ABSTRACT
Cultivating Mathematical Affections:
Re-imagining Research on Affect in Math Education
“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 noncognitive 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
Cultivating Mathematical Affections: Re-Imagining Research on Affect in Math Education
Josh Wilkerson
Texas State University
www.GodandMath.com
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.
a. Think through a list of adjectives to describe that scene
b. My wager is that those adjectives are more likely to do with the affective learning
taking place in the classroom rather than the cognitive
c. 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.
d. 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.
a. 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
b. 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.
a. 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.
b. 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:
a. 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.”
i. 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
ii. The perception is simply: let’s address the cognitive demands of
mathematics and meet those standards and then the students will value
the experience.
b. 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.
a. Research: if we go back to the original formation of the affective domain of
learning we see that affect is not synonymous with emotion.
i. 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.
ii. 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.
b. Classroom:
i. 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.
ii. 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).
iii. The business as usual approach to teaching math will continue to
produce these affections. How do we change this?
6. What if we take a different approach to affections?
a. 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.
b. 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?
7. A new perspective on education (from Smith):
a.
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.
b. 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.
c. 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.
d. 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.
e. Behind the veneer of a “value-free” education concerned with providing skills,
knowledge, and information is an educational vision that remains formative.
f.
There is no neutral, nonformative education.
8. So how do we cultivate mathematical affections? Through the practices, habits, and
rituals of the classroom.
a. To try to convince you that affections are shaped through practice, consider the
iPhone (or smartphone in general)
b. 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.
c. 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:
i. Feeling as though we deserve immediacy, being used to information at
the push of a button, thus reducing our patience to solve a problem
ii. 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.”
d. 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.
9. Here are two areas we should focus on re-imagining our research (and practice) of
affect: technology and assessment.
a. 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.
b. 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.
10. 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. 575596). 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.