Mosquito Netting and Open Sewers

A story that’s a metaphor for course and curriculum design.

A couple of years ago I was at a math-biology workshop and during the reception, I had a great conversation with a student from Ghana. Somehow we got on the topic of how it was popular for mathematicians (mostly westerners) to study diseases that are prevalent in Africa. He then mentioned how often the solutions that are promoted are not really practical as they don’t take into account certain elements of local society. He mentioned several examples, but the one that stuck with me was the work on malaria. At that time, it was popular for some organizations to raise money to provide mosquito netting in areas that are suffering from malaria.  My colleague said that the netting helped, but there were still mosquitoes all around, and the real problem was that there was standing water in open sewers. So it would be more helpful to fix the sewer problem rather than just supply mosquito netting.

I was left with thoughts about the appropriate of some modeling approaches I had taken, but I was more struck with wondering how often we try to solve teaching/learning problems by providing the equivalent of mosquito netting when there’s a bigger problem (open sewers) that we are ignoring.

The first example I thought of was College Algebra. I’d bet that if you talked to any math department from an open or less-selective university, you’d find that the low success rate in College Algebra was a big concern, especially with recent concerns about completion rates. I’ve been to conferences where passing rates were reported at 40-50% and ‘improvements’ to 60% were considered outstanding. In most cases, efforts to improve the pass rate focus on improving the quality or quantity of instruction (e.g. moving to open computer instruction, adding hours or study sessions, etc.).  This makes some impact, but usually doesn’t really fix the problem (and can be very costly). The ‘open sewer’ in this case comes from looking at how a typical student arrives at College Algebra.  Since the course has basically the same content as Algebra 2 in high school (which is required at my school for admittance), all students have seen the material at least once. However, something happened to bring them into College Algebra. Maybe they don’t have good math study skills, or they struggled with some prerequisite math skills (e.g. fractions). Whatever the issues may be, until they are addressed no amount of improved teaching will be able to make a significant impact.

I haven’t spent a lot of time thinking of other situations in detail, but I do use this framework when something comes up, and try to make sure that we aren’t just taking the easy/popular solution, but are considering all the issues and maybe doing the unpopular work of sewer-fixing.


Duck Theorems

Another difference between Math and the rest of STEM: the strong use of if and only if relations which lets us determine something exclusively by its properties.

A follow-up to Does Math Really Fit in STEM?

An example: Suppose you wanted to show that for an arbitrary value a, a\cdot 0 = 0, assuming you have a proper context for all the other symbols, etc. The first problem is that 0 is only really known as an (or the) additive identity, i.e. b + 0 = b. And we don’t know anything about how it works with multiplication. But there’s a way:

a + a \cdot 0 = a \cdot 1 + a \cdot 0 = a \cdot (1 + 0) = a \cdot 1 = a,

using the rest of the field axioms.  But why is this good enough? We rely on what I call a/the Duck Theorem, which in this case is the fact that a + x = a has a unique solution (0), and since a \cdot 0 also satisfies this equation, we must have a \cdot 0 = 0.

Aside: Why do I call it a Duck Theorem?  From the old saying: If it looks like a duck, walks like a duck, and sounds like a duck, it must be a duck. In this case a \cdot 0 acts like 0, and since 0 is unique, a \cdot 0 = 0.

There are many things in the background that make this work, all of which rely upon the exceptionality gained from using if and only if and the corresponding deductive arguments. The definitions involved exclude any other options. The Duck Theorem provides a distinctive uniqueness (which could be rewritten, in our case, as x + a = a if and only if x = 0.) I’m not an expert on the other STEM fields, but I’m not aware of many (or any) cases where some physical result can be used to 100% identify some particular object or property. Most definitions are descriptive and, unless they’ve been abstracted into mathematics, are not given in an if an only if form. (A mammal is warm-blooded (as well as other things), but being warm-blooded doesn’t make you a mammal). Of course I’m only talking about how math relates to math, this doesn’t always extend to the results when math is applied (e.g. statistics is as precise within its own context, but it doesn’t extend the if and only if nature to the results of a statistical test as understood in the application area, i.e. statistical results are not if and only if/causal).

This all is not to say math is better (I’d argue that this feature makes math worthy of the many jokes made about math); it is just different, and maybe different enough that method proven to be good for teaching physics, biology, engineering, programming, etc., might not translate over to being good methods for teaching math.

Three Ts and an L

After living with the 3Ts guideline (Think, Talk, Try) for teacher development for some time, I’ve decided that I need to add an L for Listening.

In conversations about improving teaching, I’ve always given the advice of using the 3Ts to guide growth. They are

Think – review and reflect on ones teaching; developing ideas from the big, involving identity as a teacher, down to the small, involving specific classroom activities

Talk – discuss those ideas with others, for feedback and for accountability

Try – do it, in some small way, or if necessary, all in

As I’ve used the 3Ts and have grown as a teacher, I’ve started to realize that I should add (or really, recognize) another: L for Listening.

In making the big jump from being just a teacher to being a more mindful, reflective teacher, I thought that I was enough to generate those good ideas about teaching. Now I realize that listening, in both conversations and through reading blogs, papers, books, etc. is an important part of developing as a teacher and an activity that I do and value very much.

So, I now recommend paying attention to the teaching that is going on around you. Pick a blog or two, and read what someone else is thinking about teaching. Listen to a podcast, to see what ideas or practices are out there. Pick a topic and search for some research articles. Read a book on education. Find things that challenge your assumptions about teaching and learning and ones that confirm you were on the right track. Then think about how they fit with your development as a teacher, talk to others sharing these ideas, and try something you’ve learned.

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.

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.


Question Tree

A method related to Usiskin’s Problem Analysis for looking and enriching math topics.

I saw some elements of this somewhere (probably on the web, but what a vague reference!) and have developed and used it in a couple of classes with high school and community college teachers. Recently I saw an article in Mathematics Teacher about using Problem Analysis from Usiskin and noticed some similar ideas and so I thought I’d put the idea of the question tree out here. It is also related but at a different scale to a Wu story, and it might remind you of a Frayer model, just to throw out another buzzword.

The fundamental purpose of a Question Tree (or q-tree) is to provide a structure for developing an in-depth understanding of a specific topic. It also provides insight into innovative ways to approach the topic, and can be used to extend ideas across a curriculum. It is primarily a tool for instructors, but might have some use with students in developing higher level thinking skills.

I’ll describe it from the inside and working outward, but the usual way that I’ve used it is by starting at one of the leaves and working both in and out.

First the q-tree lives in some environment. This is the context for the q-tree. It can include grade-level and content information. An example: 3rd grade, geometry, proportionality.

The trunk of the q-tree is an instance of the context and is the topic.  As we form the branches, you’ll see how the topic should be formed, but for now think of it as defined term, e.g. similarity of geometric figures.

There are three branches on the q-tree and each branch has a leaf. The leaf is an instance or example of the branch, and is usually formed as a question or activity:

  • theory branch – a theorem or statement revealing some abstract property of the topic; the corresponding leaf is a proof, as a problem that requires the theory to resolve
  • application branch – a class of applications that use the topic; the corresponding leaf is a problem, a word problem connected to the application
  • process branch – a specific method related to the topic; the corresponding leaf is an exercise, a ‘textbook’ problem that requires the process to solve

Although it would be easy to imagine a topic having many different branches with multiples of each type, but for the q-tree, there are only three – one of each. And those three branches should be related, so the application should depend upon the process, which should be a consequence of the theory, etc.

The q-tree also has two roots, which are connections or instances outside the context or topic:

  • generalization root – an extension or abstraction of the topic
  • X-problem root –  a motivational or summary problem related to the topic, usually more of an open task or a longer-term project

As I said starting out, the usual approach to forming a q-tree is to start with a leaf then work to the trunk and then form the other branches and roots. So let’s see it in action:

Starting point: A typical calculus problem about finding the extrema of a given function: e.g. Find the extrema of f(x) = x^3-2x+1 on the interval [0,5]. This is the exercise.  And the corresponding process is the usual procedure of finding the critical points and testing them and the end points for extrema status. The topic could be ‘application of derivatives to graph properties’ in a context of a first-year calculus course, derivatives and applications. From here we can form the theory as the extreme value theorem, or a conceptual statement relating derivatives to extrema, and the proof could be either a proof or a consequence of the theorem, or an abstract problem that requires a careful use of the theorem to resolve. The application is constrained max-min problems, with a problem as one of the classic maximize the volume of a box under some limits. The generalization could be extrema in higher dimensions or characterizing extrema with the 2nd derivative when you have an unbounded domain (moving from global to local extrema). For an X-problem we could start with a practical problem, say of taking a load of something in a truck and needed to make a box to hold as much as possible, so that the constraints either have to be assumed or discovered separately.

The end result is a collection of related topics that give us a deeper understanding of the basic ideas. The roots give us links that we can use to present the material in class or to connect to other ideas. In a classroom driven by standards and/or learning objectives, the individual q-tree could be grouped around the class objectives. The process of forming the q-tree is a nice exercise in thinking about a topic from multiple perspectives, and is valuable for both teachers and students.