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Topic: Getting answers Vs Deep understanding (Read 4162 times) |
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Noke Lieu
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Getting answers Vs Deep understanding
« on: Oct 1st, 2008, 1:38am » |
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I have an almighty task ahead of me.
It's fun and rewarding, but not easy.
Without going into too many details, it's a series of quick revision tips for parents to help their kids with Maths at home. (oh glorious day). 
It's sparked a curious question in my mind- and I guess it one that is mostly academic, but it would be interesting to hear points of view on it.
For the average punter (which most of us have been at one time or another) is it more important to
be able to answer questions, and thus build confidence in what is typically accepted as Maths Ability (the ability to correctly answer routine questions) and then maybel start to play with the why and develop insight...
or
to be shown what's actually going on in the first place?
I know that with my schooling there was a bit of flipping between the two states, and it worked for me. Not so for my brothers, despite having mostly the same teachers.
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| « Last Edit: Oct 1st, 2008, 1:39am by Noke Lieu » |
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towr
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Re: Getting answers Vs Deep understanding
« Reply #1 on: Oct 1st, 2008, 2:06am » |
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I don't know who or what the average punter is; but I would think the best teaching approach varies on a student to student basis.
Using what works on average might not work very well. For example it's fine and dandy to be on either side of a river, but on average you'd have to swim for your life.
So perhaps to reduce the "swimming distance" you might use two or three approaches, so students/parents can find the one most suited to their situation. The first revision tip might actually be to find out what type of learner they are.
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Noke Lieu
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Re: Getting answers Vs Deep understanding
« Reply #2 on: Oct 1st, 2008, 4:04pm » |
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on Oct 1st, 2008, 2:06am, towr wrote:| For example it's fine and dandy to be on either side of a river, but on average you'd have to swim for your life. |
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I like it!
whilst it hurts my sensiblities sometimes, in Aus we have a habit of using "average" as a synonym for "terrible".
By "average punter" I mean "struggler" (as a chemistry teacher of mine used to say)
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Sir Col
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Re: Getting answers Vs Deep understanding
« Reply #3 on: Oct 2nd, 2008, 12:12am » |
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As you know I can only speak for England, but if the recent rot trend is anything similar in Australia then it has to be "just tell me the answer".
[rant]
It makes me laugh (in a manic, losing my sanity sort of way) that our government continues to introduce new initiatives into our teaching as a knee-jerk reaction to any criticisms.
In a response to the overwhelming research that places UK graduates near the bottom of the recruitment pool in Europe, the latest initiative is "thinking skills". Sounds good? Read on...
In our lesson plans we are required to deliver a 3-part lesson: starter, main, plenary, and we need to explicitly state the lesson objectives (which must be written on the board at the start of the lesson). In addition we need to explain how we aim to incorporate content that addresses citizenship, morality/spirituality, the gifted and talented, special education needs, literacy, numeracy (I joke not, as maths teachers we have a numeracy section to complete!), ... , and now our lesson plans need to add how our lessons address "thinking skills" too.
Looking at the "recommended" lesson plan structure (i.e. do this or our government appointed inspectors will fail your school) is like a geologist looking at strata to determine what happened at different points in time. Each "box" demonstrates yet another degree of incompentancy of our current government to properly address a need. I applaud the idea of putting back into our teaching what we used to do before the other crap was introduced. Can you believe that the latest drive from our government is called "teaching and learning"? What would we do without them? For all these years we never realised that is what was supposed to be happening in our classrooms!
You might ask the question, how could you already be teaching a subject like mathematics without developing thinking skills? The answer is simple: the sum of the previous initiatives. Ironically the new initiative, which is called "thinking skills", aims to stifle even futher. It is nothing more than a paper filling exercise which reduces to identifying areas that should develop thinking, flagging them in your lesson plans (and students' books if appropriate), but then doing nothing more about it. The tragedy is that most of the content that would develop thinking skills has been removed to make room for the crap, so there is very little that genuinely addresses this area.
[/rant]
The bottom line from my perspsective is that there is very little in the curriculum that requires thinking. Certainly our government has stripped the curriculum to the type of content that lends itself to rote/methodical learning.
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teekyman
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Re: Getting answers Vs Deep understanding
« Reply #4 on: Oct 3rd, 2008, 12:23am » |
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I've tutored college-level students in the US in math and introductory probability (Math 1a-54 + occasional Upper Div Math and Stat 134, for you Cal people) for much of the last 4 years, and this is what I've learned.
A deeper insight into something will always allow a greater range of problems that can be solved and is of great value to a student, but many people are not ready for the insight. Most of math education in the US before college involves rote computation and little to no problem solving, and so entering college most students are unprepared for math that involves thinking. Its very easy to learn how to do a very specific problem by looking at examples, but get lost when some small aspects of the problem change (dealing with triangles instead of squares, or a different type of function as an input, or the variable is now called y, etc.) So always go for the deeper insight if you feel the student is ready for it, but if the student isn't then it will usually only serve to confuse them, and once a student is in the "this is too complicated for me, I will just stare at the problem until I get more hints or give up" phase, then it is almost impossible to get them back for quite some time
Many of those who have not even gained competency at the rote exercises absolutely hate "math" and don't have any desire to do more than pass their class with the bare minimum of effort (Usually freshmen taking introductory classes). My own life experience is far enough removed from their world view that I don't feel qualified to guess as to how to give any deeper insights to them and a love for problem solving - At best I have been able to give them the "insight" to perform the flavor of computation that they are currently "learning" over and over, but often they still forget this after a while.
[rant]Increasingly, there are more and more of these people in the US, and it seems in at least Australia/UK as well. I've read somewhere that Australian education is becoming very touchy-feely with respect to quantitative skills. By this I mean that students who are lacking in quantitative skills are taught that any such lack is "part of who they are" and that they should focus their efforts in other skills that they hopefully have like skills in the arts, literature, etc. The goal is so they are not made to feel bad about their failures but consequently they make no efforts to improve, and are made to believe from a young age that they will never be skilled at math or science. I think the same article recounted other western nations (it was probably the UK) that was adopting these policies because they were "better" for children, citing the fact that these children on the whole had more positive self esteem. As a consequence, entire countries can become somewhat stupider because of this desire to spare children and teenagers from feeling bad about themselves. In my opinion, positive self esteem is great, but I think many western countries have completely forgotten that there usually exists more joy in conquering challenges than in avoiding them. I guess these beliefs come from the fear that one will be unable to succeed (or more often that one's child will be unable to succeed). [/rant]
Hey look! - it seems like most of my post was ranting as well (rant now tagged), and unfortunately my final answer to your question will also be: it depends on the student
For the vast majority of people, deeper insights are only possible after being able to answer questions. Thus, I would first teach the how, and then the why for more fundamental topics that one learns in elementary / junior high school. As one moves up in the math world though, you need to do it the other way around, so also keep in mind what you're trying to teach.
I'm also pretty far removed from the younger kids, so I realize I may be totally wrong about this, but I suspect that we're not too different 
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| « Last Edit: Oct 3rd, 2008, 12:25am by teekyman » |
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rmsgrey
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Re: Getting answers Vs Deep understanding
« Reply #5 on: Oct 3rd, 2008, 2:27pm » |
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I spent ten minutes earlier this evening with a friend doing a largely self-taught maths A-level in their spare time, who could reliably get the "right answer" but wasn't happy about it because they had no idea why they were doing it.
It's hardly a statistically significant sample, but there's at least one person out there who is more interested in getting the deeper understanding than in being able to churn out rote answers...
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Sir Col
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Re: Getting answers Vs Deep understanding
« Reply #6 on: Oct 3rd, 2008, 4:09pm » |
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I think you've hit the nail on the head there, rmsgrey. Whilst learners are intrinsically motivated they will seek a deeper understanding, but as long as they are doing it because they have to they become much more utilitarian and simply want to know how to get the answer.
We should be dealing with quite a different audience when teaching A level as they have chosen to be there and generally want to learn mathematics. However, you will have those who chose maths because they couldn't think of a fourth AS, or heard that having maths would help them get on a course in X (where X is not maths related), and they will want the more "just give me the answer" approach. The tragedy, at least in England, is that even A level has now been diluted to make it accessible to this type of student. Consequently the reordering of the specification means that the more natural foundational approach we used to adopt in teaching A level no longer exists. Instead everything is bite-sized and we do "a little bit of this and a little bit of that". Deriving things from first principles has been abolished and instead we teach A level now by giving some vague notes, providing a couple of worked examples, and consolidating by repetitious exercises. Due to the modular approach, proper considered structure is out of our hands and we are forced to teach this way. Boo hiss!
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JiNbOtAk
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Re: Getting answers Vs Deep understanding
« Reply #7 on: Oct 10th, 2008, 8:39pm » |
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Let's take this discussion out of the academics world, and see how it applies to the industrial world.
I don't know about you guys, but one of the basic things I've learned on the job is that, "You don't have to understand how it works, you're just there to make sure it does." Granted, having an understanding on why and how stuff works is satisfying, but that does not guarantee it will get the job done.
For example, being a mechanical engineer, I could explain why diesel engines are not equipped with spark plugs. I could go on about the piston head design, the turbulent air flow within the combustion chamber..but if the a diesel engine breaks down, I wouldn't even know where to start looking for the problem. Compare that with my knowledge in diagnosing computer problems; A computer which boots up for 5 minutes and suddenly shut down on its own, well, it's probably caused by a faulty power supply or the processor fan might not be working, among other things. Why does the computer stops working when the processor heats up ? Why does it even heats up in the first place ? Heck if I know, as long as they just work.
My point is, understanding of a subject matter is good, it gives a sense of belonging to the knowledge. However, we could not deny the fact that there are times when knowing how to fix it is more important that knowing why it broke down.
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towr
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Re: Getting answers Vs Deep understanding
« Reply #8 on: Oct 11th, 2008, 4:50am » |
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on Oct 10th, 2008, 8:39pm, JiNbOtAk wrote:| I don't know about you guys, but one of the basic things I've learned on the job is that, "You don't have to understand how it works, you're just there to make sure it does." Granted, having an understanding on why and how stuff works is satisfying, but that does not guarantee it will get the job done. |
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That reminds me of the infamous case of the therac-25 (a machine for radiation therapy). Where someone with little real understanding of the machine "fixed" an error, to get it to "work", and consequently killed the patient with an overdose of radiation because of it.
Granted, the deeper cause of the malfunction was a programming error that allowed this; but if the operator had not tried to fiddle with something he didn't understand it wouldn't have happened either.
It's plain unsafe behaviour to try to make things you don't understand work in situations you don't understand.
Is this an issue with maths? Well, if you don't understand if the numbers you're dealing with are metric or emperial, you may crash a nasa-probe or two.
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| « Last Edit: Oct 11th, 2008, 4:53am by towr » |
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rmsgrey
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Re: Getting answers Vs Deep understanding
« Reply #9 on: Oct 11th, 2008, 9:28am » |
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on Oct 11th, 2008, 4:50am, towr wrote:| Is this an issue with maths? Well, if you don't understand if the numbers you're dealing with are metric or emperial, you may crash a nasa-probe or two. |
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Or the conversion between binary and decimal in the programming of smart missiles in Gulf War I causing rounding errors to accumulate and the missiles to not be so smart after a couple of weeks sat on the launch racks...
Taking it back to the original question, which is more useful depends to some extent on what you plan to do with the knowledge in future - if you just want to play Quake, or to pass the exams at the end of the year, then knowing more about the computer than how to turn it on, or more about the topic than how to get the right answers to certain classes of question, is redundant. If you want to take things further - to write Quake mods, or to pass next year's exams, then having a deeper understanding of the underlying principles will help.
And if something unexpected happens, having the deeper understanding can allow you to figure out a good fix, while having a rote knowledge means you either need to appeal to someone who does have the understanding, or fall back on trial and error and guesswork...
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Noke Lieu
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Re: Getting answers Vs Deep understanding
« Reply #10 on: Oct 21st, 2008, 10:48pm » |
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I think that's another salient point rmsgrey.
It can come down to intent.
(I suspect this of getting a bit ranty, sorry- I meant to staay on topic)
The way that I've tried to explain in the past is that maths is only ever an option. Market forces (perhaps?) have shaped our world in such away that you can opt out of actively doing maths and still lead an normal enough existence.
And it's not a problem, until it is.
If you can do something, and don't need to, there's no problem. On the other hand, if you need to be able to do something and can't, then it IS a problem.
The convoluted and unrealistic (or mundane) examples that people tend to provide to give an example for maths relevance, on the whole, I feel are deleterious.
I mean, there is an certain degree to which students are actually asking "what is it for" though I suspect a much greater degree is looking for a an excuse to get out of learning, looking to push boundaries, generally be the annoyance that many teenagers like to be.
That been said, there is a story that I like to relay to students about the value of maths in unexpected situations (given that the majority of them think that maths is arithmetic- maybe with a bit of geometry tacked on maybe- the unexpected applications are pretty numerous...)
It invovles Katie Bender (for a quick, though detailed synopsis, points 1,2,3,5,10, 15b pretty much cover it). The fact that she was a school kid, not unlike the student themselves, pretty much amplifies the effect.
(I typically only trot this one out when there's been a particularly irrating session though, I sure you can appreciate why)
More usually, I get them to answer these questions:
Can you think of a time when it's really a good idea to be bad at Maths?
What about when it's a bad idea to be bad Maths?
(though it's a little stupid- when would it be a good idea to bad at... cake decorating? but the point is clearly made)
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Re: Getting answers Vs Deep understanding
« Reply #11 on: Oct 22nd, 2008, 1:33am » |
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on Oct 21st, 2008, 10:48pm, Noke Lieu wrote:| (though it's a little stupid- when would it be a good idea to bad at... cake decorating? but the point is clearly made) |
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I think some understanding of symmetries would be very good for cake decorating; and you may have to bear in mind in how many pieces it will be cut, such that each get's an equal number of, say, strawberries.
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Noke Lieu
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Re: Getting answers Vs Deep understanding
« Reply #12 on: Oct 22nd, 2008, 5:22am » |
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you miss get my meaning.
I can't think of a (realistic)skill set that it would be a good idea to lack.
hence, when would it be a good idea to be bad at cake decorating. Or flower arranging. Or onion throwing. or bird spotting.
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| « Last Edit: Oct 22nd, 2008, 5:25am by Noke Lieu » |
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towr
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Re: Getting answers Vs Deep understanding
« Reply #13 on: Oct 22nd, 2008, 5:53am » |
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on Oct 22nd, 2008, 5:22am, Noke Lieu wrote:
you miss get my meaning. |
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Yes, yes I did.
Quote:| I can't think of a (realistic)skill set that it would be a good idea to lack. |
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How about the skill of misinterpreting what someone means? 
I suppose in the grand scheme of things it might not be bad if everyone lacked the skill of fighting wars, perhaps? Or would that just make them worse, if they're fought without skill.. hmm.
I'd say give it a few days, and people here'll come up with a few better example.
Quote:| hence, when would it be a good idea to be bad at cake decorating. Or flower arranging. Or onion throwing. or bird spotting. |
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When you don't want to do it.
If you're bad enough, someone else will take over in frustration. (Perhaps not as applicable to the last two; but certainly to things like cooking and cleaning, home improvement, etc)
I suppose you can argue whether it's really a good idea to lack the skill in those cases; and certainly the person stuck doing the chore because of it might disagree. So it stands to cause tension and strife.
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| « Last Edit: Oct 28th, 2008, 6:21am by towr » |
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JiNbOtAk
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Re: Getting answers Vs Deep understanding
« Reply #14 on: Oct 27th, 2008, 11:12pm » |
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At the risk of changing this discussion into a thelogical one ( that happens quite a few time with me  );
In Islam, no skills or knowledge is forbidden. It is permissible to learn absolutely everything about everything. This includes witchcraft, beer-making ( is that the correct term ? ) and yes, differential equations. Practicing what you've learnt, however, is another story.
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ThudnBlunder
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Re: Getting answers Vs Deep understanding
« Reply #15 on: Oct 27th, 2008, 11:30pm » |
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on Oct 27th, 2008, 11:12pm, JiNbOtAk wrote:
In Islam, no skills or knowledge is forbidden. |
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But the problem is whatever is not forbidden is compulsory.
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JiNbOtAk
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Re: Getting answers Vs Deep understanding
« Reply #16 on: Oct 27th, 2008, 11:34pm » |
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By all means T&B, if you are capable of learning them, why not ?
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ThudnBlunder
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Re: Getting answers Vs Deep understanding
« Reply #17 on: Oct 27th, 2008, 11:39pm » |
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on Oct 27th, 2008, 11:34pm, JiNbOtAk wrote:| By all means T&B, if you are capable of learning them, why not ? |
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Does not compute. Learning what? But perhaps you had a momentary lapse of reason and meant 'teaching them'. LOL
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| « Last Edit: Oct 27th, 2008, 11:56pm by ThudnBlunder » |
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JiNbOtAk
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Re: Getting answers Vs Deep understanding
« Reply #18 on: Oct 28th, 2008, 5:59am » |
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on Oct 27th, 2008, 11:39pm, ThudanBlunder wrote:| Does not compute. Learning what? But perhaps you had a momentary lapse of reason and meant 'teaching them'. LOL |
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Err..no. I meant learning them. By them I meant the skills Noke had iterated,
Quote:| hence, when would it be a good idea to be bad at cake decorating. Or flower arranging. Or onion throwing. or bird spotting. |
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Though on second thoughts..onion throwing ?
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Noke Lieu
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Re: Getting answers Vs Deep understanding
« Reply #19 on: Oct 28th, 2008, 10:34pm » |
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I was trying to think of *general case* skills by concocting a bizarre one. I could have been leopard juggling. Or keypad mashing. Or potato floating. Seed spitting, cowpat tossing...
As for being good at it if you don't want to do it...
just because you're able to do it doesn't mean you have to. Until you do. But then you could opt to do it badly so that you don't appear to be able to do it well. (and the onions don't go very far)
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| « Last Edit: Oct 28th, 2008, 10:36pm by Noke Lieu » |
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towr
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Re: Getting answers Vs Deep understanding
« Reply #20 on: Oct 29th, 2008, 1:53am » |
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on Oct 28th, 2008, 10:34pm, Noke Lieu wrote:| Until you do. But then you could opt to do it badly so that you don't appear to be able to do it well. |
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But then you'd have to be dishonest. And it may clash with your personality to underperform. Or it might be an somewhat automatic skill.
I can remember when I was younger, on a trip abroad in England, I met some German girls. Now, I really didn't like talking in German. So I pretended only to speak English. Until I slipped up and replied in German to something. Oops.
Sure, you can try to pretend you lack a skill; but chances are when you have it, and it's half automatic, then you'll slip up and look like a prat.
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JiNbOtAk
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Re: Getting answers Vs Deep understanding
« Reply #21 on: Oct 31st, 2008, 3:29am » |
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on Oct 29th, 2008, 1:53am, towr wrote:| And it may clash with your personality to underperform. |
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Well, I don't think there's a shortage of those here.
on Oct 29th, 2008, 1:53am, towr wrote:| I met some German girls. Now, I really didn't like talking in German. |
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towr, you do realize that most guys would have done almost the exact opposite ? What's the use of "useless" knowledge if you don't use it to score girls ?
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| « Last Edit: Oct 31st, 2008, 3:30am by JiNbOtAk » |
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towr
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Re: Getting answers Vs Deep understanding
« Reply #22 on: Oct 31st, 2008, 4:09am » |
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on Oct 31st, 2008, 3:29am, JiNbOtAk wrote:| towr, you do realize that most guys would have done almost the exact opposite ? What's the use of "useless" knowledge if you don't use it to score girls ? |
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It was about half my age ago, but as I recall they weren't really my type, nor particularly bright. Besides, my group (it was a school excursion of sorts) already had a girl I was infatuated with. (Not that I pursued that, either.)
[edit]Also; very, very glad not to be like "most guys"[/edit]
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| « Last Edit: Oct 31st, 2008, 4:11am by towr » |
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