19th ACMS Conference
Bethel University, 2013
(Click on the image above for the presentation slides)
I. The Need for Affective Learning
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.
II. The Place of Affective Learning in the Math Classroom
Let me take a moment to define what I mean by mathematical affections. The title of this talk is in homage to Jonathan Edwards’ Treatise Concerning Religious Affections. Edwards’ goal was to discern the true nature of religion and in so doing dissuade his congregation from merely participating in a Christian culture (a mimicked outward expression) and motivate them to long for true Christian conversion (an inward reality of authentic Christian character). The purpose of this talk is to engage us as math educators in discerning the true nature of mathematical assessment and how we use it in the classroom: does it simply mimic the modern culture of utility by requiring outward demonstrations of knowledge retention and application, or does it aim deeper at analyzing true inward character formation? For Edwards, affections were not synonymous with emotions as they tend to be in today’s culture. Because today’s culture sees affective learning as simply an emotional state the culture classifies affective learning as a purely subjective domain and therefore not worthy of developing objective standards or assessments. But Edwards understood affections as aesthetics – a way of orienting your life via a mechanism that determines what was beautiful and worthwhile. If we see affections as character producing then Edwards’ definition leads to a more objective perspective that provides more potential for assessment rather than viewing affections as emotions.
It is Edwards’ definition of affections (orientation of life, determining worth) that actually appears in popular math education literature. According to the NCTM Standards for Teaching Mathematics, “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.” Adding it Up: Helping Children Learn Mathematics, a report published by the National Research Council 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.” Both of these foundational documents in the area of math education plainly portray mathematics as beautiful, of value, and affecting the habits of the learner to see math as worthwhile. However, neither of these documents develops how we as teachers are to go about accomplishing this task. It is almost as if these phrases are included in these documents as a courtesy – as a way of saying “this is how we as math teachers feel about math and it would be nice for our students to feel this way too, but feeling is subjective so there is no real way for us to objectively instruct or assess students in this regard.”
This is a point of connection that we as Christian educators can make with educational system as a whole – we can answer the questions of how. We have much to contribute here and we don’t have to be overtly religious in the presentation. What we should really be emphasizing is the classical approach to mathematics. The mission statement of the classical Christian school where I teach states that: “The mission of Regents School is to provide a classical and Christian education, founded upon and informed by a Christian worldview, that equips students to know, love and practice that which is true, good and beautiful, and challenges them to strive for excellence as they live purposefully and intelligently in the service of God and man.” Non-Christians will obviously reject the beginning and end pf that statement but the key phrase here for the purposes of contributing to fixing the problem in education is “equipping students to know, love and practice that which is true, good and beautiful.” The true, The Good, The Beautiful – these are ideas from Plato who believed in a rigorous development of mathematics. Legend has it that above the door to his academy was the phrase “let no one ignorant of Geometry enter here.”
Beyond Plato, the truth, goodness, and beauty of mathematics has been attested throughout history by famous mathematicians who were not espousing a religious view. In regards to truth, Richard Feynman said “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” In other words, the truths of mathematics are written into the very core of nature. In regards to goodness John Von Neumann wrote “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” I believe no one would argue the good that comes from learning mathematics. There would not be nearly the outcry that we see today over fixing the educational system and increasing our place in mathematics scores if people did not think knowing math was good. And finally, in terms of beauty, G.H. Hardy wrote “Beauty is the first test: there is no permanent place in the world for ugly mathematics.” Every mathematician will attest to the beauty of mathematics. If we can argue that mathematics is objectively true, good, and beautiful, then I believe we can argue that these qualities, which contribute so greatly to our affections, can be more objectively assessed.
III. Cultivating Mathematical Affections
According to the NCTM Assessment Standards for School Mathematics: “It is through assessment that we communicate to students what mathematics are valued.” If we are going to increase our attention on the affective domain of learning we need to do so by writing rigorous learning objectives and developing quality assessments just as we do for the cognitive domain. Assessment is always tied to our objectives; by definition objectives are quantifiable learning outcomes of a lesson. Below, I would like to offer some examples of learning objectives in a statistics course followed by a list of methods for assessing affective learning that very from traditional cognitive assessments.
The following examples of affective objectives use the current taxonomy of the affective domain of learning developed by Bloom and Krathwohl (middle column of the diagram above). I am not endorsing this taxonomy, as I believe there are improvements that can be made to it. For now though, I would like to propose how we can start working within the current system. Notice that in each objective there is an affective verb, a cognitive verb, and a vehicle for assessment.
- Receiving: The student will (TSW) differentiate (affective verb – AV) between valid and fallacious statistical arguments and argue (cognitive verb – CV) their reasons in a written response (method of assessment – MOA). Assessment should account for the initial discernment between truth and falsehood in addition to the correctness of the argument.
- Responding: TSW engage (AV) in class discussion (MOA) comparing (CV) and contrasting (CV) religious and statistical knowledge. Assessment should account for the level of engagement in addition to the determination of proper similarities and differences.
- Valuing: TSW support (AV) the mission of a local non-profit organization through the design (CV) of a statistical study (MOA) done on the organization’s behalf. Assessment should require student to defend the worthiness of the cause motivating the project and not simply report valid data analysis.
- Organizing: TSW define the limitations (AV) of statistical inference procedures and accordingly make recommendations (CV) to the acting agency in an oral presentation (MOA).Assessment should account for recognition of shortcomings (humility) in the oral presentation.
- Characterizing: TSW be evaluated (CV) in their intellectual integrity and rated positively by their peers (AV) through a written reflection survey (MOA).Assessment allows for students to communicate their personal reflection and evaluation of others.
The key to the above objectives is the integration of affective and cognitive learning. We need to be creative in our assessment techniques to allow the affective domain an opportunity to be assessed. Some examples of assessments include (but are not limited to): Math Journals, Reflection Assignments, Personal Interviews, Class Discussion/Debate, Oral Presentations, Open-Ended Group Problems, Historical Reading and Response, and Service-Learning Projects. If you are still under the belief that textbook assignments, quizzes, and tests provide a more objective measurement of student ability, let me pose the following questions: Who chooses what problems to assign? Who writes the quiz or test? Who grades the quiz or test? How is partial credit handled? I can ask many more questions in a similar vein that will hopefully allow us to realize that even the cognitive skills that we assess have some level of subjectivity involved. I am not arguing that affective learning is completely objective, but it is at least as equally objective as the cognitive domain.
There is much work to be done in this area. I hope that I have convinced you of the need to engage whole-heartedly into this work. For now I will leave you with a student quote on the impact of affective learning: “I was more dedicated because I saw a deeper purpose.”