Challenging Questions

A former colleague of mine, someone who inspires me to think carefully and critically about what I do as a teacher, sent me a great email recently. I am going to excerpt it below. (Note – I added the question numbers so I can more easily refer to them later.)
I’ve been doing a lot of thinking about teachers and data. More
specifically, how teachers are surrounded by all kinds of data every
single day in their schools and classrooms – qualitative and quantitative.
This data can be life-changing for their practice and for student
learning. I’m growing more and more curious, personally, about how to help
teachers tap into that.
I’m considering writing something about this. Not sure of the venue or any
of the specifics yet. Right now I’m just exploring these ideas with
teachers like you.
I’m particularly interested in data related to teaching skills for
transfer. For example, I recall my students learning lots of great math,
but then being unable to transfer their math learning to a science
classroom where the context is different and where the problems read
differently. It’s a question I’ve always wanted to tackle – how to help
students get better at this type of transfer.
That’s where my question for you comes in –
1) What’s a skill(s) for transfer that you see come up again and again for
your students, a skill where they seem to learn it well in isolation, but
they then struggle to apply it to a new/different context? The more simple
it is, the more interesting I’d actually find it to be, if you have an
example like that.
2) What about grit and/or persistence in problem solving? What challenges do you see there?
3) How about engagement? Anything that comes up there for you?
I am so curious to dig deeper on these questions in relation to info/data
gathering. Really curious about this. I know it smacks of action research
(not a bad thing) but I’m really eager to find a way to simply it and help
teachers work on something like this.
Welcome your thoughts! Let me know if anything I’m asking is unclear.

 Quite a bit to chew on, I have to say. She sent me this email four days ago and I think I am finally able to attack some of these questions coherently. Hold on tight, this might be a long post.

Regarding question 1 – the question about transfer – there is one big area that occurs to me. I think that it is related to the difficulties in bridging the gap between the equation solving skills that students want to rely on and the worlds of graphical representation and written explanations where most teachers seem to want their students to take up residence. A recent example is sticking in my craw right now. I have been giving my Calculus classes weekly in-class problem sets to work on. One of the problems this week was: Given that cos 80 = 0.173648…, find the following values without the use of your calculator (a) cos 100   (b) cos 260   (c ) cos 280   and (d) sin 190. When I chose this problem, I wanted them to recall the symmetries inherent in the unit circle and recognize the relationships between sine and cosine measures of angle that are all 10 degrees off of an axis. While many students (not as many as I’d hoped) seemed to breeze through this, I saw a wide variety of REALLY bad mistakes. Mistakes of the nature of cos 100 = cos 180 – cos 80. Now, it’s important to note that these are my AP Calc BC kids. They are among the best math students we have at our school. In their defense I will note a couple of important points. (i) I was not there the day they were working on it. Some of them probably saw this as ‘busy-work’ to a certain degree. Also, with me being gone they may not have been collaborating as much as usual. If I’m here and see confused looks or hear whispered questions, I’ll dive in an nudge them. (ii) Precalculus was quite a while ago for these kids who went through AP Calculus AB last year. Not in their defense, I will note that they had each other and their texts as EASY references. I’ll also note that the mistakes I saw really made almost no mathematical sense for functions more complex than linear ones. They should know (flat out) that the distributive property probably does not hold over subtraction with a function like the cosine function. However, I have seen over and over again in my 20+ years of teaching that symmetries based on graphical representations of functions do not seem to stick with my students the way that I think that they should. I use Geogebra and Desmos regularly with my students, but their Precalculus teacher did not. I wonder how deeply they tried to internalize this behavior when they were studying their trig. So, this seems to me to be an example of what my friend was asking about. The trig skills I referred to above were certainly ‘mastered’ in their precalculus days, however future applications of these ideas rarely go well in my experience. I fear that the students see trig (and many other aspects of math) as facts to learn, rather than concepts to master.

Another skill that is ‘mastered’ in precalculus is working with the definition of the intermediate value theorem. Another area where graphing sense really comes into play. However, in many of the cases where this theorem is invoked in Calculus, my students seem flummoxed. To a great degree what I see here is that my students thrive when they need to remember a formula or a definition. Where they struggle is applying this idea in the context of a problem such as identifying an interval where a root for a function might lie. In each of these cases I see students who can recall facts when they are taught, who can recall definitions when they are presented, but they struggle with the applications of these ideas. This seems especially true when graphing ideas are involved.

Since the introduction of graphing technology in the classroom (the TI and now web-based graphers) I have had the habit of looking at graphs on my own when working AND when working in front of my students. I know that they usually have computers nearby when working, but they don’t seem to have inherited this habit. I am at a loss as to how to help instill that.

For question 2 I have the following observations. When the work is graded for completion, there is little sense of determination to work through a perplexing problem. Simply writing something down is good enough. When the work is graded for correctness, most of my students will then dig in and really challenge themselves to get the problem right. However, this often is accompanied by a lack of discipline about time. Most of the grit and determination I see is directly related to the impact of that work on the student’s grade. This, of course, is not true of all my students but it is true of the majority of them. There are certainly students who are inspired by challenging problems to explore, but more often I find that my students don’t feel that they have the time for this. The challenge I see here is that the students who make it to these highest classes in the curriculum are often the most ambitious and involved students. I believe that there is an inherent curiosity, but I see it diminished under a heavy workload.

I am sure that I have some more thinking to do along these lines, but here is my first major swing at these interesting questions.


One thought on “Challenging Questions”

  1. very interesting – now you’ve got ME thinking. am wondering whether the 10K of practice rule applies here esp as the concepts and skills become more complex (as they would in higher level math courses). how much “practice” does it take to master these concepts and skills in ways that support the level of transfer needed to go from skills and concept in one form – equation – to other forms – graphical and visual representations. that’d make for an interesting study, for sure.

    your thoughts on grit/resilience/engagement are interesting, too. have to admit that i’m typically juggling lots of projects and a heavy workload. i do tend to prioritize those projects that influence the most people, have the quickest deadline, have the greatest impact on certain areas of the company. in short, i prioritize based on some feedback loops around $$, impact, time, colleagues’ work, etc. in a way, it seems that students do the same thing only their feedback loops center on grades, no? as you aptly mention, they have to choose from lots of assignments and heavy workloads (they’re BC calc kids), so they have to make choices given all the work in front of them.

    i, too, will keep thinking about these questions and your responses. thanks for thinking with me!

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