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)


C4C: Agility and Forgiveness

All about the role of agility and forgiveness in learning.

Continuing the Changes for Change series, here’s some thoughts on agility and forgiveness:

Agility is a combination of being able to respond quickly and flexibly in response to changes and new opportunities.

Forgiveness is a necessary companion to agility and is the acceptance and recovery from mistakes and failures.

Agility and forgiveness are very necessary teacher attitudes for the day-to-day function of an interactive classroom. One can’t (and shouldn’t) plan for every possibility, and so you have to be able to respond to new questions and directions that show up in class.  Even being agile, there will be times when things don’t work out as you had planned and even your adjustments can always correct things.  So, you need to be able to forgive yourself for you mistakes, promise to correct them (if necessary), and go on. I think preparing for being agile and forgiving consists of always looking for new ideas anjaslfasdlfjasldfjaslfdj

For my students, agility and forgiveness are probably the highest skills.  They should learn to plan and organize, but they should also be able to adapt.  This is especially true with respect to how they are applying themselves to the course.  Every assessment is an opportunity for them to evaluate the success of their study skills. I can play a role in this by asking them to reflect on the effectiveness of their preparation. This could also

I was reading an article in the local paper about a new program at my university1.   As part of this new program they included some cool feature (X). An administrator who was involved in the development on the program said (see the footnote again): “Students have asked for X for years and this shows how we respond to student input.”  The intent was to highlight student input, but it made me think about some other conversations I’ve had about whether or not our university or higher ed in general is can respond quickly to new ideas.

This is interesting because universities are always talking about being cutting edge, and of their important role in the economy for developing new ideas, yet, in practice we are pretty slow.  I’ve highlighted some examples below:

Publishing: It takes 1-5 years for an idea to go from creation/discovery to final publication through traditional means. High stakes results may come out quicker, at least between formal write-up and publication, and there are always informal pathways that are quicker, but overall it is a slow process.

Curriculum Change: 7 years (catalog life)

Forgiveness is an important companion, but is much harder to implement in a class. Forgiveness often means forgetting, which means redoing or substituting or just dropping. I think universities as a whole allow more forgiveness than the private sector, with repeat policies, second chances, and petitions. Individual faculty have some wiggle room for experimental attempts in the classroom and for unproductive research agendas.

For my classes, the next step for forgiveness is to move towards more standards-based or specification grading which rewards movement towards achievement and minimizes the consequences of failure. I’ll post more about specification grading later.


1The details are going to be vague because I actually slightly misread the article and I don’t want to embarrass myself or the other people involved. The inspiration is valid, even if the details aren’t.

A Question of Prerequisites

The initial idea was simple: come up with a hypothetical question about how performance in a prerequisite course should correlate with performance in a course with a goal towards identifying whether or not we inherently limited students achievements.  The problem, as always, was in the details.

Here’s the initial version of the question (more or less): For a course, Math 2, which has Math 1 as a prerequisite, what are the probability that a student who made an A in Math 1 will make an A in Math 2. Repeat for all the possible grades in Math 1 and in Math 2. Limiting to only grades of A, B, C for each, why aren’t all the probabilities 1/3?

Then the problems began.

  1. What does it mean that Math 1 is a prerequisite for Math 2?  How much of Math 1 actually appears in Math 2, and is not (re)covered in Math 2? For example, say Math 1 is an algebra course and Math 2 a calculus course. How many of the algebra topics in Math 1 show up in any detail in calculus? At what point does not being fluent or knowledgeable about algebra start interfering with learning calculus, or is the problem just in the algebra portion of the class, i.e. algebraic simplification of results? In our hypothetical situation we can assume that the answer is ‘all of Math 1 is needed in Math 2 to understand Math 2’, but that’s pretty unrealistic.
  2. What does A, B, C mean? We can change this to a numerical score and base it on a set of specific learning objectives (think of a 100 question critical concept final). So we can say A means 90+% knowledge, but then how do we know that is current.  And then it gets more complicated for multiple grades averaging to a C.  Is a student who made 70% on every assignment the ‘same’ as a student that made 100s mixed with 50s on the various assignments, or even the student that made 60s on tests but pulled out a C on homework or other ‘friendly’ assignment.
  3. Then we get closer to the real issue. If a student makes a C in Math 1, can they make an A in Math 2? I don’t mean numerically, or what’s possible, I mean does whatever led to the student making a C in Math 1, limit them so that it is very unlikely they could make and A in Math 2. For example, save Math 1 is Differential Calculus and the student made a C because they struggled with the algebra and/or trigonometry. If they don’t increase their skills in those areas, do they really have a chance for an A in Math 2 (Integral Calculus).

So, although I never came up with a good form of the question, I did form an opinion: I think we too often curse students who performed poorly in prior courses, well beyond how that performance is relevant to the current course. I’d suggest that forming and being faithful to some carefully formulated learning objectives would help out, but that’s for another post.