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The problem is that if you don't fill the period with busywork? The students don't build up the intuitive, gut-level skills they need to handle the higher-level stuff.

Math requires building muscles in your head, and muscles take reps. Lots of reps. Let's look at the elementary level: it's not enough to simply introduce BEDMAS or least-common-denominator and write a few problems that get the student to check off "okay, I know that" from the list of stuff they know.

The student has to be so practiced at these concepts that they could do it while trying to juggle 9 other extremely complicated brand-new concepts in their head because that's what you need to do when you build calculus upon factoring upon algebra upon fractions upon arithmetic.

If you fail to do this, you get kids walking into grade 9 math class who've forgotten everything they were supposed to know about fractions and teachers have too much new material to cover to re-teach fractions (my wife is a math teacher. This happens way, way too much - too many students enter her class missing necessary prerequisite skills). Students struggling with applying the basics cannot properly grasp the new material - so the basics have to be rock solid.

To a certain extent that's a product of a bad curriculum though. If you teach something, tick it off, teach something else, tick that off. And build a large number of things ticked off in that manner - that don't reuse the things taught earlier - then of course you're not going to get the reps in.

As compared to, say, the syntax in a programming language. Where you don't write out hello world sixty times to get the bracket rules, you just reuse the rules in other things you're doing often enough that they become ingrained.

personally, busy work just never sticks for me, i don't remember almost any of the math i did in HS and uni that was just "do these problems over and over till you remember it" because it just wasn't all that interesting, and this goes for any subject.

However some concepts, limits, derivatives and integrals i remember perfectly well (5+ years later) because my math teacher explained how the all fit together, what they represented wrt the real world and how they can be useful and that actually got me interested.

Yea you need to know the basics, but i daresay busy work isn't the best way to teach them.

It's the difference between skills and knowledge. Limits/derivatives/integrals are interesting knowledge so you can discuss them verbally, even if you don't have extensive practice in using them.

The basic, core stuff is the opposite. Fractions, bedmas, etc. you don't care enough to talk about, but if I handed a problem that used those concepts you could do it in your head, because they're skills that have been developed down to an instinctive level.

Common factoring is dull. There's nothing to say about common factoring. But if I show you an equation where you can apply it, you'll spot it instinctively because you've had common factoring ground into your reptilian brain.

Dull work can be made fun with a little effort. One of my high school teachers had us form lines, we refactor on the fly, said the factors of a quadratic out loud, and then were able to shoot the opponent (like a duel).. After a few shootouts everyone got very quick at refactoring. That little fun boost made the grind feel meaningful.

There is quite a lot to say about 'common factoring.'

For example, nobody knows whether there is an efficient algorithm to factor integers (where efficient depends on the number of digits in the integer). However, there is an efficient algorithm to factor polynomials, and you can write all irreducible polynomials of a certain degree as factors of one mother polynomial.

Moreover, being able to factor integers efficiently would allow you to break a lot of encryption schemes.

So many incorrect words, tired analogies and pithy statements.

The problem with your comment is that busywork is not (and never has been) mathematical weight lifting. You don't gain thinking skills by memorizing multiplication tables or practicing adding fractions, you gain it by solving problems you don't know how to solve.

Of course it's not enough to state something, but making the students reiterate what you've stated a hundred times is only marginally better.

Another problem is that no student really learns ANYTHING until they need to use it to do something else. No student learns fractions until they need to use it to do algebra, and they don't learn algebra until they need it to do calculus. And they don't learn calculus until they need it to do engineering/physics or higher mathematics (real analysis, which they don't really learn until they do measure theory...).

I think you're absolutely right that math has to be motivated. A lot of the effort to motivate it is to provide problems that students understand relate to the real world. Most of them come across as contrived.

You hit on a good concept with the idea of incorporating the history of math. I'm working on a Geometry curriculum that does just that. Understanding math in the context of a grand story of mankind trying to understand the world around us is a lot more interesting than as a set of received wisdom that must be memorized.

I would love to see more math material along these lines, but there isn't much out there, and it takes a long time to produce. I've been working on mine for around 6 months and am just up through looking at how the discovery of incommensurable lengths influenced Plato's philosophy. The idea is to work through the history and philosophy in parallel with Euclid's Elements, relating points back and forth where possible.

I've tried teaching a history of math class to local homeschool kids along those lines, focusing on Egyptian - Greek history, with some success. It takes a lot of research and work though.

> It takes a lot of research and work though.

Yeah, absolutely. In fact, I'd say that it's just as important that students receive confirmation of this history and motivation in their other subjects. For example, in various history classes, we learn about kings and wars and important philosophers. But we rarely learn about mathematicians and why their contributions were important, of if they were polymaths, we discuss their non-mathematical contributions, but omit their mathematical ones entirely.

Then in writing and literature we spend endless hours on appreciating tiny non-relevant symbols, but don't read something like Ringworld and spend part of a session calculating and relating the size of it. Conversely, in Math class, writing a paper on the history and influence of Platonic solids or similar would have been an interesting break and let kids struggling with crunching numbers take an interest and shine a little.

Or imagine a math course that analyzes things like agricultural output from Roman times to modernity, involving geometry, arithmetic, algebra, percentages (taxes to the king!), nutrition, etc. Extend out to how greater agricultural output per unit land produces surpluses that enable people to take up other labors like politics or art or music.

Off the cuff examples that would need more careful thought, but hopefully the idea is sound.

Bringing math into other non-math subjects, and bringing arts into non-art subjects I think helps prop up what students see as isolated subject "silos" that have nothing to do with anything.

Literature and history courses mutually reinforce each other, and math and science course mutually reinforce each other, but there's a tremendous gap between these two groups of courses where they don't reinforce each other.

This really needs to end because it all is connected and builds relevance for students.

Is your work done in the public? It would be nice to put it on a public repo and get others feedback and help.

I'm still struggling with how to handle this. On the one hand, it makes a lot of sense for it to be public. On the other, I've given up a lot to spend time developing it (including going part time on my day job), and I would love to be able to devote more time to it, which would require making some money off of the material in some way.

I've been thinking about doing a kickstarter, but I'm interested in exploring other models. I don't have a ton of time, and I'm always torn between spending time trying to figure out how to publish/market/exploit the material versus spending the hundreds of hours reading history and math and trying to actually write the curriculum. So far I've just been punting on it till I get this first course done.

I don't have a lot of knowledge, but there are websites that allow for the monetization of lesson plans. Maybe workbooks and videos to help teachers present the materials?

I do think a kickstarter would definitely help as would getting it out in other teacher's classrooms to get earlier feedback.

I totally support this.

Just checked out your site, it is like you are brother from different mother. Awesome stuff. I have a project I am working on that I will ping you with to get your feedback, I think you will like it.

I just realized it might not be that easy to find my contact info. I added it to the website, but it is my username here, no caps, at gmail.com

> Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to.

Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications.

Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?

> Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications.

It does, but for average Joe and Carla, who's day job involves making sure the box of sprockets sitting on the loading dock matches the P.O., and who's most pressing math problem is balancing their check book every few days and using turbo tax once a year, they'll never encounter anything more complicated than very basic arithmetic.

I think the hardest math problem most people encounter is figuring out percentages so they can fill out the "tip" on the credit card receipt when they eat out. Yeah it's "algebra" but it's a very limited class of problem and people just use a calculator or a tip calculator to do it these days.

>Trig

Yeah, trig really requires good motivation. The answer to most trig questions really is "buy the appropriate measuring tool" not "do a bunch of calculations and look up the result of the trig functions in a table". I don't ever remember having the calculations for cos(), sin() explained to me, they're just magic functions, number goes in, number comes out.

What motivated trig functions for me was writing 5 line BASIC programs as a kid to draw circles, spirals, etc. Realizing that COS was the X coordinate and SIN was the Y coordinate made it all clear.

Unfortunately there's no easy, low-friction way for kids today to do that with their PC.

Fortunately, TI continues to provide this to many students.

I had the same experience, but with a TI-83.

I feel like the educational and software worlds have failed to work together to provide a modern, powerful, 21st century answer to the TI-83.

Being able to animate and play with your math helps a lot. GeoGebra is a good example, but it's not the holy grail.

You might also enjoy https://www.desmos.com

With that kind of attitude we are just going to make more average Joe and Carlas. If you take what you are saying to a logical conclusion it leads to the exclusion of the many in fear of over education which I would argue can never occur.

There's a really, really simple way to get kids to love trig. Get them to write a video game.

Seriously, as a video game programmer, you use trig _every day_. If you want to find the distance between two points when you have the angle (eg. I'm firing a projectile at a certain angle), use cosine for the length of the x axis and use sine for the y axis. You get the hypotenuse from using pythagoras' theorem. If you want to find the angle between two points, use arc tangents.

Once kids have mastered all of that, you can teach them law of cosines and get into linear algebra; simple rotational matrices which can let you do collision detection and a whole bunch of other really, really cool stuff.

So yeah, Bob the builder might be boring as shit as a word problem, but Bob the builder might be a pretty cool video game. Without trig, as a game, Bob the builder is probably going to be boring as shit too.

"If you want to find the distance between two points when you have the angle (eg. I'm firing a projectile at a certain angle), use cosine for the length of the x axis and use sine for the y axis. You get the hypotenuse from using pythagoras' theorem."

Using cosine and sine and then Pythagorean theorem to get the length of the hypotenuse doesn't make sense - in order to get the length of the X and Y components with cosine and sine you need to already know the length of the hypotenuse.

You're right, I did state it a little awkwardly, but the statements are correct. I'll give a more concrete example:

I'm firing a projectile at 52 degrees and it needs to move 10 units. I'm at (5,7), where does the projectile end up?

delta x = cos(theta) * 10 delta y = sin(theta) * 10

So my final position is (5 + 6.16, 7 + 7.88), or (11, 15).

If I already knew how far the delta X and delta Y were, I could get the length of the hypotenuse with sqrt(6.16^2 + 7.88^2). That equals roughly 10, of course. If you want to try this out yourself, just remember that most trig functions are done in radians, so convert theta to radians with pi * theta (degrees) / 180.

Yes, I certainly wasn't disagreeing that trig is useful for games. The "if you want to try this out yourself" comes across a little patronizing in this context...

It was more trying to be helpful. A lot of people's heads explode when they see trig, and they stumble on things like radians vs degrees. You can do some really cool stuff with just a little bit of knowledge.

Yeah, I don't get why more people don't get trig - it's always come very naturally to me, but I have to work to recognize that's not the case for everyone, particularly when confronted with experiences from my past like working on homework with math majors and finding them confused when I applied a trig identity to the problem...

> Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?

You can also throw in some economics: Bob would be fired and/or get better equipment.

The irony is that all mathematics up to the high school level was historically motivated by practical problems. Algebra began as a system of rules to fairly partition land, for irrigation or inheritance, for example. But most of the high school math teachers are not actually mathematicians, so they themselves have no clue what algebra and calculus are used for. There ought to be incentives for professionals of all fields to go into teaching.