yeah man, helped a girl with her homework a few days ago and saw how horrible these new textbooks are...

Is it just one paragraph? Otherwise I'm not seeing it all on my mobile. What's the problem?

What it is is a ten minute lecture of a maths guy slating textbooks for being unsuitable for teaching new maths guys. His point essentially revolves around the idea that students are given questions that are merely how to process a formula rather than how to work out how to solve a real problem.

they learn Not to think .

Not really. You need to know how to use these formulae and how they work before you can attempt the thinking problem stuff. Otherwise you just get a class full of frustrated pupils who have no idea what to do building up a deep hatred of the subject. It would be a bit like asking pupils to race to the other side of a swimming pool before teaching them to swim.

This is why I dropped out of high school. I have no faith in the American education system.

disagree, Octavious, "knowing how to use the formula" can only come from someone who understands the concepts embedded in it. I think that understand the underlying math well enough to be able to independently come up with a formula to solve a problem is the only true way to have learned something

I am training someone pretty dim at work right now. I can show him how to copy the process, but he doesn't learn anything when I do that - only how to mirror other people's work. It's a lot more important to show him the "why" so he can actually start to apply problem solving skills. That is the only way to actually learn and retain it.

Dharmaton, this is a pretty big issue for us in the news right now. Yours actually fares the best. http://www.theglobeandmail.com/news/national/education/making-the-grade-the-provincial-push-for-better-math-scores/article16290351/?page=all

Being able to come up with a formula is hugely more difficult than being able to use one. This sort of teaching will only work iff the student is naturally good at maths. For the vast majority of pupils, who would never give the skiers a letter on their own initiative in a million years, it would be a disaster

It is settling if you are satisfied with a population who can do lots but knows and understands little.

Londo, are you kidding?, I saw textbook full of errors... from Quebec, ... anyways Cheerios to you

Has it not always been thus? How many people understand how a fridge works? Hell, how many understand how gravity works?

It is important to understand gravity in it's basic form. It is not important to know how a fridge works. The reason being, how many times do you need to apply "refrigerator mechanics" on a day to day basis? None, never. It is, however, important to understand the importance of refrigeration of perishable foods as well as ways to achieve this goal if your fridge breaks down (read: coolers and ice packs).

But math is applied every day and is important for contributing to society. It is crucial that people understand mathematics and can apply problem solving skills to every day situations.

It is also important to understand gravity in its* basic form. Knowing how a fridge works is important when repairing it. Mathematics, at least not on the level you're talking about, are not necessarily applied every day.

You have to learn how to write words before you can learn how to write sentences. You have to learn how to write sentences before you can learn how to write essays. You have to learn how to manipulate formulas before you can learn how to solve algebra problems.

IMO Dan Meyer's approach may help students figure out why they would want to learn how to manipulate formulas in the first place, which is good for their motivation, but it doesn't change the fact that there are certain prerequisites to each next level of maths knowledge. As such his approach can't be a substitute for the more traditional approach to teaching math, it can at best complement it.

Before you can write words & compose sentences you need to know how to communicate in a language (mostly verbally, but I say communicate because sign-language is also communication). What Dan Meyer is proposing is a problem based creative approach to teaching math. Rather than simple ROTE learning formula's he is talking about initiating curiosity in his students.

In his vision students are co-creating a solution, it doesn't matter if the skiers are called A,B,C etc... what matters is that they have named these characters in the image in order to discuss the problem.

Seriously I am aghast at how some people here seem to think that ROTE learning is actually a good thing! Education is about learning, and there is very little learning to be had by simply memorizing formulas. The students who are already far ahead of curve mathematically have already become interested in the subject. Dan Meyer is aiming this method at students who have disconnected from learning about maths in a school environment.

@kaner

Students don't co-create a solution. A couple of the more gifted students take the lead and everyone else takes the opportunity to daydream or play hangman. And learning by rote has its place. I learned my tables by rote, for example, and it has been of great use throughout my life.

This topic is of particular interest to me; I'm glad someone brought it up. I am completely incapable of performing even the most rudimentary mental math. As you might image, that made learning elementary math (times tables, long division, etc) a nightmare. Once we got past all that, though, I started to excel in math. This was partly because classes started shifting away from memorization towards problem solving. More so, though, was that I had developed the discipline and rigor that is required to study math at a higher level. Through highschool and college, I watched as other students started leaving math because they hadn't developed the level of discipline required.

So, I absolutely think we need to spend more time teaching how to solve real-world problems with math and less time how to plug numbers into formulas. Yes, math computation has a very important place, but with the ubiquity of calculators and computers, it is not what should be focused on. By focusing on real-world application, too, hopefully we can get more people *interested* in math, which is paramount to them actually wanting to study it.

I've been teaching math to kids and I've seen intelligent 15 year olds punch stuff like 1 times 6 into their calculator on many occasions. I'd like to see the argument for how that's a good thing.

Yes, you can use a calculator to do a division. But that doesn't mean that learning long division is unimportant. By learning long division, even though it's just an algorithm, kids get a better understanding of what division actually is (as opposed to just punching in a few numbers and hitting enter they acquire an intuition for how the answer relates to the operands of the division).

In addition calculators can't divide polynomials (http://www.purplemath.com/modules/polydiv2.htm), which you can do using basically the same concept as long division. As such depriving kids of learning long division will make it harder for them to learn polynomial division later down the line. This is just one example of how using the existence of calculators as an excuse to dismiss learning basic math and arithmetic skills is just plain stupid.

That being said, arithmetic and algebra require different skills and it is not at all uncommon for kids that have trouble learning arithmetic to later turn out to be quite talented at algebra or the other way around. Apparently you, abgemacht, were one of those kids. That says more about your individual talents than about teaching methods though.

I punch stupid things into my calculator all the time. Punching 1*6 into your calculator doesn't mean you don't know 1 is the multiplicative identity; it just means you multiple with a calculator by habit. Since there's no realistic scenario in which you'd ever have to do math without a calculator, I don't see what this is a problem.

I'm not saying it's unimportant; I'm saying it's not as important as other math skills nor should it be so heavily focused on in school. There are more important things to be teaching.

I was always relatively decent at math through primary and highschool. But it wasn't until I learned about the historical context for why various formulas were invented in the first place that I finally *got* it, unfortunately it was a little to late for me as I had already started to gravitate towards art and design - but I appreciate the cross over and try to expand on this whenever I can in a classroom.

My brother, who now flies planes in the Australian airforce, was absolutely useless at math until year 10. He has said to me a number of times that he only began to understand math after he took philosophy and grasped the reason for math.

In my mind both examples clearly illustrate the need to provide context in a math classroom. Otherwise like abgemacht math skills will be unlikely to be carried into adult life. And if subjects are unlikely to be used in adult life, why the hell do we teach it in school?

@Octavious - of course students co-create meaning. Unfortunately for the vast majority of *compulsory* subjects that meaning happens to be "it's useless, I don't get it, and it has no impact on my life now, and will have no impact on my life after school"

1x6 = sex
.
http://arxiv.org/archive/math

As my Chemistry professor says...
You've gotta remember something.

It's little use being able to think critically without actually knowing enough to think with. It's true that mathematics is problem solving, but you can't solve your trigonometry problems without knowing the definitions of sine and cosine for example. If I ask you to find a way to determine the height and distance of a building across the river without crossing the river, then while there are many ways to achieve that, you probably are going to have to use similar triangles, and if you don't know how similar triangles allow you to use ratios of sides then you aren't going to have much luck. Ultimately you need to have knowledge before you can apply it.

For those who are wondering how you find the height and distance of a building across the river without crossing it, have a go at it! There is at least one solution.

@Gobble, I agree entirely, but the question is how you to get people to remember. In your example you use trigonometry. How is it best to get students to remember the formula? Is the best way to get them to repeat the function 1000 times from (let's face it - quite boring) text books, or do is the best way to get students to remember to function by providing an engaging situation where they are *tricked* into utilising the formula because they arrived at it *themselves* (with a little prompting from the teacher)?

I don't agree with you that you can find an 'engaging way' to get students to learn the basic concepts of mathematics. One of the reasons why many students find mathematics hard is because of the difficulty of the basic concepts. Once you get the basic concepts down, it becomes much easier to build on that knowledge. You are suggesting that you want students to arrive at those difficult, abstract ideas by themselves. It took Isaac Newton and Leipzig, two of the best mathematicians in all time, to formulate calculus. By your logic everyone is capable of what Newton and Leipzig did, and we would be full of mathematicians. The fact is your average student is not Newton and this sort of teaching would never work in anything other than an elite class.

it took most of human history to figure out writing, and look at us now. Why, we are easily reading letters and all of those abstract magical scribblings...

Yes, memorizing basic trig identities, of course, is important. But, what do you really need to memorize to be good at trig?

sin(0)=0, sin(pi/2) = 1
cos(0)=1, cos(pi/2) =0
sin is odd, cos is even
tan=sin/cos
c^2=a^2+b^2-2abcos(C)

Remembering those, you can pretty much rebuild all of Trig on the fly. I would much rather give students a sheet with trig identities and have them apply them then test them on basic questions that require the memorization of dozens of identities.

And, quite frankly, if a student enjoys and uses trig, they'll naturally memorize the identities without being forced to.

^ that's way more than you need to know. All you need to know is that
e^i*theta = cos(theta) + i*sin(theta).
If one truly understands that relation, all of trig follows trivially.

Sure, but most students don't learn complex variables the first time they learn trig...

But they should

Maybe. That seems like a lot to tackle all at once.

Thx Kaner ! I really appreciate that lune of yours !

yeah kaner, and we don't let prep kiddies figure out writing by themselves, we teach them the alphabet.

And hope to inspire them to read books along the way. I should hope we don't simply teach the alphabet, and have them recite this 1000s of times in the hope that some day they will spontaneously pick up a novel and read for their enjoyment.

I'm saying that for my interest on a purely symbolic universal language for math & science.

They can't be inspired to read books until they bloody know the alphabet, and that's my point. You can't teach students problem solving when they haven't completed the 'basic knowledge' part of the curriculum! Your suggestion that we completely cast aside rote learning is flawed as there are just things that you have to memorize. Just as everyone eventually has to learn the times tables one way or other no matter what way you teach them with.

Except that people carry calculators with them at all times, so why does it matter if you don't remember what 6*7 is?

How could anyone forget The answer to the Ultimate Question of Like, the Universe, and Everything?

"Life" not "Like" Grrrrr.

sure whip up a calculator every time you need to do dome simple multiplication. like how much half a dozen eggs cost at seventy cents each.

You look stupid and waste time.

Who buys a half dozen eggs?

We do. Two people who just want to male a batch of egg salad or omelets or scrambled eggs on a Sunday morning don't even need six eggs. But 70 cents is more like the price for the entire half dozen.

Think for a second about how you do your mental arithmetic. Take the 5x tables.
Some people will multiply the number by 10 and then halve it,
some will halve the number and then move the decimal point,
some - especially the older generation, will simply know the answer like they know the tune to jingle-bells.
There are other ways as well.

The point is each person does their mental arithmetic slightly differently, usually in a way that they find comfortable. If we try to put a straight jacket on student's arriving at their knowledge by *forcing* them to learn in only one way, then of course they will loose interest and cause trouble. In fact teaching the times-tables for primary school is more effective if the students themselves problem-solve their own mental tricks to figuring out how multiplication works.

So while saying that some things only work if you *know* certain facts is - in a limited sense - true. Actual understandings only really occurs when student construct that knowledge for themselves.

Sure you can teach the sin, cos etc... and require them to remember those functions, but wouldn't it be that much more interesting if you created a curriculum where they arrived at this knowledge through their own investigation (directed by teacher set problems) and then build upon their knowledge in leading them towards what can be done with these functions?

I know what sort of math teacher I would have wanted.

To a certain degree, this topic does not need to be a zero sum game, an either or situation. Memorization and fluency with traditional algorithms and facts are important. So is creative problem solving.

In my experience as a student and teacher, rote memorization, developing fluency with numbers and algorithms, is important. This approach is not a barrier to developing a deeper level of understanding in mathematics. It does train people to think.

http://www.projectpro.com/icr/research/di/summary.htm

The above is a link to an article that reviews the largest, and most thorough education study in American history. While the article is geared towards the cyclical debate in reading instruction, the underlying points cross over to mathematics as well. Parts-to-whole, direct instruction that incorporates memorization produces greater results not just with lower level core knowledge, but with the higher level cognitive processes as well. There will certainly be exceptions to this. Abgemacht, I found your account to be interesting because my experience has always been that higher performing math students were almost always also fluent with facts and figures.

I think there is something to be said about incorporating elements of Dan's approach. Education is not merely a search for the most efficient way to convey knowledge. It should capture the students' imagination as well. Including activities and problem sets such as mentioned in the video could certainly help achieve this. I am skeptical, however, that these project based activities alone would be adequate.

This being said, my focus is on lower level education (K-8). High School and College math classes perhaps try to follow a rote system that is not as effectual at those levels. Kaner, I have heard from a number of people who took college ,level math courses, at Great Book schools etc., that really enjoyed course where math was taught in its historical/philosophical context and found it to make much more sense to them. Having never experienced this, it is difficult to comment, but I am intrigued.

I must respectfully disagree then, I just do not believe basic concepts can be 'guided' to with no memorization. It's certainly possible and often done, to create class exercises to introduce students to a new topic, but that cannot substitute for the whole course.

Understanding.