Compartmentalized learning?

By Marcus Wilson 29/03/2011

One of the benefits of me undertaking a teaching qualification is that I am now a lot more conscious of the kinds of thought processes my students are using. (The best way to do that is to talk to them). This year I’ve noticed how ‘compartmentalized’  students’ learning appears to be. What I mean by that is that students appear to find it difficult to drag a concept from one area of physics/maths and apply it to another.

An example I had recently was that, having had students analyze a mechanical problem, they obtained a quadratic equation.  Now, all of them I’m sure, if they were just given a quadratic equation and asked to solve it in a maths class, could do it without a problem. But no-one in the class (to my knowledge) spotted that what they had was a quadratic equation, so could be solved. The mathematical concept didn’t carry across to the physical problem.

If I had more time, I’d like to analyze this a bit more.  (I’m sure others have gone down this road with their research, so there’s probably a lot written about it.) First of all, I’d really ascertain whether this is a problem, or just something I have a ‘feeling’ about.  It’s incredible how many scientists lecture based on ‘feelings’ of what their students might or might not be experiencing, without actually finding out for sure. A scientist would never do their research based on ‘feeling’ – they’d always look to do a properly controlled experiment – but it doesn’t happen in teaching as often as it should.  (I got that from Eric Mazur).

Then perhaps I might look at what’s going on. For example, is it a result of our semesterized teaching, where everything is arranged in neat little packages called ‘papers’ – you do the paper, sit the exam, and then promptly forget everything in it? I’ve often thought that NCEA encourages compartmentalized learning; a concept from one assessment standard absolutely can’t be used in a problem in another assessment standard.  The trouble is that real physics isn’t like this – the spider’s web of concepts is tightly linked together. That’s one thing I like about the approach of the  NZQA physics scholarship exam – a student needs to pull together concepts from across physics to make sense of a problem – just like real physics.

However, applying Mazur’s rule here, I shouldn’t speculate about whether semesterized teaching / NCEA encourages this – I should actually go and find it out. So maybe I should just stop writing on this blog entry, fire up the literature search engines and see what I can find – and, if the answer’s ‘not much’, then actually think about doing some proper research on it.

0 Responses to “Compartmentalized learning?”

  • Yeah it would be interesting to know if this problem is worse under NCEA or not. I certainly see lots of students who struggle to perform basic algebra on equations (or even use their calculators at all) because they are in “chemistry” not “maths”.

  • As an old fossil who had to do end of year exams, and did quite well in them, I have often looked on topic-by-topic exams with some skepticism. However, a topic-by-topic approach does leave a lot of room for better, more complex questions to see how well the material has been learned.

    It never occurred to me that the topic-by-topic approach might prevent students letting things filter through their brain from one course into another.

    And what is it that lets us absorb stuff from one area of work so we may use it in another? Perhaps time to reflect? Time for the subconscious to pick up some hints? Yet we make students fully utilise their time nowadays. Perhaps a wee bit of down time isn’t a bad idea?

  • Now that would be a very interesting research project indeed! Wanna do a joint one – physics & biology?

    With the NCEA there’s certainly more opportunity for a disconnect between the curriculum & the assessment, probably more than existed under the ‘old’ system. I do know when NCEA was set up it wasn’t intended that the assessment would drive the teaching, & in fact I’ve had some really interesting discussions with teachers around how you could teach multiple standards in a coherent structure that allowed you to bring everything together & show how it’s all linked (both within & between subjects).

    Unfortunately that sort of approach is less amenable to ‘measuring’ than working through each AS in turn, plus teachers are under a lot of pressure (from school admin, parents, & students) to deliver good results & thus to teach to the assessment. And because not all students taking the external exams will have studied all the examinable standards, an examiner isn’t able to – using biology as an example – set a question for the ‘patterns of evolution’ standard using human evolution as the context, because many students may not have taken the latter standard. (A pity, because it would be a rather elegant question & would certainly achieve the need to get students to integrate knowledge & concepts from different parts of the curriculum.)

    paulfalloon – I used to have the same problem when teaching in secondary school bio & expecting students to use good grammar, spelling, punctuation etc in their bio essays. “But miss,” they’d wail, “this isn’t an English class!”

  • Marcus said…
    The best way to do that is to talk to them

    I agree. I have coached students at both senior high school level & undergrad university level (mostly children of relatives/family-members) over the past 9 years in maths & physics (mostly evenings n my spare time) and the problem that I find is they usually vague about what they don’t understand. If a parent asks me, that his/her son or daughter needs some help in 7th form math’s, then I usually ask that student what/which topic he/she wants help on. If it is about complex numbers, then I would then ask what specific complex number problems you don’t understand. The answer usually came back that he/she doesn’t understand anything about it. So, this gives me a specific topic to start with, but it also gives me a headache, since it means that I have to start complex number afresh again from the beginning (irrespective of what’s he/she’s already being taught/learnt at school). I like students who ask specific questions on specific topics, (since you can pretty much work out the gap in their knowledge) rather than a general blanket question saying he/she doesn’t understand a whole topic or whole subject. It makes it difficult in this situation to pinpoint of where the help is needed or where to start.

  • Marcus said…
    Now, all of them I’m sure, if they were just given a quadratic equation and asked to solve it in a maths class, could do it without a problem. But no-one in the class (to my knowledge) spotted that what they had was a quadratic equation, so could be solved. The mathematical concept didn’t carry across to the physical problem.

    Marcus, see the first video (top left hand) from the following link, where as a 10 year old (who is still at primary school – year 6) is solving standard quadratic and also a non-standard one, which is being transformed into a standard form in order to be solved via the standard quadratic formula. Spotting a quadratic that needs to be transformed may be tricky, but once it is transformed, then standard formula can be used.

    The kid above is one of 3 school primary school children that I started coaching them math since the beginning of last year (2010). Their parents brought them to me and asked if I could help out with their fractions & integers. We did those 2 topics (fractional & integer operations) over in 2 weeks and since they fully grasped the topics I was asked to help them with, I thought that was it, everything is done.

    The dad of the kid in the YouTube video above (which I call enthusiastmathkid) asked if I could teach him math topics that I think would be of interest to his learning (the father meant his son’s own level). Since I’ve been coaching senior high school & undergrad university students before in maths/physics, I immediately thought, why don’t I give it a try here (well to be honest – an experiment) by giving him higher level mathematic concepts and see how far he can go. I started introducing him simple linear equation and how to solve it, eg: 10 – x = 3 (answer: x=7) and he had no problem solving them after giving him a few rules of how to do it & also showing him a few examples. After that, I changed the linear equation from having nice integer coefficients into fractional, something like this,

    2/3 + x/6 = 3 (answer : x = 14)

    and again, after giving him a much simpler rule to use (scale the whole equation by the denominators LCM = 3*6 = 18, which rids all the fractions, thus it becomes easier to solve : 12 + 3*x = 54).

    By the last term of school last year (2010), I have introduced him to (calculus) polynomial differentiation & indefinite integration. As I saw that he has potential, I asked his parents if they’re keen on their boy sitting/attempting the year-13 (7th form equivalent) math CIE (Cambridge International Exam) at the end of this year (November 2011), they said yes, only if I think he has a chance. Now he has been accepted to one Cambridge affiliated school in Auckland to sit one year-13 CIE math paper in November as a private candidate. All he does is to turn up on the exam dates, because his preparation is done by me and not by the school.

    The plan for enthusiastmathkid in 2012 is get him to sit 2 more advanced pure maths and one physics CIE papers. That will get him an A-Level certificate.

    The other 2 primary school kids (one is 9 year old girl and the other is 10 year old boy) that I’m also coaching, their parents also wanted them to sit some external exams as enthusiastmathkid will be in November, but I told them, not this year but they may be ready for next year in 2012 for CIE year-11 (for the 9 year old girl) and CIE year-12 (for the 10 year old boy).

    My students think that they’re unique, but I showed them on YouTube that there have been kids as young as 8, 9, 10 11 , who have done similar learning in the past, especially an 11 year old got a college degree in Astro-Physics (he started when he was 8).

    Anyway, I think that all kids have this absorbent & receptive capability to new concepts being introduced to them. So, what I think I’m doing is not special. Anyone can do maths with young kids because of the absorbent & receptive capability of their brain.

  • The LCM in my previous message is supposed to be 6 rather than 18. The product of the 2 denominators (3*6 = 18) is the general method taught to enthusiastmathkid to use because it works on algebraic fractions as well as numeric fractions (see his video #3 from the left on the top row on solving algebraic equation – If the denominators don’t have a common factor then the LCM is just the straight multiplication of those.