Our Biases versus What Students Need

So we are thinking about our fall classes. Shall we structure them like we’ve done for the past n years? Shall we do something different? Perhaps we recently saw a presentation that inspired us. Such questions deal with the fundamental problem of teaching: Are the methods I use reflecting my biases about what ‘should’ be’, or are the methods designed to meet the deeply-understood learning needs of my students?

Perhaps it is your belief that collaborative learning is the key. Why is that? You might have an analysis which looks like this:

In turn, this type of image is based on research conducted by people who have definite ideas about how learning ‘should’ take place. Much of the basis provided for collaborative learning is based on the faux theory of ‘constructivism’ in which each student ‘creates’ their own learning. See http://archive.wceruw.org/cl1/CL/moreinfo/MI2A.htm ] In the radical form of this philosophy (it is definitely not a theory), there is no external standard for the learning being correct or complete — it is an individual process with internal criteria. Many advocates of collaborative learning — whether in K-12 or higher education — have a strong constructivist bias (unstated, in some cases).

What do our students need, specifically? Often, we rely on easy images like this:

In fact, I have colleagues in my department for whom this image is critical to their classroom practice which has remained unchanged for ten years or more. Just for fun, do a search for “learning pyramid research” or “learning pyramid myth”. The fallacy of the pyramid is obvious, yet it holds influence over our practice. [Ironically, most people remember learning about the pyramid in a ‘lecture’.]

What do my students need? What do your students need? Start with the obvious answer — they need to learn mathematics. Specifically, we have a defined package of mathematics illustrated by a set of learning outcomes. Do they know some of it before our class? Very likely. Do they have misconceptions about it? Almost always. Can we identify those misconceptions? Oh, yeah. What does it take to reduce the misconceptions and build a better understanding? Well, that is the fun part … because the needed treatment varies with the material being learned and the set of students in front of us.

We all approach the teaching problem with some biases — mine involve a socratic process, which is (like constructivism) a philosophy not a theory. Perhaps you think the key is to make sure every student is responsible for completing their portion of a group process, and to shift this portion over time. Perhaps you think a crystal clear presentation is the most important element. These are all ‘wrong’ to some extent.

My point is that the situation is simple to describe:

Learning is always an active process
Talking and explaining (by students) is linked to the quality of their learning
Our expertise is used to structure a successful learning process

In other words, don’t let your own biases guide or limit your instructional practices. Instead, focus on key principles like those above and use your expertise to design a learning process.

I’ll give an example from our Math Literacy class, to illustrate what I mean. We were learning “dimensional analysis” (DA). As you know, becoming skilled at DA involves basic fraction concepts along with some analysis. Here is the instructional plan:

Students work in teams with the directive “no student left behind” (everybody learns)
A series of questions is provided; each one is answered with the direction that everybody agrees; of course, I’m checking in with each team
Early questions deal with reducing one fraction; few students understood ‘common factors’ in reducing.
One of those questions is then presented in factored form (they are all constants) to get more students thinking about common factors.
Another question puts a unit (like “ft”) in the position of the common factor to see if students recognize that a unit can cancel.
A dimensional analysis problem is presented with a structure (steps) included; students fill in the blanks to see how to ‘get the answer’.
A brief example shows the idea of analyzing units.
A dimensional analysis problem (simpler variety) is presented without any structure.
A ‘lecture’ component involves the articulation of the principles involved with more examples. [By the way, the analysis part of DA is easier for my students than the fraction part.]
An ending team activity checks to see if every student “got it” so they can complete the homework.

You can see elements of ‘scaffolding’ in this plan. What may not be as visible is that there are opportunities for identifying misconceptions; for example quite a few students did not think the factored form of a fraction product was equivalent to the ‘original’ form. Either team members, or the teacher (me), will re-direct in this situation. Of course, it is important to be realistic — a 5 minute conversation will not overcome years of misconceptions about fractions (or any other mathematical topic). What I want you to notice is the level of instructional analysis involved; in the case of this DA lesson, I spent a solid 3 hours working on the design and the documents … for a topic I have ‘taught’ many times before.

In the time I have been using this “no student left behind” team approach, I have not encountered a student who did not participate in the process. Nobody has the additional stress of more leadership on a team than they are ready for; many students develop those skills and become more comfortable. My team assignments are stratified random samples — each team has low, mid, and high skill students; these teams are shuffled twice later in the semester. Each class tends to form a learning community, and I end up not being sure how much they would need me. [One student commented on the course evaluations that “we could not help but learn”.]

Remember, my goal today is not to boast of my great teaching prowess (I try, but I know my weaknesses). My goal is to suggest that you build a process designed for students to learn each day where you apply your expertise to uncover misconceptions as every student is engaged with the mathematics. Use teams. Use lectures. Most of all, make sure your math class is inclusive for every student regardless of their mathematical ‘worth’.

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