that seem particularly elegant often prove to be the most successful. In 1931 Gödel destroyed earlier hopes of purely mechanical methods of generating mathematical theorems and proofs, making the fundamental role of intuition even more necessary. Modern researchers are beginning to understand that intuition is not a fuzzy feeling, but rather a rigorous source of insight. Robert Root-Bernstein (2002) makes a powerful argument that all scientific thought occurs first as an aesthetic intuition, and is then confirmed by verbal logic. Therefore aesthetics guides our mathematical exploration and is the basis for our mathematical reasoning. But we often show only the final algorithmic logic to our students.
  • Results: Mathematicians don’t often discuss aesthetics explicitly, but when they do, they usually point to theorems and proofs, which they insist should be elegant. The American mathematician Morris Kline observed that “Much research for new proofs of theorems already correctly established is undertaken simply because the existing proofs have no aesthetic appeal” (1964). Mathematicians especially appreciate results which are surprisingly simple or have significant connections or visual appeal. Such results are said to be beautiful. The Mandelbrot Set, for example, is beautiful partly because its definition is surprisingly simple and partly because it has great visual appeal. It is interesting to note that criteria such as significant connections indicate that beautiful results will be among the most useful and important. Criteria such as surprise suggest that beautiful results may be important for insight and understanding, and therefore also for education.

So mathematics is seen to be aesthetic “almost entirely.” At this point, some would say the discussion is merely philosophical and has no real world implications. Indeed, most mathematicians give aesthetics little explicit thought unless questioned about it. It is perhaps for this reason that many educators have not picked up on the importance of aesthetics in mathematics.

Simple, visual “proof” of the Pythagorean Theorem by Bhaskara II (12th century AD)
Simple, visual “proof” of the Pythagorean Theorem by Bhaskara II (12th century AD)

The Christian View

Throughout history, most mathematicians have been Platonist, at least in practice. We tend to think that mathematical ideas are discovered, rather than invented. In more recent times, some have questioned this, claiming that mathematics is simply the brain’s way of understanding how the universe is structured, and mathematics could be very different for an extraterrestrial species. (See, for example, Lakoff & Núñez, 2000.) Others disagree, pointing out how mathematics inexplicably predicts new discoveries. Of course, all agree that certain things, such as notation, conventions, and choice of axioms, are man’s invention. But where do the beautiful results we admire come from? The Greeks cannot be said to have “invented” the Pythagorean Theorem. Most would agree they (and other cultures) “discovered” it.

Most Christian theologians, from Augustine (354 – 430 AD) onwards, as well as Christian mathematicians, have agreed with a Platonist perspective, believing that mathematics is in the mind of God, and we discover these eternal truths. Mathematics cannot be part of Creation because it is not a physical part of nature—it is a collection of abstract ideas. One does not physically create abstract ideas, one conceives them. And God must have always known these ideas, so they have always been part of his thoughts. Mathematics preceded Creation and is untouched by the Fall. It is perfect and beautiful and contains awe-inspiring ideas, such as Cantorian infinity, which is part of God’s nature but not part of our physical universe. However, we ourselves are fallen, so our understanding and use of mathematics is imperfect.

Some modern Christian thinkers have proposed other possibilities similar to those of Lakoff and Núñez, making mathematics a human activity, or only one of many possible systems of mathematics in the mind of God. Nevertheless, all Christians affirm that mathematics is not independent of God. Even if there are other possible systems of mathematics, the one we know is the one God chose for us as good, and it has always been known by God. It is not some arbitrary invention. I like to think that when I am studying mathematics, I am studying the very thoughts of God, that mathematics is part of God’s attributes. God did not “create” love; God is love (1 John 4:8). Likewise God did not “create” one and three; God is one Being in three Persons. God did not “create” infinity; God is infinite. And so on.

But whatever position you take, whatever the ontology of mathematics, it should not surprise us that mathematics is beautiful, because God is beautiful. Mathematics is indeed “an aesthetic subject almost entirely.” Mathematical beauty and usefulness is a mystery only if we do not believe it comes from God. (See, for example, the classic article by Wigner, 1960.)

Euler's Identity, relating five fundamental constants and three basic operations, is often called the most beautiful result in mathematics (Wells, 1990).
Euler’s Identity, relating five fundamental constants and three basic operations,
is often called the most beautiful result in mathematics (Wells, 1990).

Education

Though aesthetics is part of the very foundation of mathematics, it is largely neglected in math classrooms. As mathematician Seymour Papert pointed out, “If mathematical aesthetics gets any attention in the schools, it is as an epiphenomenon, an icing on the mathematical cake, rather than as the driving force which makes mathematical thinking function” (1980, p. 192). However, an increasing number of researchers (including myself) have been noting important consequences of mathematical aesthetics for how we teach mathematics at all ages.

The interested reader can turn to researchers such as Nathalie Sinclair to see how modern research has been discovering the importance of aesthetics in mathematics education. Aesthetics is a “way of knowing” mathematics prior to verbal reasoning and should be an important part of our mathematics classrooms. Indeed, Sinclair (2008) has found that good math teachers tend to use aesthetic cues in their teaching implicitly, though they may not realize it. For example, teachers who reveal a “secret weapon” or present a surprising fact or note simpler ways to express certain solutions are modeling a useful aesthetic to their students. In my own research (Eberle, 2014), I have found that even elementary school children come with their own aesthetic ideas and use them in valid mathematical ways when given the opportunity to do open-ended math problems. And this is true of all children, not just those that are gifted in mathematics. Children’s initial aesthetic ideas are far from those of mathematicians, but through experience they are refined. Educators from John Dewey to the present day have argued that aesthetics is important for all of education, and now we are discovering how this is true for mathematics.

Nathalie Sinclair (2006) has proposed that mathematical aesthetics has three roles in education:

  1. Aesthetics gives intrinsic motivation to do mathematics. This is in contrast to the extrinsic coaxing we often use with students. Instead of “sugar-coating” math problems by placing them in artificial contexts, we should allow students to explore the natural symmetry and patterns found in every branch of mathematics. I sometimes challenge teachers to see how many patterns they can find in the “boring” multiplication table. They are usually very surprised. Students can also engage in mathematics in a natural way by pursuing projects they themselves have suggested. Such genuine contexts are highly motivational. (See these posts by Josh Wilkerson for a Christian perspective on this idea.)
  2. Just as with mathematicians, aesthetics guides students to generative paths of inquiry. When allowed to explore freely, children use their own aesthetics to find valid mathematical insights, though this may take time. Students need opportunities to pursue their own ideas and conjectures.
  3. Aesthetics helps students to evaluate their results. Often math is presented as black-and-white with only right and wrong answers. But if students are allowed to do more open-ended inquiry or project-based mathematics, they can use their growing sense of aesthetics to evaluate the solutions found.
Solution to an open-ended geometry problem found by a 4th grader by using aesthetic symmetry
Solution to an open-ended geometry problem found by a 4th grader by using aesthetic symmetry

Christian Education

As Christian educators, we should realize that God gives common grace and we should always be open to learning from the best results of secular research, filtered through the worldview shaped by our faith. Throughout history, Christians have often been at the forefront of recognizing the importance of aesthetics. God gave us our ability to appreciate beauty and patterns for a reason, and what is math if not the study of patterns (Hardy, 1940)? We Christians should be among the first to recognize the importance of educating the whole child, even in mathematics, and embracing research showing the importance of allowing aesthetics to have a deep role in education, including our mathematics instruction.

Even more importantly, we should be careful not to make a sharp dichotomy between “secular” knowledge and “spiritual” knowledge. Mathematics is often taught as if our faith had nothing to do with the knowledge we are learning. Though it is wrong to artificially “spiritualize” every lesson, at the very least Christian students should understand the relationship between their faith and their studies. One way to do this is to let students know that math is not just a series of arbitrary algorithms and heuristics to be memorized, but a rich, creative, beautiful subject to be explored and appreciated. And when students see some of the beauty of the subject, we can lead them to reflect on the Source of that beauty. Indeed, we are doing a great disservice to Christian students if we lead them to believe that a subject that is in the mind of God is somehow boring or ugly.

I have to admit I am distressed sometimes by certain popular views of mathematics. I remember reading one author who wrote that mathematics was part of Creation, and as such, the author seemed to believe mathematics was purely arbitrary, as if there were no special reason God created 2 + 2 to be 4. I often come across this idea that math is not understandable, a result of learning by rote. All we can supposedly do is grit our teeth and memorize the mysterious methods. This author’s solution was to teach students to plug away at exercises and learn to praise God every time they correctly found God’s answer, and be thankful that God, in his faithfulness, had not changed the answer in the meantime. I fear that such instruction will not generate praise for God but rather fear of mathematics. My hope is that we can learn instead how to teach that mathematics is a deep, joyful, meaningful, beautiful subject. It is a reflection of God’s nature.

Flower with spirals in Fibonacci sequence Helianthus flower, Bannerghatta Bangalore by L. Shyamal / CC-BY-2.5
Flower with spirals in Fibonacci sequence
Helianthus flower, Bannerghatta Bangalore by L. Shyamal / CC-BY-2.5

Conclusion

For Christians, mathematical aesthetics must not be an optional extra-credit topic, but must rather be at the very foundation of our mathematics teaching. As Christian educators, aesthetics should guide our understanding of mathematics, inform the way we teach, and be a goal for our students’ learning—and this from the youngest ages. Just as students learn to appreciate poetry or music, Christian students should learn that mathematics is beautiful, and why.

References

Burton, L. (1999). The practice of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37(2), 121-143.

Eberle, R. S. (2014). The role of children’s mathematical aesthetics: The case of tessellations. The Journal of Mathematical Behavior, 35, 129-143.

Hardy, G. H. (1940). A mathematician’s apology (1967 with Foreword by C. P. Snow ed.). Cambridge, UK: Cambridge University Press.

Kline, M. (1964). Mathematics in Western Culture (Electronic version ed.). New York: Oxford University Press.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.

Poincaré, H. (2000). Mathematical creation. Resonance, 5(2), 85-94. (Original work published 1908)

Root-Bernstein, R. S. (2002). Aesthetic cognition. International Studies in the Philosophy of Science, 16(1), 61-77.

Sinclair, N. (2006). Mathematics and beauty: Aesthetic approaches to teaching children. New York: Teachers College Press.

Sinclair, N. (2008). Attending to the aesthetic in the mathematics classroom. For the Learning of Mathematics, 28(1), 29-35.

Spencer, J. (2001). Opinion. Notices of the AMS, 48(2), 165.

Wells, D. (1990). Are these the most beautiful? The Mathematical Intelligencer, 12(3), 37-41.

Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1).

Advertisement

From the Classroom to the Community (and back again): Stories of Statistics, Significance, and Service

joshwilkerson6 Comments

This week I am giving two different talks at the 2015 Joint Mathematics Meetings in San Antonio, TX. What follows is information relating to the second talk. You can find my first talk here. My first talk was on cultivating mathematical affections – how we can change the way we understand affect in math education to produce students who value their mathematical experiences. My second talk is actually one practical example that can you can implement in the classroom to instill in students an appreciation of mathematics.

This talk was for a session on best practices for teaching introductory statistics. The focus of my talk was on integrating service-learning projects into the statistics curriculum, a topic that I have written about numerous times here at GodandMath. In addition to the resources that you will find below, feel free to check out some of the prior posts on service learning:

  • Serving through Statistics: the first (and largest) service project that I implemented complete with video summaries and interview with students.
  • AP Stat Reading Best Practices Presentation: Short presentation I gave on service learning in AP Statistics at the 2014 AP Statistics Reading in Kansas City, MO.
  • CAMT 2012 Presentation: Presentation I gave at the Conference for the Advancement of Mathematics Teaching based on the first statistics service-learning project mentioned above.
  • Geometry and the Homeless: the first service-learning project I did with my geometry students. An updated version from this last school year should find its way onto the site by mid summer.

ABSTRACT:

This presentation will outline the design, implementation, and evaluation of service-learning based statistics projects in which students partner with non-profit organizations in their local community. These projects synthesize the major concepts of experimental design, data analysis, and statistical inference in the real-world context of community service. Through these projects students integrate their conceptual understanding of statistics with the practical functioning of their local community, ultimately gaining a deeper appreciation for the role statistics plays in the organization and evaluation of service societies. Successful examples and practical resources will be provided.

PRESENTATION:

Below you will find the PowerPoint that accompanied my short 10 minute presentation (click on the image below to access the PowerPoint). Due to time constraints, the meat of the information can be found in the resource documents that I have also included below.

serving

OUTLINE:

The main question I aim to address  is this: what is the best resource that a teacher can introduce into his/her statistics classroom to help students make meaningful connections between course material and the true value of statistics?

I don’t think it is technology (be that calculators, iPhone apps, online applets, or statistical software packages) which is often discussed as a teaching aid in statistics. I don’t even think that is integrating current articles and published studies into classroom discussion.

Don’t get me wrong, both technology and current events can be powerful pedagogical tools and there certainly is a place for them in the classroom. As a teacher who regularly uses technology and “real-life” articles in my lessons, I would like to submit to you that there is actually something else, something better, that when used well can really cement the value of statistics in the hearts and minds of students. That something: service-learning. As it turns out, I think the best resource that you can introduce into a statistics classroom is to actually get the students out of the classroom and into the local community.

Why I think service-learning is an effective vehicle for communicating the significance and value of statistics to students:

  1. Students are actually doing statistics. 
    • There is something about the physical practice of getting outside the classroom to collect and analyze data that implants an appreciation for the processes of statistics into students.
  2. Students are actually doing statistics in an unfamiliar/uncomfortable (read: human) way.
    • In service-learning there is interaction with actual human beings. The data on the paper now has a face and the analysis becomes a little messier and less clinical. I find this tends to stretch students out of their comfort zone in a good way. It also encourages their focus to shift from individualistic outcomes (such as what grade they might receive) to more altruistic aims of education.
  3. Students are actually doing statistics in an unfamiliar/uncomfortable (read: human) way and they (as well as the community) are experiencing firsthand the fruits of their labor.
    • I require students to complete their project by giving an oral presentation to the service agency. Interpreting confidence intervals/levels, p-values, and significance levels becomes so much more meaningful to students when they have to explain these concepts to a service-agency and build connections for the agency as to what to do with this information practically moving forward.

DESIGN:

  • A non-profit service agency which requires survey research for program evaluation, grant applications, or client needs assessment is identified by the students.
  • Students will participate in a group which will provide the following services:
    • Meeting with agency and developing a survey instrument
    • Piloting and conducting survey*
    • Compiling, organizing, and analyzing data
    • Presenting final results to the agency
  • The teacher acts as a consulting facilitator outside of the direct chain of project command

KEYS TO SUCCESS:

  • The Power of Choice
    • Students have a vested interest in a personal topic
    • “How can we apply the concepts learned in statistics to benefit our local community/service agencies?”
  • Meaningful Applications
    • Real life scenario with real people
    • The “Aha Moment” – Deep connections drawn from course material to project implementation
  • Improving Civic Mindset, Professionalism, and Presentation Skills
    • Obligation is to the community/organization, not just a grade
    • Comfort levels stretched through community interaction
  • Required Reflection Beyond Calculations
    • Students chose the topic so they have to defend why it matters
    • Importance of statistics cemented

NEED FOR REFLECTION:

  • “Some people may think that this reflection process refers to a kind of ‘touchy-feely’ exercise that might be quite foreign to the mathematics classroom. I prefer to think of it as the processing of a rather complex set of experiences to assure that students share and solidify their insights and thus obtain maximum lasting benefits. This has actually been one of the most important contributions of the service-learning initiative.”
    • Hadlock (2005)
  • “Service-learning in its most effective and well-developed sense is more than another name for ‘real-world learning’ and consists of more than applied work in the public/non-profit sector. It involves a multilayered reflection process that can substantially increase its educational value in a broad sense…. Service-learning reflection asks the learner to become more aware of what he/she brings to the learning process: values, assumptions, biases – many of which are unexamined and potentially problematic….To leave these aspects unexplored would be to miss a vital educational opportunity, for they invariably stir up thoughts and feelings highly deserving of reflection and discussion.”
    • Zlotowski (2005)

Check out the presentation and the resource documents for more information. Always feel free to contact me through this site if you have any further questions or want to discuss the topic in more detail.

RESOURCES:

EXTERNAL RESOURCES:

Cultivating Mathematical Affections: Re-Imagining Research on Affect on Math Education

joshwilkerson2 Comments

This week I am giving two different talks at the 2015 Joint Mathematics Meetings in San Antonio, TX. What follows is information relating to the first talk. I will post information on the second talk in the coming days.

This talk is taken from my forthcoming article “Cultivating Mathematical Affections: The Influence of Christian Faith on Mathematics Pedagogy” for a special, math-themed issue of Perspectives on Science and Christian Faith. Once that article publishes I will link to the full text. For now I will simply share a copy of the presentation and encourage you that if the topic interests you, I will be posting much more on this concept of affective learning in mathematics as it is the main focus of my dissertation.

jmm15

ABSTRACT:

“When am I ever going to use this?” is a common question in mathematics. It is also more typically presented as a statement. It is a statement of frustration. It is the culmination of confusion and stress and typically serves as an exclamation by the student of their withdrawal from the mental activity at hand. I argue that the real question being raised by students is “Why should I value this?” We as math educators must do a better job of addressing this non-cognitive question. We need to do a better job of cultivating what I term as mathematical affections.

Affective language permeates national policy documents on the teaching of mathematics as an ideal we should strive to inculcate into students, but there is little discussion on how to go about doing this. This talk will examine the specific passages of the policy documents in question, discuss the shortcomings in the current body of research that exists on affect in math education, and outline a new framework (based on recent work in cognitive psychology and contemporary philosophy) for understanding how we might cultivate mathematical affections. Practical classroom resources and exercises will be offered.

OUTLINE:

Word Document of Outline

  1. Take a moment and visualize the best classroom experience that you have had as an educator. Or visualize what you would consider to be the ideal mathematics classroom.
    1. Think through a list of adjectives to describe that scene
    2. My wager is that those adjectives are more likely to do with the affective learning taking place in the classroom rather than the cognitive
    3. You certainly may have listed “students performing higher level critical thinking” but much more likely (and much more memorable) are phrases that describe student engagement and attitudes.
    4. If you did focus more on the affective outcomes of the classroom in this visualization exercise, you wouldn’t be alone…
  2. In a foundational article on affective learning in mathematics in the Handbook of Research on Mathematics Teaching and Learning, Douglas McLeod states: Affective issues play a central role in mathematics learning and instruction. When teachers talk about their mathematics classes, they seem just as likely to mention their students’ enthusiasm or hostility toward mathematics as to report their cognitive achievements. Similarly, inquiries of students are just as likely to produce affective as cognitive responses, comments about liking (or hating) mathematics are as common as reports of instructional activities. These informal observations support the view that affect plays a significant role in mathematics learning and instruction.
    1. While we can argue about the importance of affective learning objectives from many different angles, for now, based on our own experiences as educators, we understand the significant impact of affect in math education
    2. Education is inherently affective; it is inherently value laden. It is not a question of “Are you teaching values?” but rather “Which values are you teaching?”
  3. Value language is scattered throughout national policy documents on the teaching of mathematics.
    1. We see this language in national standards such as the NCTM Standards for Teaching Mathematics (1991) when it states that “Being mathematically literate includes having an appreciation of the value and beauty of mathematics as well as being able and inclined to appraise and use quantitative information” (emphasis added). Mathematical literacy, according to the NCTM, involves not merely using quantitative information (remember the “When am I going to use this?” question) but also giving the discipline of mathematics its proper value.
    2. Another national policy document, Adding it Up: Helping Children Learn Mathematics, a report published by the National Research Council (2001), argues that mathematical proficiency has five strands, one of which is termed “productive disposition.” Productive disposition is defined as “the habitual inclination to see mathematics as sensible, useful, and worthwhile.” To be mathematically proficient (not just literate, but proficient) the valuation of mathematics must lead to a habit of seeing mathematics as worthwhile – that is, valuable to justify time or effort spent. Math education is inherently value-laden.
  4. While we want to see these ideals in our students, nowhere (in these policy documents or elsewhere) is it discussed how we go about achieving this (How we cultivate these mathematical affections). Why is that? I think there are two perspectives, one from the classroom and one from research:
    1. Classroom: Veatch (2001) notes that “There is a prevalent attitude that one learns what is good mathematics by seeing and doing it, not by discussing values. The knowledge needed by the person entering the field will rub off on her. The classroom clearly reflects this attitude.”
      1. This seems to me to be the reason why there is no explanation in national policy documents as to how to go about forming affections in students
      2. The perception is simply: let’s address the cognitive demands of mathematics and meet those standards and then the students will value the experience.
    2. Research: In a special issue of Educational Studies in Mathematics devoted entirely to affect in mathematics education, Rosetta Zan states: Affect has been a focus of increasing interest in mathematics education research. However, affect has generally been seen as ‘other’ than mathematical thinking, as just not part of it. Indeed, throughout modern history, reasoning has normally seems to require the suppression, or the control of, emotion.
  5. Both of these represent incorrect perceptions of affect in mathematics. We will address these in reverse order, dealing with research first and then the classroom.
    1. Research: if we go back to the original formation of the affective domain of learning we see that affect is not synonymous with emotion.
      1. The affective (heart/feeling) domain of learning is more specifically referred to as “Krathwohl’s Taxonomy,” due to the work of David Krathwohl. The affective domain is not simply based on subjective emotions (though emotion may play a small part in affective learning), rather it’s about demonstrated behavior, attitude, and characteristics of the learner – all of which are deeply rooted to success in the mathematics classroom, and all of which are largely misunderstood in math education research.
      2. Affect then is not equal to emotion. Rather affect is an aesthetic, it is an orientation of life, a mechanism for determining what is worthwhile. This perspective matches better with the phrases from the policy documents cited above: Consider once more that being mathematically literate involves having an appreciation of the value and beauty of mathematics, and being mathematically proficient involves a habitual inclination to see mathematics as worthwhile.
    2. Classroom:
      1. From McLeod again: As it stands our current methods of teaching mathematics are producing untold numbers of students who see mathematics more about natural ability rather than effort, who are willing to accept poor performance in mathematics, who often openly proclaim their ignorance of math without embarrassment, and who treat their lack of accomplishment in mathematics as permanent state over which they have little control.
      2. How many of you, when you introduce yourself to someone for the first time and inform them that your work involves mathematics, receive the other person’s condolences? Or the person says something to the effect of “I was never any good at math.” Math is the only profession that I know of where this occurs (nobody every meets a dentist and then unprompted admits to never flossing).
      3. The business as usual approach to teaching math will continue to produce these affections. How do we change this?
  1. What if we take a different approach to affections?
    1. James K.A. Smith: “Behind every pedagogy is a philosophical anthropology.” Before you can teach a human being you must first have a notion of what a human being is.
    2. What if human beings are primarily affective creatures before they are cognitive ones. How might this change our understanding of how students learn in the math classroom? How might this change how we approach research on affect in math education?
  2. A new perspective on education (from Smith):
    1. Education is nor primarily a heady project concerned with providing information; rather, education is most fundamentally a matter of formation, a task of shaping and creating a certain kind of people.
    2. What makes them a distinctive kind of people is what they love or desire – what they envision as “the good life” or the ideal picture of human flourishing.
    3. An education, then, is a constellation of practices, rituals, and routines that inculcates a particular vision of the good life by inscribing or infusing that vision into the heart (the gut) by means of material, embodied practices.
    4. This will be true even of the most instrumentalist, pragmatic programs of education (such as those that now tend to dominate public schools and universities bent on churning out “skilled workers”) that see their task primarily as providing information, because behind this is a vision of the good life that understands human flourishing primarily in terms of production and consumption.
    5. Behind the veneer of a “value-free” education concerned with providing skills, knowledge, and information is an educational vision that remains formative.
    6. There is no neutral, nonformative education.
  3. So how do we cultivate mathematical affections? Through the practices, habits, and rituals of the classroom.
    1. To try to convince you that affections are shaped through practice, consider the iPhone (or smartphone in general)
    2. Look around the conference today at how many people are engaged in conversation versus having their heads buried in their phone. Also notice how many people this week will demonstrate frustration at having to wait in long lines.
    3. While we don’t spend time thinking cognitively about our smartphones (we just use them as a regular practice), the technology is still engraining affections in us. Two specific ones might be:
      1. Feeling as though we deserve immediacy, being used to information at the push of a button, thus reducing our patience to solve a problem
      2. Feeling of inflated self-worth, of being in a social situation and responding by saying “I am not having my social needs met by this scenario so I will retreat to my phone where I can look at what I want to look at.”
    4. Approaching affective studies in this way will not be an easy task: according to Goldin Mathematics educators who set out to modify existing, strongly-held belief structures of their students are not likely to be successful addressing only the content of their students’ beliefs…it will be important to provide experiences that are sufficiently rich, varied, and powerful in their emotional content to foster students’ construction of new meta-affect.
  4. Here are two areas we should focus on re-imagining our research (and practice) of affect: technology and assessment.
    1. Similar to the iPhone example above, we need to consider what technological practices of the classroom we participate in that mold our student’s perceptions. We need to be careful not to implement the newest technological accessories in our classroom just because students are used to having technology in their lives outside of school. If we are trying to offer mathematics up as being the technologically savvy discipline and therefore worth the interest of students, I would argue that we are largely going to lose that battle. We are offering math as a competing interest against the newest apps, games, and electronic devices that students are inundated with on a daily basis. As much as I love math, I know that this is a competition it won’t win. What if instead we focused on technological liturgies in the classroom that utilized mathematics as a way of examining and critiquing technological advancements rather than simply using those advancements to try to make math more fun? What if these liturgies could instill in students a sense of mathematics (and education as whole) as being something other than just a competing product for their attention and rather a foundation for their life that informs the product choices and decisions they make? What if we stopped feeding the culture of immediacy that technology has engrained in us and purposefully use the classroom as a time to step back and reflect? Perhaps then students won’t automatically jump to the calculator when faced with a difficult problem and proceed to give up if the answer is not achieved in under a minute.
    2. More consideration needs to be given to assessment. The NCTM Assessment Standards for School Mathematics (1995) state that “It is through assessment that we communicate to students what mathematics are valued.” If our goal is cultivate mathematical affections (values) in students, assessment is the primary means by which we do so. We need to consider what liturgies of assessment we participate in at both the formative and summative level. For instance, is the emphasis on correctness of a student response? Perhaps a teacher poses a question to the class and a student answers incorrectly. The teacher responds with a simple ‘no’ and moves on to call on another student who they know will provide the right answer and move the lesson along. If we fall into this pattern (liturgy) of formative assessment we are instilling into students the notion that math is only about getting to a correct answer and we ignore the productive struggle that it takes to get there. At a summative level, as long as high stakes standardized exams exist where the main goal is to achieve a certain percentage of correct responses, we will always be fighting an uphill battle in getting students to value mathematics for its creative processes.
  5. I hope this offers a starting point for us to re-imagine the research on affect in math education and begin the process of cultivating mathematical affections in our students.

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.

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

McLeod, D.B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575-596). New York: Macmillan.

National Council of Teachers of Mathematics. (1991). Standards for teaching mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (1995). Mathematics Assessment Standards. Reston, VA: NCTM.

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.

Veatch, M. (2001). Mathematics and values. In R. Howell & J. Bradley (Eds.), Mathematics in a Postmodern Age: A Christian Perspective. GrandRapids: Eerdmans, pp.223-249.

Zan, R., Brown, L., Evans, J., & Hannula, M.S. (2006). Affect in mathematics education: An introduction. Educational Studies in Mathematics (Affect in Mathematics Education: Exploring Theoretical Frameworks: A PME Special Issue), 63:2, 113-121.