> Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000). I also provide an explanation of why those same people, when presented with the very same mathematical challenges in a traditional paper-and-pencil classroom fashion, perform at a lowly 37 percent level.
This. I love that this argument is being made. There is this notion among academics that the student should bend to the (fabricated?) stringent regulations on how math is taught or expressed. As a society, we accept the fact that there are those who grasp concepts, learn and develop sensibilities of material in various different ways.
For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor. How far Jamie has to walk to get 3 bags of milk is not execution, it's practise. This is a gross simplification of a beautiful subject - but this is the point where many get lost: Purpose.
For those who absorb material differently, this is where the conversation needs to start.
> How far Jamie has to walk to get 3 bags of milk is not execution
Fewer examples are that planted in reality, although you still encounter crossing trains and leaking bathtubs. But even if it's just about finding the reason of a geometric suite or calculating the derivative of some function, the teaching process is the same as in the Jamie example:
step 1: those are the Rules and Theorems.
step 2: IF $situation THEN apply $rule42
step 3: solved!
Basically this is about the same as someone telling you that when you encounter screws you need to use a screwdriver. Thus the whole teaching of basic mathematics currently is akin to learning to build all Ikea furniture directions by rote and expecting people to get something out of it that would propel them to carpenter level.
Math is not about driving screws with the screwdriver given to you, it's about understanding what a screwdriver is made of, why it is shaped this way, and building your own appropriate screwdriver should you encounter an unknown screw.
The current system is so hopelessly wrong that when I was faced with it my personal solution was to solve stuff in the quickest way possible so that I'm not bothered with such crap anymore. Incidentally to achieve this I taught myself to truly understand mathematics. The majority of others though could not be bothered and simply learned the rules by rote, because it was so complicated (and of course it is if you understand zilch about what you're doing). The irony is that the whole teaching process ended up this way because some people up above deemed mathematics too hard and "simplified" teaching. Then the whole thing got out of hand in a self-sustaining loop.
Painting, modelling and theory; writing, reading & comprehension. These are practice. Getting your painting into a gallery or your book into the hands of an unobligated reader - that is execution.
In Civil Engineering, we ran problem after problem in school. Nothing I do now professionally in civil design resembles the practice I did in school.
Which is just to say, execution should be the goal of ALL programs, but none seem to come close.
[edit] it's Friday morning and I'm done; any suggestions for a synonym for 'unobligated'?
I agree that they are practice in some sense, but the end result is valuable immediately. When I was a young student, I would take my painting, science experiment or short story home to show my mom. When I did well in math I showed my grade, not the actual work. Here is where we find perception of activity vs. successful comprehension.
Math is not valuable unless it has purpose and if some are not given purpose they will stop investigating it. It's easy to categorize those who have that viewpoint. As someone who found math later in life on my own terms, I can say that it was much more enjoyable than what was being taught as a student.
This was foretold in each math class I had in the schools I attended. There was always a "Math is necessary for..." poster on the wall in the classroom, and yet, no other class had a "Art is necessary for..." or "History is necessary for..." poster. We've identified the problem. Now what do we do about it?
Language Studies (aside from immersion) also could benefit from being more execution-oriented. For me, 4 years of French class was easily lapped by a month in Quebec.
I completely agree. I found math later too, at 29, and had it been taught better it might have clicked with me earlier.
The only academic program I've experienced that got right to the crux of its execution was History: but only because I feel its only real purpose is self reflection.
I found math interesting in school, but then I'm also virtually certain I was the only one in my high school classes actually reading the math textbooks. Even at an engineering college, I don't think reading the calculus textbook was the norm.
I like the notion of 'unobligated' in that it stresses that, for this instance, in school one's readers are obligated. With 'volunteer' I don't feel that condition is evoked; in fact, I can imagine people volunteering to read one's as yet unpublished work, but still feeling obligated to volunteer as a friend. But 'unobligated' is a mouth full of tongue to say, and if it should express freedom and willingness then it should lyrically sound that way. 'Volunteer' accomplishes that part, but leaves me wanting to tack on another qualifier.
For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor.
If science fair projects or English class writing assignments count as "execution" then certainly solving word problems counts as "execution." None of those things have real-world value, except in the very rare cases of extremely talented students who write a publishable story or produce valuable scientific results. The purpose of English class is to develop skills that will be applied outside English class, just like math.
What I personally saw in high school was that teachers were pedagogically obsessed with application in context, exactly what you call execution, but were at a loss as to how to apply the principle in practice. A project to create "context" for solving math problems could take many hours of class time and result in students using math only a handful of times. Solving a particular math problem is rather like hitting a curveball or playing a piece on the piano. You have relatively few chances to execute in a meaningful context -- a few dozen times per year, if you're lucky. Those natural contexts are just not sufficient for developing skill. So skills are developed in artificial contexts: hundreds of swings in batting practice, hours of practice alone at the piano, and many, many math problems with no real context.
It's easy to provide imaginary context, of course. I'm solving for the side of this triangle because I'm building a bridge and people could die (or I could get fired) unless I can figure out how long this side is. People don't find that very compelling, and that isn't unique to math. Kids taking batting practice or playing etudes aren't immersed in vivid major league or concert fantasies every time they swing or strike a key -- a lot of the time, they're fighting with their minds to fully engage with the task. But they take it for granted that they have to practice to succeed, while in math we have almost reached an attitude that practice is inimical to understanding. It's time we admitted that math is just the same as any other skill: little understanding can exist without competence, practice deepens understanding, and the mind is not freed to combine basic skills fluently until the basic skills become second nature.
It would be nice if it were possible to create a compelling context for every practice problem, but it isn't, not any more than you can create a compelling artistic context for every musical scale or writing exercise.
I experienced one really good example of application in high-school. We had one assignment that was coordinated between Science (in this case Biology) and English where we wrote a paper and the content was given a grade by the Biology teacher and the style/formatting, etc. was graded by the English teacher. It would have been nice if there were more things like this, as it directly addresses a concern pg expressed in one of his essays:
" Certainly schools should teach students how to write. But due to a series of historical accidents the teaching of writing has gotten mixed together with the study of literature. And so all over the country students are writing not about how a baseball team with a small budget might compete with the Yankees, or the role of color in fashion, or what constitutes a good dessert, but about symbolism in Dickens."
My school tried to adopt that approach. It was called "writing across the curriculum." It didn't last long at my school, which I thought was too bad, since it sounded like a pretty cool idea to me.
You may have point there. However, I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me. Execercising it more afterwards does improve my usage, but more in the sense, that I'm quicker to spot situations where I can (or can't!) use that particular feature/theorem. Sometimes I happen to see a new angle which enables new tricks. Learning to play piano on the other hand is a lot about muscle memory, which you train by repeating the same thing again and again. Yes, there is part of this in math too, but thats the handicraft-stuff. Arithmetics. Things a computer can do better.
This. I love that this argument is being made. There is this notion among academics that the student should bend to the (fabricated?) stringent regulations on how math is taught or expressed. As a society, we accept the fact that there are those who grasp concepts, learn and develop sensibilities of material in various different ways.
For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor. How far Jamie has to walk to get 3 bags of milk is not execution, it's practise. This is a gross simplification of a beautiful subject - but this is the point where many get lost: Purpose.
For those who absorb material differently, this is where the conversation needs to start.