The Need for Affective Learning

Later this week I will be giving a talk at the 19th ACMS Conference entitled “Mathematical Affections: Assessing Values in the Math Classroom.” Overall I argue for 1) the need for affective learning, 2) the place of affective learning in mathematics, and 3) how we cultivate what I’ve termed as “mathematical affections.” I will post the talk in its entirety once I have given it. For now, I thought I would share a teaser from the introduction. Enjoy.


How many of you, as math educators, have heard the question “When am I ever going to use this?” be uttered by your students? If you have been teaching for more than 5 minutes then it’s safe to assume that phrase has been mentioned in your presence. Occasionally it is posed as a valid question; the student is genuinely interested in the future career application of the topic at hand. However, I believe the majority of the time the phrase “When am I ever going to use this?” is spoken it is not as a question, but as a statement. A statement which implies that the obvious answer is “I will never use this so learning it is a waste of time.” The real issue being raised by these students is not one of application, but rather one of values. If we could translate their question into what they are really trying to communicate then “When am I ever going to use this?” will become “Why should I value this?” Students express their inquiry in terms of mathematical practicality because that is the language in which their culture, including their math teachers, has conditioned them to speak.

To illustrate how we as math educators have contributed to this misconception that value equals utility, let us turn our attention to the foundational document for composing the learning objectives and outcomes of an academic course: Bloom’s Taxonomy (pictured below).


A quick glance at this chart will reveal that ‘application’ falls under the cognitive (mental/knowledge) domain of learning while ‘valuing’ falls under the affective (heart/feeling) domain of learning. The cognitive domain is almost exclusively emphasized in the preparation of teachers within the modern educational system while the affective domain is largely ignored. So while we ‘improve’ our teaching and questioning to make mathematics less abstract and to focus on real-life applications so that we can address the question of “When am I ever going to use this?” before it is even asked, we are actually implicitly teaching students that mathematical value is to be found only in application. If we really want to help those students address the true foundational question of “Why should I value this?” then we need to do so through increasing our attention on the affective domain of learning; writing rigorous learning objectives and developing quality assessments just as we do for the cognitive domain.

Now, application is certainly useful in the teaching process and it should not be ignored. I am not advocating the promotion of the affective domain over and above the cognitive. My goal is to simply bring the affective up to the same level as the cognitive. The best learning is done when both domains are utilized in conjunction with each other. In The Abolition of Man, C.S. Lewis writes “Education without values, as useful as it is, tends to make man a more clever devil.” I believe this is a fairly accurate statement of the modern day system of education. If we don’t focus on values, if we don’t focus on the affective learning of our students, then their education will still be useful – they’ll increase in cognitive ability and learn to apply their thinking. But is that really valuable in and of itself? Without a proper sense of values to guide their application, aren’t we really just making students “more clever devils”?

You see, you can never actually remove values from education. Education is inherently value laden, and I believe Lewis knew this. It is not a question of “Are you teaching values?” but rather “Which values are you teaching?” Lewis’ point is that the value we instill in education should be affective – loving learning for its own sake and valuing wisdom. If you don’t focus on affections, then you still have usefulness, but is that really beneficial? In the words of the Bishop in Victor Hugo’s Les Misérables: “The beautiful is as useful as the useful…Perhaps more so.”

Application is indeed useful but it should be presented in a way that promotes the development of what I’ll term mathematical affections. Learning has little meaning unless it produces a sustained and substantial influence not only on the way people think, but also on how they act and feel.

A Mathematician’s Lament

The excerpt below is taken from the paper, “A Mathematician’s Lament,” by Paul Lockhart. Lockhart also has a recent book by the same name, that is based off of the original paper. This excerpt is a little less than what you get when you sample the book on iTunes, so I believe it is safe to share here.

I must say I don’t agree with Mr. Lockhart’s perspectives on how to fix the problems prevalent in math education. However, he does an excellent job of giving you a mathematician’s perspective on the problems. It is clear that he feels very passionate about this issue, as do the rest of us who are paying attention. That is why I share this here.

A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.

Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

As for the primary and secondary schools, their mission is to train students to use this language— to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or
transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”

In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”

In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.” Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact
that they will never hear one. “To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable— every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician someday.”

Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. “Of course!” he reassures himself, “No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!”

15 Ways to Support Math and Science Education

List courtesy of the National Math and Science Initiative.

(I’ve highlighted in red the items that I especially agree with or am trying/will try to implement for my own students)

  1. Think and speak positively about math and science. Never again say to your child, “I wasn’t good in math either. Math is hard.” Rather say, “Learning math is critical for everyone today. I sure wish I had studied it more.” Encouragement and praise can be the first steps toward success.
  2. Pay attention to math and science teaching in your child’s school. Have you reviewed your child’s science or math homework lately? Are assignments or projects creative and tied to real-life situations or your child’s interests? Ask the PTO to schedule a presentation by the math and science departments at your child’s school so that you can better understand what’s being taught – and school leaders will see that parents expect quality.
  3. Support more modern lab equipment in local schools. Does the equipment in your child’s science lab look like what you used when you were in school? Times have changed and many school labs need to be updated. Do you belong to a club, or another type of organization that might help raise funds for more supplies and better equipment at your school?
  4. Stand up for algebra. Does your child’s school offer 8th grade algebra? If not, speak up to your school leaders and school board. Algebra is the gateway class that enables all students, no matter what their fields of interest, to move forward in school and college. If you don’t feel comfortable going to school leadership and/or the school board alone, you can see if other parents agree. Are there other parents like you who would support either signing a letter or going with you to a PTO meeting to try to enlist support from other parents?
  5. Become an advocate for Advanced Placement* Program courses that give students strong preparation for college work. Find out if your child’s high school offers Advanced Placement courses in math and science. If not, see if other parents would also want more information on these rigorous courses. Together you can show that parents are interested in having their children participate in AP courses. If the school already offers AP courses, find out how hard it is for students to get into the advanced classes. Advocate for a system that registers students for advanced classes unless their parents opt them out. You also should check and ensure that they are offered at non-conflicting times so students won’t miss out on helpful classes.
  6. Encourage your school leaders to provide incentives for students to successfully complete AP exams. Better yet, you can become a donor yourself – or recruit donors to defray the cost of taking exams and provide financial incentives for students to pass the exams. Do you have contacts with corporations that appreciate the need for a math-literate workforce? Do you have ties to sororities, fraternities or faith-based organizations? Look for people you know who might be willing to partner with the school to provide incentives for students who are taking rigorous AP courses. Might they also be willing to provide incentives for the teachers involved who are doing so much additional work with the students in the AP courses?
  7. Offer to mentor students in local schools who may be struggling or want to do advanced work. Recruit others to help. Is your service organization, professional group, or faith-based organization involved with your school? If not, are you willing to ask for their participation? Education ministries and service organizations can be a great source of support from retirees and others who might be willing to assist with tutoring/mentoring/grant writing.
  8. Encourage more colleges and universities to provide math and science recruitment programs for high school students. If you are a member of a civic or service organization, you could encourage the group to help organize a summer instititute for middle and high school students in math and science at a local university or college. Or, urge the university or college that you attended to reach out to high school students with special math and science programs.
  9. Volunteer to help organize a science fair if your middle school does not have one. Get the parents who are in science, health, engineering and computer fields to serve one afternoon as judges. Get local businesses to offer prizes.
  10. Rally local business support for math and science careers. Encourage your employer or chamber of commerce to form a partnership with a local school to support students who are interested in careers in math, science, computer science, and engineering. Summer internships and scholarships can make young dreams come true.
  11. Encourage more college students in math and science to become teachers. Find out what your local colleges and universities are doing to graduate more qualified math and science teachers. Are there programs in place to encourage more math and science majors to become teachers? Would a teacher training program such as UTeach be feasible there?
  12. Encourage foundations in your area to provide greater support for math and science education. Is there an organization in your community that might be interested in helping with funds for new textbooks, school supplies, lab equipment, scholarships for math and science? You should give them a nudge.
  13. Support more opportunities for girls and women in math and science. Find out if your school district offers the option of single-sex public schools. If not, encourage your school to look at the success of schools like Austin, Texas. Experience has shown that female students perform better at math and science in single-gender schools. Or, you can help more women pursue math and science careers in college by encouraging the creation of more scholarships for female students to enter those fields. Do you know any women’s service groups and foundations that might be helpful? Although more than half the students in medical schools today are women, the percentages of women in schools of engineering and most other sciences is still low.  You can help by providing a grant through a local foundation, non-profit or university that will help pay for childcare help for female graduate students and post-docs.
  14. Support more opportunities for underrepresented groups in math and science. Hispanic and African-American students are still under-represented in AP courses and in math and science classes in college. In 2000, only 4.4 percent of the science and engineering jobs in the United States were held by African Americans and only 3.4 percent by Hispanics. More minority participation is needed to provide the infusion of talent that our country will need in these critically important fields. You can help by supporting grants, scholarships, summer programs, and internships that bring more diverse students into math and science study.
  15. Urge your representatives in Congress to fund the America COMPETES legislation. The legislation was overwhelmingly passed by Congress in 2007, but has not yet been funded. The legislation cannot make a difference without funding.