That's not a *typical* approach? Mr Harkey sounds like every science or maths teacher that I had (sans chemistry, unfortunately) since fourth grade upwards.

Is mathematics really taught in the same way as some capital memorization for geography? I recall that you were homeschooled, just like I was in a rather weird high school so the perspective of both of us could be somewhat skewed… but (almost?) any approach other than the one in the article strikes me as a weird choice.

My highschool education did a decent job of this (albeit not quite as strict on explicitly writing out each step), but, as you mention, I was homeschooled and my dad took the time to find textbooks that took this approach. EDIT: I also spent a year studying classical and symbolic logic, which was frustrating at the time but incredibly beneficial in retrospect.

On the other hand, my wife was homeschooled but with a different math curriculum (Saxon math, if you're curious) that heavily focused on memorizing the mechanics (i.e., "this is how you do long division", etc.) but wasn't good at promoting an overall understanding of the material. Instead, it shows math more as a "bag of tricks" and solving a problem means picking the right trick out of the bag and then applying it.

At the beginning of this school year I taught "Algebra in a week"--a review of algebra for students that didn't do well on their math placement exam. I'd sneak in stuff like proofs that (a^b) * (a^c) = a ^(b+c) and comments like "we can rearrange the order of these terms because of the commutative property". At the end, several of my students thanked me for doing this!

(Unfortunately, having TA'd for "Calc II for engineers", I can say that even at the university level not every math professor teaches this way. It's incredibly frustrating to watch students struggle to understand the material because they're being taught a process instead of why that process works. On the other hand, I estimate that most expect math class to be taught that way, and view it as a hurdle that they have to pass in order to get to the classes they want to take. Furthermore, teaching focused on the process does little to discourage the "I can just do this on a computer in the real world" mindset. I could go on for hours on this topic, since it's hardly specific to teaching math...)

- "we can rearrange the order of these terms because of the commutative property". At the end, several of my students thanked me for doing this!

and

- It's incredibly frustrating to watch students struggle to understand the material because they're being taught a process instead of why that process works.

Plus after looking through some of the books (both Polish and American) with maths for engineers I can only muster "why?!". It's so idiotically rushed at places that I don't know how anyone could make any sense out of them. Here's an abbreviated 'contents' page:

1. Limits and continuity of functions (not a word about fucking *sequences*)

2-5. Derivatives of one and multiple variables, applications. Implicit functions.

6-7. Antiderivatives and integrals (practically no applications!).

8. Approximations and numerical methods.

9-10. ODE, 2nd order ODE, 1st order PDE.

11. Series. (Yes, *after* approximations!)

12-13. Multivariable integration.

14. Complex functions.

15. Integral transforms (Laplace and Fourier).

And believe it or not, that's *Calculus 1*! I don't even want to look what's on Calculus 2, you probably have one week of numerical PDE followed by Hankel's transform followed by Green's functions. No wonder that 1st year engineering has something close to 75% drop-out rate.

I can't fathom which is more frustrating. Teaching or being taught this curriculum. And that shit was squeezed into the 15-week long semester. I think that you can imagine how I was feeling when I got a few people from mechanical engineering to tutor and realised that most of their current knowledge of maths is pretty much the first two weeks from about six 101 courses. They only lacked a week of probability to make it a complete gibberish :/.

- Furthermore, teaching focused on the process does little to discourage the "I can just do this on a computer in the real world" mindset. I could go on for hours on this topic, since it's hardly specific to teaching math...)

If you have time and energy, do it. There's always a #rant tag and it's there for a purpose ;). Although I have to say that while I'm strongly opposed to people going to the computer to get their answers… it's at least useful to know the basics of both approaches. Maxima and Cadabra are just such amazingly useful tools for error catching or simplifying some of the really wacky formulae that I don't even know where to start singing their praises. But I'm not going to advocate people substituting linear algebra or calculus requirements with a phrase "can't I just run it through Mathematica?". CAS is something that you should earn as a privilege, just like in mr Harkey's class ;).

Here, it's a little different:

Calc I:

- Limits

- Derivatives, antiderivatives for single-variable functions

- Riemann sums

- Basic integration, up to u-substitution

Calc II:

- Trigonometric substitutions

- Partial fraction decomposition; applications to integration

- Natural logarithm/Euler's constant

- Logarithmic differentiation

- L'hopital's rule

- Indefinite integrals

- Sequences & series

- Taylor series

- First-order ODEs

The most infuriating part of Calc II is that students are expected to be able to do basic convergence/divergence proofs for the series & sequences part without ever having learned how to write a proper proof in their lives. Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.

So many students give up on that section since it's not well connected to any other material in the class and some of the concepts (especially the various remainder theorems) cannot easily be rotely applied to a problem.

And, yeah, "I can do this on a computer" does you no favors when you're trying to understand why some technique (even in another field of study) works, rather than just "oh I have this technique, let me apply it to some problems".

Anyway, you and I should be thankful that we've had something of a nonstandard education in math, and do our best to help others see what we see.

- The most infuriating part of Calc II is that students are expected to be able to do basic convergence/divergence proofs for the series & sequences part without ever having learned how to write a proper proof in their lives. Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.

First day of Calculus II happened Tuesday. Professor stated we are going through series and sums first. It's going to be a pain on top of all homeworks being online. This is my first time taking a math class in two years as well which is a bit of a pain having to go back and relearn (read: brush up on) many rules. Thanks for the article. I couldn't help but relate to this section of the comments. Thankfully, I'm also taking Proofs as a class, hired a tutor and plan to attend all the extra supplementary instruction sessions I can. As you can see, Devac, it really can be an odd system [from the outside].

- Thankfully, I'm also taking Proofs as a class, hired a tutor and plan to attend all the extra supplementary instruction sessions I can.

Regardless of the tutor, feel free to pester me if you would happen to have any problems! And even if your class about proofs does not follow Daniel Solow's *How to Read and Do Proofs* it's not a bad book to have on hand. Aside from one treatise on vector bundles, it was the sole English-only book that seemed to actually benefit my studies (at least so far). And from me, that's practically a glowing recommendation of Western textbooks. ;)

I think sequences and series are quite cool, and they tend to pop up all over the place in other fields, including probability, combinatorics, computer science, and communications.

I'm interested: what all does your proofs class cover? Is it something like a foundations class, where you start off with a few definitions and axioms and prove a bunch of stuff, or does it focus on logic and different proof techniques, or a mix?

Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.

According to the syllabus, we'll be going through Logic, Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, and Functions. Some Proofs in Calc.

Bless my HS Calc teacher for getting our toes wet early on since some of this sounds familiar. Just a matter of dusting of memories. My tutor has gone through the class so he's my first line of defense in the comparison department. If not, I'd like to start a study group (especially so if some from my Calc class are in the Proofs as well).

- and they tend to pop up all over the place in other fields

I would not be myself without adding "you forgot science". I think that I have seen a series or two in physics. ;)

By the way, one of my favourite uses of sequences in physics is a proof that no matter how many times you will boost your speed (ν) by some constant (Δν) you are still going to be restricted by the speed of light. That's a pretty cool exercise to solve in class.

- Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.

That's a really cool way to learn proofs. After getting some practice it gets better when you can play the "I may or may not have deliberately made an error, can you spot it?" reversal.

The critical point of no return is when you can do this with category theory :P.

- Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.

Damn, my most sincere wishes of good will toward you should manifest in some sort of alcohol in front of you about now. Not to mention that this format just feels wrong. Making an exam on something that most people never attempted before must look to students completely arbitrary on top of that. Like a medical school that ends with a pie eating contest to determine who gets to become a doctor.

- Anyway, you and I should be thankful that we've had something of a nonstandard education in math, and do our best to help others see what we see.

The thing is that I wasn't all that *aware* about the gap. But I do agree, the least we can do is to show that maths is not about learning some schema but about actually deducing solutions and formulating proofs. Hell, now I feel like I should rewrite most of the complex numbers primer that I made since I have assumed that the reader is prepared to do some heavy lifting instead of (at least that's how it seemed to me) being 'spoon-fed'.

On that note, it's probably for the best that I didn't share most of those Maths for Project Euler sheets. After about second part I have started shortening proofs and fell into the mathematicians mindset of "but that's a trivial step, removed". XD

- Like a medical school that ends with a pie eating contest to determine who gets to become a doctor.

This is a disturbingly apt comparison.

- Hell, now I feel like I should rewrite most of the complex numbers primer that I made since I have assumed that the reader is prepared to do some heavy lifting instead of (at least that's how it seemed to me) being 'spoon-fed'.

I say leave it as-is. At least part of the problem I see is that bad math education tends to hand people properties that seem to have appeared from thin air. On the other hand, if you say, "why is X true?", then that gives an explicit clue to people that they should (if interested) sit down and try to work that out.

Proofs are tricky things to write, but I do prefer a proof written to explain, rather than simply to derive the conclusion. A little exposition here and there can go a long way towards showing others the relationships you see. I am hardly innocent here, though!

I do need to work out a good way of writing proofs in LaTeX. It feels like mine always get compressed into a paragraph of math symbols. Maybe I just need to focus on writing a sentence or two of exposition per step and make one paragraph per conceptual step.

- I do need to work out a good way of writing proofs in LaTeX. It feels like mine always get compressed into a paragraph of math symbols. Maybe I just need to focus on writing a sentence or two of exposition per step and make one paragraph per conceptual step.

I assume that you want it for notes and not publications, right? I have been using this style for student physics club handouts and it seems pretty well received. Sometimes with margin notes, if I had some general remark about the proof as a whole but could not figure out how to fit it into the text.

An interesting perspective on learning. I wonder why don't more teachers follow it. Is it because it isn't actually taught to them? Or maybe it is, but teachers being already adult, it's more difficult to follow and, therefore, uphold?

I think it's a mixture of things:

- Teachers weren't taught the material this way, so they don't know to teach it like that.

- There aren't a lot of textbooks that rigorously follow this approach (or at least I haven't seen them), so you'd have to write your own curriculum.

- It definitely takes more work, and students won't like it if they're used to math being about the answer.

I had a physics professor who used the same philosophy, and I enjoyed that class--analyzing a physics problem was all about understanding how the different laws related to each other.

As a middle school math teacher, I can say that I try. From day one, our first and foremost class norm is "Our reasoning is more important than our answer." This unfortunately, for many, goes over like a lead balloon. This is my first year that I've made this norm as explicit as naming it every day. Some are buying in, which is good, while some are echoing Mom and Dad at home and complaining about this "new" math.

It's a slow go, but we're moving forward. With testing culture as prevalent as it is, the focus on using tricks to find the answers is at a premium.

In my classes, I do have a great number of students conditioned to proving their claims with evidence and reasoning. They're beginning to recognize the value of sound reasoning and problem solving.

Now, if I can find a way to remove the cultural acceptance of math illiteracy that occurs around me. Why is it that parents will meet with me at conferences and condemn their kids to math ignorance? For example,"I know little Billy is struggling with math, but I was never a math person myself anyway...can you just give him some extra credit?"

No. No I cannot give him extra credit. You are not supposed to undermine his education by asking that, or get all ticked off when I try and explain that this is a bad idea.

Can you instead give him a stronger work ethic and convince him that he will understand math better when he actually listens and follows through with his math? Oh, he'll have the grades and test scores you want too.

Yes. It's a slow go.

- From day one, our first and foremost class norm is "Our reasoning is more important than our answer." This unfortunately, for many, goes over like a lead balloon. This is my first year that I've made this norm as explicit as naming it every day.

I can count on one hand how many of my math teachers explicitly enforced this idea through encouraging such work ethic via their rubric/grading. That alone was enough for a later instructor's motto, heard at least twice a week for the 3 years I had him, to resonate: "a problem well-defined is a problem half solved." For those it sticks with, I can assure you it will be worth it. It was for me, at least.

- while some are echoing Mom and Dad at home and complaining about this "new" math.

It isn't new, though, is it? Solid, traceable reasoning has always been the basis of mathematics. The fact that a man once spent more than two hundred pages just to prove, with fundamental footwork that even religious fanatics shall envy, that 1 + 1 = 2 seems to prove it. It's the lack of mental rigor that undermines the solid structure that mathematics is based upon, which is what your kid's Mom and Dad have aplenty of.

Tell them "Your lack of mental capacity to understand the necessity of hard mental work when it comes to hard science shouldn't undermine your child's future capabilities" next time someone complains about "not being a maths person". I'd love to see their faces when they hear that.

You're doing divine work. Bless you, and best of luck with your work. Don't give up because of a few bad apples: dust accumulates all by itself, but people who make effort to shine are what's worth looking at exactly because they don't stand to stay filthy.

Thanks. I appreciate the support. It's a nice reminder that so many folks like you are out and around as well.

I have tried, on various levels, to give the math and reasoning ignorant a piece of my mind, but that never goes over well. Venting my feelings here on Hubski helps a bunch.

In the meantime, I know that I must win their hearts before their minds. I'll be killing them with kindness as they slowly recognize that reasoning increases the happiness in their lives as it does mine and yours.

Thanks. Let's keep that dust from settling where we can.

- I'll be killing them with kindness

That's a reasonable way to deal with it. Mine was cheeky, merely to create spite; not a wise one, and certainly not the one to act on when dealing with antipathy. You can't subdue fire with fire.

- In the meantime, I know that I must win their hearts before their minds.

Do you have any particular methods or lines of thought that you attend to in order to do that? Sounds like a good thing to learn from someone who's dealt with the issue of pupils' parents before, since I aim to be a teacher myself.

Glad to hear that you plan on teaching. It can be a bumpy, yet rewarding journey. There is no formula, or maybe actually, too many formulas to actually generalize what works. That's the art of teaching I guess. Knowing your audience is definitely primary to figuring out your strategy. I don't know you too well, so I knocked out as general a plan that I've found to works for me right now. Best of luck to you.

Firstly, in order to win over the parents and students, you'll want to fully recognize that change will come slowly. It's like watching a tree grow in your yard that you planted as a kid. It seems like nothing is happening until one day, you gain a new perspective and find the little tree has a trunk you can barely wrap your hands around.

Secondly, you'll need to up expectations by educating them on the importance of their endeavors. For example, you might set up a study into fractions, "Have you guys ever realized what portion of the hours of your week you spend, sleeping, in school, etc... Let's check that out." Relate the subject to what they know.

Thirdly, generate interest and enthusiasm through open dialogue with the subject matter as the center. This is generally more difficult in math than say science which is likely to be immediately relatable. I find that for math, continually relating, "Math is just patterns and puzzles about the world that people have discovered. Today we're checking out how _________ pattern works. What do you guys know about it so far?"

From there, I'd say, always have a clear and fair instructional system and give them lots of feedback. Students, especially my middle schoolers, as they are working to take full responsibility for their learning, when frustrated, will look for excuses out of their responsibilities in learning.

By having a system that is clear and fair, they will less likely to take the easy way out, and then learn to adapt to the expectations you have of them.

Thanks for asking the question. It's nice to rethink my reasoning for my teaching.

Glad my question helped, and thanks for giving me some guideline. I find it difficult to work without even a faint idea of what I'm supposed to do, and to have some ground makes it seem far more tangible.

- Firstly, in order to win over the parents and students, you'll want to fully recognize that change will come slowly.

Man... Patience. Always had problems with it. Though I'm growing into a better model of thinking, I'm sure it'll still be difficult for me to grasp the consequential nature of learning and teacher-student interactions until a few years later, when I do see the trunk grow. Any advice you have on keeping my mind on target through the daily drag?

- Secondly, you'll need to up expectations by educating them on the importance of their endeavors.

I can relate to that. It's much easier to grasp something you can relate to, and for me, it's like telling a story. I've been doing this for a long time, so it shouldn't come as a problem. I am, after all, going to teach languages: they are what makes stories, so as you learn the language, you learn how to tell a story, and the importance of that... Well, I'm sure I'll be able to relate the kids to that. :)

- Thirdly, generate interest and enthusiasm through open dialogue with the subject matter as the center.

That's where things get blurry. Where do you draw the line between having a dialogue and having to teach the curriculum? From what I hear, it's a big deal, to finish the curriculum before the end of the year; and of course, I don't want *not* to teach them something if the subject is concerned with it. And yet, dialogue *is* extremely important. So, how do you make it work, and how you do reign in a discussion gone rogue?

- From there, I'd say, always have a clear and fair instructional system

Are there any examples you can give, good or bad? Again, it's about tangibility for me; to learn where to go and what to refrain from. I have a vague idea about it - I did have some good teachers at school - but I'd like some concrete examples.

- By having a system that is clear and fair, they will less likely to take the easy way out

It probably depends on the person as much as most everything about teaching is, but - how do you reign in the pupils who fall angry at being tired of failing to learn or understand something? Simply explaining it won't do, because that's not the point, important as it may be afterwards. How well does talking straight ("I know you feel frustrated because you can't understand it...") work, in your experience? What else might work well?

Again, thank you for answering that. The more I learn before diving into the world of teaching, the better.

I wonder if this could be applied to linguistics. Mathematics and natural sciences have structure in them that you can easily follow, given similar rigidness to what the blog post describes. Linguistics is about a subject that's in constant change, miserable in size as it may be to an observer; the only "solid" parts about it are grammar and phonetics, and even then...

I'm sure it can--phonics is a good example: once you know it, you can almost always pronounce words that you haven't seen before. Same thing with algebra and equations/derivations/proofs you haven't seen: if the axioms are part of how your mind works, the math comes naturally to you.

Interestingly, I think computer science has some things to learn from linguistics about how to teach programming. I'd like to take the time to properly develop a curriculum for teaching a programming language as a written language one of these days.

- Interestingly, I think computer science has some things to learn from linguistics about how to teach programming.

That's something I came to think as well. There's a reason they're called "programming *languages*", even if people relate them closer to mathematics. The grammar might not be particularly human-friendly relative to English, but then - it isn't often with languages you're only starting to learn.

If you'd ever like to discuss the matter - feel free to message me: I'd be excited to merge my two passions for a purpose. If you already have some observation on the matter that you could share - please do, here or in IRC: I'd be delighted to hear it.