Constance Kamii has devoted her career to explaining - and proving - the value of teaching math for understanding. Kohn offers this from The Homework Myth:

Lots of practice can help some students get better at remembering the correct response, but no tot get better at - or even accustomed to - thinking. "In traditional math," says Kamii, "kids are given rules that don't make sense to them, and repetition seems to be necessary to memorize rules kids don't understand." She generally recommends steering clear of homework, "partly because what kids do at school is enough, and repitition is neither necessary nor desirable," partly because when parents try to help their children with math assignments they tend to teach them what they've been told are the "correct" ways to solve problems. Again, this shuts down children's thinking.Part of the problem with automacity is that by definition it invokes a kind of mindlessness - a kind of auto-pilot. But in learning and in life, rarely do situations remain stagnant long enough for us to engage in this mindless, automatic state IF we wish to remain successful.

The next time you are thinking of assigning homework so the students can practice, ask yourself how likely is it that your students will mindfully engage in what you are asking them to do? Or how likely is it that they will do the homework in a way that just goes through the motions?

I think the basic question here (though correct me if I'm wrong) is, "Are students learning for the test or for themselves?" If they're made to do memorization drills, they'll do what they have to do to get by on the test. If they're encouraged to develop a deeper understanding of what exactly is going on, and when this kind of math is useful, they'll memorize for themselves if quick recall of math facts is useful.

ReplyDeleteThink about poor calculus, which I harp on so frequently. I just did a Google search for "What is calculus useful for?" and the first answer I found was "Calculus is useful for solving non-linear equations." With that kind of answer, it's no wonder so few people who learn calculus actually use it in their lives. Now here are some actually cool applications of calculus that would make me want to learn and develop a deeper understanding:

"So if you have a hankering to know the exact force a 2,127ton meteorite with a 26degree angular velocity of 63,000Kph, spinning at 3,800kph will cause when it strikes a planet moving at -64 degree angular velocity of 40,300Kph, striking it at an angle of 42 degrees on a surface spinning at 4,600kph at a 12 degree angle, I suggest you break out a calculus book." You could even turn this into a cross-content investigation, examining whether the effects of the meteor in the movie "Deep Impact" were really realistic.

"What is the charge in a 47uF capacitor in a particular circuit at 3.5uS? This is NOT linear. Calculus is the quickest and most accurate way to measure this and makes all these HDTV's, PDA's, and the occasionally successful Mars landings possible." so students can learn about practical electronics at the same time as practicing calculus, but I never got this chance in my education.

Doing this quick search, I'm starting to change my mind, thinking that calculus is actually VERY useful. But I, like probably most others who learn calculus, lack the deeper understanding and other practical skills that could help me know when to use it.

She sounds like a person who is very remote from the field of mathematics. There is no contradiction between practice and understanding. Maybe instead of devoting "her career to explaining - and proving - the value of teaching math for understanding" Dr. Kamii should have studied and practiced mathematics herself. Then maybe her advices would be more relevant.

ReplyDeleteAnonymous, I'm not sure you'll come back to see this commment, but in case you do, here's a TED video that talks about a lot of the same ideas, but perhaps in a way you'll find more convincing.

ReplyDeletehttp://www.youtube.com/watch?v=BlvKWEvKSi8

Nowadays, endless worksheet drills and multiple choice questions are called "traditional methods". "New (?) methods" are of course "discovering mathematics". "Discovering" sounds like a great idea for someone like Dr. Kamii who does not realize how large the body of mathematical knowledge is, how long it took to accumulate it and what kind of people developed this knowledge for all of us. It is simply ridiculous to expect that most school students will be able to “discover” all required knowledge in the limited time allocated for mathematics in school between PE and marching band. And I am pretty sure that streaming video on an iphone and other multimedia distractions will not help.

ReplyDeleteMathematics is not a new discipline. It was taught successfully for centuries by lectures, complex problem solving, proofs, construction problems, etc. These methods work just fine.

Anonymous, I don't think anyone will argue that short lectures, problem solving, proofs, construction problems, etc don't have their place. These are useful and even necessary tools. But problems can still be introduced to students in ways that better emulate the real-world situations they could be applied to. See this video below for more on this.

ReplyDeletehttp://www.youtube.com/watch?v=BlvKWEvKSi8

Also remember that multiple choice questions aren't used because they're useful for learning, but rather because they're useful for grading. And mathematics has been around for a long time, but do you think Plato's Academy used multiple choice questions or worksheet "drills"? No way. These are relatively new inventions of the factory-style education system of the past 100 years.

And by the way, I do think worksheets can be useful in the classroom, if students are the ones that choose whether, when, and how they are used. Otherwise, students just end up doing busy work or trying to solve problems they're not ready for.

Thanks for continuing the conversation. If you visit this thread again, I look forward to hearing your thoughts!

Thanks, I saw the video, but I am not sure if I agree with this approach. Most of the mathematics problems are already examples of how mathematics applies to real life situations. If you want to make it "more real" by showing a videos of physical experiments you will spend significantly more time. As a result, kids will have less time to solve problems and consequently apply their mathematical knowledge to fewer real life situations.

ReplyDeleteHowever, I understand the difficulties that math teachers are facing and I appreciate the attempts to improve the situation.

Interesting conversation :-) Thank you.

ReplyDeleteIn reference to Anonymous's comment on June 13th "will have less time to solve problems and consequently apply their mathematical knowledge to fewer real life situations". I'd like to respond (respectfully) as follows.

I was one of those kids who struggled with maths at school - I was the 'dumb' kid who was given extra help by the ever more despairing teacher. Now I have left school and have the freedom to apply the maths to real life situations that are relevant and authentic. I have now reached the realisation that maths can describe and make sense of the world, and I am teaching myself through concepts and application (as opposed to abstract workings out on paper) - now I 'get it'. Maths is not just a subject made up specially to torture me ;-) with apparently no relevance to anything outside of the classroom / assessments.

So I would argue that although, apparently, there is less time to cover the content in the syllabus, the approach and use of multimedia suggested by Dan Meyer will mean that there is more and deeper understanding. Is this not what learning is about?