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

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.

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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.

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Teaching a Love of Mathematics

A few weeks ago, NCTM President Linda M. Gojak posted her final message as president entitled “A Reflection on 25 Years in Mathematics Education.” You can follow the link to read the article in its entirety. In this message Gojak outlines from her perspective what the mathematics education community has accomplished over the last 25 years and what challenges still need to be addressed. I will let you determine for yourself how much you agree with her assessments. What I am most interested in is her closing remark:

B. F. Skinner famously said, “We shouldn’t teach great books; we should teach a love of reading. Knowing the contents of a few works of literature is a trivial achievement. Being inclined to go on reading is a great achievement.” With apologies to Skinner, as mathematics educators we might say, “We should not just teach mathematics, we should teach a love of mathematics. Knowing the content of some mathematics is a trivial achievement. Being inclined to see the beauty in mathematics and to go on doing mathematics are great achievements.”

We should teach a love of mathematics.

Knowing the content of some mathematics is a trivial achievement.

I agree with both of these statements, as I believe the majority of math educators would. However, these two statements get to the heart of the issue with the state of mathematics education today: while the majority of educators would agree on the sentiments of these two statements, both statements run contradictory to the current system of mathematical standards and assessments.

If we really believe that our goal as educators is to teach a love of mathematics (which I should note is a very different thing than saying every student has to love math) then we as a community of educators need to actually determine how to go about doing so. Because trust me, focusing on core standards/higher order thinking/critical reasoning/whatever you want to call increased cognitive demands, will not influence people’s affections. The issue is much more complex than that. We are talking about teaching love, beauty, truth to human beings created in the image of God.

I will have a lot more to say on this issue in the coming weeks/months/years as this is essentially the focus of my dissertation research. For now I will leave you to contemplate Gojak’s closing remark and consider why the underlying sentiment of the remark does not appear anywhere else in her summary of math education’s “accomplishments” and challenges.

Why Math Works

by John D. Mays

Back in 1999 when I began teaching in a classical Christian school, one of the first books I heard about was James Nickel’s little jewel, Mathematics: Is God Silent? Must reading for every Christian math and science teacher, the book introduced me to a serious problem faced by unbelieving scientists and mathematicians. Stated succinctly, the problem is this: Mathematics, as a formal system, is an abstraction that resides in human minds. Outside our minds is the world out there, the objectively real world of planets, forests, diamonds, tomatoes and llamas. The world out there possesses such a deeply structured order that it can be modeled mathematically. So how is it that an abstract system of thought that resides in our minds can be used so successfully to model the behaviors of complex physical systems that reside outside of our minds?

For over a decade now this problem, and the answer to it provided by Christian theology, has been the subject of my lesson on the first day of school in my Advanced Precalculus class. But before jumping to resolving the problem we need to examine this mystery – which is actually three-fold – more closely.

In his book Nickel quotes several prominent scientists and mathematicians on this issue. In 1960, Eugene Wigner, winner of the 1963 Nobel Prize for Physics, wrote an essay entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner wrote:

The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and…there is no rational explanation for it…It is not at all natural that ‘laws of nature’ exist, much less that man is able to discern them…It is difficult to avoid the impression that a miracle confronts us here…The miracle of appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Next Nickel quotes Albert Einstein on this subject. Einstein commented:

You find it surprising that I think of the comprehensibility of the world…as a miracle or an eternal mystery. But surely, a priori, one should expect the world to be chaotic, not to be grasped by thought in any way. One might (indeed one should) expect that the world evidence itself as lawful only so far as we grasp it in an orderly fashion. This would be a sort of order like the alphabetical order of words of a language. On the other hand, the kind of order created, for example, by Newton’s gravitational theory is of a very different character. Even if the axioms of the theory are posited by man, the success of such a procedure supposes in the objective world a high degree of order which we are in no way entitled to expect a priori.

One more key figure Nickel quotes is mathematician and author Morris Kline:

Finally, a study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning, and the ideals which have been steadfastly pursued for over two thousand years were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen with the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explanation of this marvelous power is called for.

The first aspect of the problem these scientists are getting at is the fascinating fact that the natural world possesses a deep structure or order. And not just any order, mathematical order. It is sometimes difficult for people who have not considered this before to get why this is so bizarre. Simply put, the order we see in the cosmos is not what one would expect from a universe that started with a random colossal explosion blowing matter and energy everywhere.

Many commentators have written about this and professed bafflement over it. All of the above quotes from Nickel’s book and many, many more are included in Morris Kline’s important work, Mathematics: The Loss of Certainty, which explores this issue at length. In his book The Mind of God, Paul Davies, an avowed agnostic, prolific popular writer and physics professor, takes this issue as his starting point. Davies finds the order in the universe to be incontrovertible evidence that there is more “out there” than the mere physical world. There is some kind of transcendent reality that has imbued the Creation with its mathematical properties.

The second aspect to the problem or mystery we are exploring is that human beings just happen to have serious powers of mathematical thought. Now, although everyone is happy about this, I rarely find anyone who is shocked by it. Christians hold that we are made in the image of God, which explains our unique abilities such as the use of language, the production of art, the expression of love, self-awareness, and, of course, our ability to think in mathematical terms. Non-Christians don’t accept the doctrine of the imago Dei, but seem to think that our abilities can all be explained by the theory of natural selection.

But hold on here one minute. Doesn’t it seem strange that our colossal powers of mathematical imagination would have evolved by means of a mechanism that presumably helped us survive in a pre-industrial, pre-civilized environment? Our abilities seem to go orders of magnitude beyond what evolution would have granted us for survival.

I know all about the God-of-the-gaps argument, and I’m not going to fall for it here. It may be that some day the theory of common descent by natural selection will be able to explain how we became so smart. That’s fine, and I’m not threatened by it. All I’m saying is that for now Darwinism still has a lot of explaining to do. And getting back to the concerns in this essay, I for one do not take Man’s amazing intellectual powers for granted. They are wonderful.

The third aspect to our problem is the most provocative of all. Mathematics is a system of symbols and logic that exists inside of our heads, in our minds. But the physical world, with all of its order and structure, is an objective reality that is not inside our heads. So how is it that mathematical structures and equations that we dream up in our heads can correspond so closely to the law-like behavior of the independent physical world? There is simply no reason for there to be any correspondence at all. It’s no good saying, Well we all evolved together, so that’s why our thoughts match the behavior of reality. That doesn’t explain anything. Humans are a species confined since Creation to this planet. Why should we be able to determine the orbital rules for planets, the chemical composition of the sun, and the speed of light? I am not the only one amazed by this correspondence. All those Nobel Prize winners are amazed by it too, and they are a lot smarter than I am. This is a conundrum that cannot be dismissed. John Polkinghorne said it well in his Science and Creation: The Search for Understanding:

“We are so familiar with the fact that we can understand the world that most of the time we take it for granted. It is what makes science possible. Yet it could have been otherwise. The universe might have been a disorderly chaos rather than an orderly cosmos. Or it might have had a rationality which was inaccessible to us…There is a congruence between our minds and the universe, between the rationality experienced within and the rationality observed without. This extends not only to the mathematical articulation of fundamental theory but also to all those tacit acts of judgement, exercised with intuitive skill, which are equally indispensable to the scientific endeavor.” (Quoted in Alister McGrath, The Science of God.)

Which brings us to the striking explanatory power of Christian theology for addressing this mystery. As long as we ponder only two entities, nature and human beings, there is no resolution to the puzzle. But when we bring in a third entity, The Creator, the God who made all things, the mystery is readily explained. As the figure here indicates, God, the Creation, and Man form a triangle of interaction, each interacting in key ways with the other. God gives (present tense verb intentional) the Creation the beautiful, orderly character that lends itself so readily to mathematical description. And we should not fail to note here that the Creation responds, as Psalm 19 proclaims: “The heavens declare the glory of God.” (I have long thought that when the Pharisees told Jesus to silence his disciples at the entry to Jerusalem, and Jesus replied that if they were silent the very stones would cry out, he wasn’t speaking hyperbolically. Those stones might have cried out. They were perfectly capable of doing so had they been authorized to. But I digress.)

why math worksSimilarly, God made Man in His own image so that we have the curiosity and imagination to explore and describe the world He made. We respond by exercising the stewardship over nature God charged us with, as well as by fulfilling the cultural mandate to develop human society to the uttermost, which includes art, literature, history, music, law, mathematics, science, and every other worthy endeavor.

Finally, there is the pair of interactions that gave rise to the initial question of why math works: Nature with its properties and human beings with our mathematical imaginations. There is a perfect match here. The universe does not possess an order that is inaccessible to us, as Polkinghorne suggests it might have had. It has the kind of order that we can discover, comprehend and describe. What can we call this but a magnificent gift that defies description?

We should desire that our students would all know about this great correspondence God has set in place, and that considering it would help them grow in their faith and in their ability to defend it. Every student should be acquainted with the Christian account of why math works. I recommend that every Math Department review their curriculum and augment it where necessary to assure that their students know this story.

John D. Mays is the founder of Novare Science and Math in Austin, Texas. He also serves as Director of the Laser Optics Lab at Regents School of Austin. John entered the field of education in 1985 teaching Math in the public school system. Since then he has also taught Science and Math professionally in Episcopal schools and classical-model Christian high schools. He taught Math and 20th Century American Literature part-time at St. Edwards University for 10 years. He taught full-time at Regents School of Austin from 1999-2012, serving as Math-Science Department Chair for eight years. He continues to teach on a part time basis at Regents serving as the Director of the Laser Optics Lab. He is the author of many science textbooks that I invite you to explore further on the Novare website.