# Imitation without Understanding

A view of what might be the biggest issue in (math) education.

Okay, so maybe I’m overselling this, but I’d like to argue that there’s a big disconnect between what we think is happening and what is really happening, and we’ve bought into a scheme where we can fool ourselves (the education community) into believing that everything is ‘okay’.

The short version is the title: we think we are teaching understanding but instead are testing and, thus reinforcing, imitation. And it is even worse in that the imitation is often done without understanding.

A story: We teach a first-year general education math course (for non-STEM students) and a common feature is discussing the rational and irrational numbers. In this, the big result is that $\sqrt{2}$ is irrational. In most sections, this is done in class, then tested on a quiz and then likely shows up on an exam and maybe even the final. The proof isn’t trivial, but is not so complicated that a student can’t learn to reproduce it. With all the ‘coverage’, many of the students can successfully write this proof, and we can ‘claim’ that they understand. However, in cases where we push this, for example by asking them to show $\sqrt{3}$ is irrational, or even better (or worse) that $\sqrt{4}$ is irrational, we often find that they have very little understanding, and have only been successful by copying or imitating the correct proof for $\sqrt{2}$.

So why does this happen? I think all parties have contributed, and will speculate as to how each comes about:

For students: this has been going on for them since kindergarten. If they made the grades, etc. to get into college, they have mastered this system. If a teacher gives a practice test, they know to do it, memorize it, and expect to be asked to reproduce the same sort of thing on the test. I suspect that some aspects of math anxiety are due to the honest fear that just being able to reproduce given results isn’t enough and eventually the student will be ‘caught’.

For teachers: it’s  both what they’ve experienced as students, and most likely what they’ve unintentionally found to be most efficient and effective in their teaching. Even for the good teachers who are trying to generate understanding, students will learn the right language and use it to appear to understand, but may only be imitating what they’ve heard.  As the teacher wants to hear that the students are understanding, they assume that the student does understand and doesn’t push further to see what’s really going on.

For both together: in my most cynical view, there’s an agreement that the teacher will count imitation as understanding and the student will pretend that they understand. The student avoids the challenge of gaining true understanding or the embarrassment of not understanding, while the teacher avoids the challenge of supporting true understanding or the embarrassment of seeing many/most student not actually gain understanding.

(When I first starting writing this almost a year ago, I was writing primarily out of frustration. Now, I have hope that it is possible to honestly address true understanding and that, in the right context, students can and will step up to the challenge)

# Does Math Really Fit in STEM?

A comparison of mathematics and the other STEM fields.

I’ve been putting together a teaching/learning philosophy for a new program we are developing and I’m trying to raise the elements from ‘it sounds good’ to ‘there’s research that supports this’. So, I’ve been going through various reports on teaching and learning studies, especially those that focus on the STEM fields. I’ve not been through much yet, but I’m starting to formulate a question, and am going to use this space to ask it and start thinking about answering it:

From a learning point of view, does Mathematics fit with the rest of the STEM fields in higher ed?

The practical version of this question is:

Can we use the results of STEM (not math) education research or K12 math education research to inform higher ed math education?

(If I start feeling generous, I’ll change the ‘Can’ to ‘How’, but for now I just want to do a comparison.)

A disclaimer: I’m not a science or engineering educator and I’m not a professional math-ed researcher. I can only claim experience in teaching math, and thinking, talking and reading about teaching math.

I’ll write STE for STEM without math, although I don’t think much is about T, so it should really be just SE vs. M.

A list of comparisons:

• STE has observations, M has stipulations (I sometimes say science is limited by reality)
• STE defines based on observations, M focuses on sharp (human-made) definitions
• STE uses induction, M uses deduction
• STE has the Scientific Method, M has proofs
• STE works more with specifics, M more with abstractions and generalizations
• STE has more history in current content, M treats most content as brand new
• STE has more vocabulary, M has more symbols
• Both emphasize process, but STE more towards application, M more towards understanding
• STE has more facts, M has more processes
• Many study STE for it’s own sake, most study M for it’s use in STE and elsewhere
• STE K-12 results are relevant for higher ed, M K-12 results don’t seem to be as relevant

Some observations:

• When I look at education research for STEM, they usually don’t include many/any examples from math, and the examples they do provide from STE seem to favor the aspects of STE that are quite different from M.
• I think maturity plays a big role in learning mathematics; I don’t know if it is needed more in M than STE, but it is a big part.
• One data point: We have Supplemental Instruction (SI) on campus and they cover math, some sciences and other areas. If you go to the SI website (just search for it; the international headquarters is in Kansas, I think), you’ll see in the data that they include math, but that the results are quite as good. Locally we get the same results: lower attendance in math sessions than other, and less of an impact. There are probably lots of reasons why this is so, but one is that the SI sessions focus on the SI Leader helping the students help each other. It is not a review or tutorial session. The example I remember from the training/promotion was for psychology, and a student expressed some confusion about some topic, and the other students were encouraged (and able) to use the book and their notes to help the student understand. I can see how this wouldn’t work in math, as a student will say they didn’t get the right answer to some problem or got stuck, and either the other students can’t help because they don’t know or they were assigned a different problem, or they just give the student the answer. There’s no discussion and no learning about learning.

That’s it for now. When I’ve progressed more on my teaching/learning philosophy, I’ll post it. And, if I figure out more how M fits with STE, I’ll do an updated version of this. However, for now, this is all I have: just a question and some ideas.

# The Best BBQ

A comparison of two approaches to selling BBQ; there must be some connection to teaching.

On a recent visit home (Texas), I decided I needed to have a good meal (which means BBQ). In discussing this with my sister, her response was a place we had gone to since I was a kid (Angelo’s) announcing that it was ‘The only BBQ in Fort Worth’ (which, of course, is not true, this being Texas, etc.). So we went, and it was really good.

But, here’s what I noticed. When I ordered the Beef Plate (Texas BBQ = Brisket), except for my drink, that was my only choice in the matter. I didn’t get asked which sauce I wanted, which sides, or what type of bread. I did notice that those ordering a Beef Sandwich (sliced or chopped) did get some say on the pickles, onions and mustard that are usually offered, and I probably could have asked for some substitutions. For my meal, the sauce did come on the side, so I guess I had an option of yes/no on the sauce, but the sides were beans and coleslaw and the bread was two slices of white bread (should have been Mrs. Baird’s Bread, but I didn’t check). I know I said this, but it was really good. The sauce was so good that I used the bread I had left to soak up some, and then what little was left after that, I drank. They’ve been in business since 1958, and show up on the ‘Best of’ lists regularly, so the folks at Angelo’s know what they are doing.

As a contrast, returning home (Tennessee) I read a review of a new BBQ place in town. The reviewer discussed all the options: several sides to choose from, 3 different breads (including Texas toast), and at least 5 sauces. Part of this is location, we’re caught between Memphis and the Carolina’s, each with a distinctive BBQ style, plus we’re Southern, and so sides are a concept all to their own. The multiple options is typical of BBQ places in town.

Both approaches are successful. People do like choice. Some like sweet sauces, some like ‘tangy’, some like slaw, some like potato salad. Personally, I like greens (with anything) and not-sweet cornbread. But people also want the best and the trust experts to produce that. I wouldn’t go to a multi-star/James Beard/award-winning restaurant and micro-manage how my meal was put together.  If the chef says that the garlic-mashed potatoes go with the seared tuna, that’s what I’ll have.

So what does this have to do with teaching? It feels like the distinction between teacher-centered and student-centered learning or direct instruction and discovery learning. Are teachers “Sage on the Stage” or “Learning Facilitators”? Maybe this says something about maturity: if you haven’t had BBQ before, limited or no options would be better, but then once you’ve developed some experience and have determined your ‘taste’ for BBQ, then a place with options would let you optimize your experience. Early on students would need more direction, but once they have figured some things out, they might be read for some options.

There’s also an interesting connection if you think about a large group going to eat BBQ, with some being more mature and some less. In a place with options, the ‘experts’ will often help the less-experience negotiate the options, or the less-experienced will just go with ‘whatever she/he’s having’.  What does this say about a typical class with a mix of maturities?

# Active vs. Passive and Learning vs. Teaching

An attempt to make sense of active vs. passive, learning and teaching.

A first definition: ‘Active Learning’ means the student is engaging, in some way, with the material.

Proclaimed from above: “Lecture is bad, active learning is good”. And then the comments begin.

In the classroom, the intent to have active learning inspires, but the practicality of time, material, life, etc. intervenes and the reality becomes passive lecture.

The studies show that classrooms where active learning happens, have better scores, higher pass rates, happier students, etc. But maybe it’s not clear what’s really going on.  For example:

A study was done [1] where 1 hour of direct instruction (i.e. lecture) was changed ever-so-slightly by inserting 2 breaks where the students were asked to take 2-5 minutes and discuss and clarify their notes with a neighbor. Results went up. But why?

Here’s some thoughts: Active learning really happened; Students took the break to rest and refocus, relative to their attention span; the Instructor put more into organizing the material into the shorter segments; the Students thought it showed the Instructor cared, so the cared more; less material was Covered, and so less was tested, etc.; added awareness of ‘Active’ made everyone more aware of the Learning; and much more.

But then there’s my experience (and likely yours, or at least your colleagues): primarily positive and prefered experiences in a lecture environment. Lately, I’ve been learning much from listening to podcasts. This is probably the most passive form of instruction as they are pre-recorded and make no distinctive effort to engage the listeners in ‘active’ learning. What’s happening here?

So, here’s my new definition or distinction about active vs. passive:

On the teaching side, the difference is a question of what is allowed, but does not address what is real: So Active Teaching means that you do things that make students do active-learning things; Passive Teaching means that you allow a passive learning approach. For example, do students have to engage the material during the classtime? If no, then its ‘passive’; If they look like they are engaging, then its ‘active’.

On the learning side, I’ll stick with the definition at the top, active learning means ‘engagement is happening’, but that doesn’t necessarily mean that it needed active teaching or that there’s necessarily any sort of visible evidence. For example, a student could be sitting in a very traditional/passive lecture and be prompted to think deeply about the material; if so, then active learning is happening. But of course, the student sitting right next to them could look the same but be completely passively relating to the material, or not relating at all.

Thus, how do we increase active learning? (Because I’m going to believe that active learning means learning). First we realize that its a horse-to-water thing and so we can’t make it happen, but with that in mind, I suggest two things:

1. Raise the Expectations: discuss active learning with the students, model it for them, give them examples (for even so-called passive times), and check up on it as the course goes on.

2. Increase the Opportunities: pick the spots where (small) changes can be made to increase the active learning type activities can happen and do them; start small and add as things go on.

[1] Ruhl, K., C. Hughes, and P. Schloss, “Using the Pause Procedure to Enhance Lecture Recall,” Teacher Education and Special Education, Vol.
10, Winter 1987, pp. 14–18.

# Some Random Thoughts

I’m back, and so here’s a list of some topics that I’ve been thinking about and might expand in later posts:

1. The transition from basic conceptual understanding to fluent understanding expressed through action can come through imitation or through true growth. Most of the time we can’t tell the difference, but we might be able to tell by the next topic that is learned.
2. There are two dimensions of learning (at least): (1) learning it (whatever it is) and (2) learning to learn better. Every course should address both of these and should expect that prior courses have done so also.
3. A challenge: addressing students mis-knowledge, by unteaching them that which they must unlearn. Two big components: moving them from bad or incomplete schemas to higher level and more effective ones and adding recognition and verify to their problem solving methods.
4. Another challenge: helping students move from extrinsic to intrinsic motivation.
5. We are good at helping students learn what to do (or at least, how to do it once they know it needs to be done), but not so good at helping them value the why we do it. For us (teacher/practitioners) this is a natural part of our being, so we need to think about how to make this invisible part of our work more visible to students.

That’s it for now.