By that I mean, start with the highest level explanation first, form an understanding of that, and then repeat the same lesson but at a lower level of abstraction. You have to be able to walk before you can run.
First, explain the concept like I'm 5. Use examples that provide real world context for why the concept is useful. Demonstrate examples of using the concept to solve a problem.
Then explain the concept again, with an added layer of complexity. Go through the same examples with this layer of complexity.
And so on. The more levels, the easier it will be for people to ease into learning the concept. When confronted with something that's too foreign, many people just shut off their brains, and convince themselves they're "bad at math" etc.
The hardest part about learning is really breaking through the wall of initial understanding. Once you know the language, everything becomes a bit easier to learn.
I honestly think math education can be hugely improved with this approach. This could even be done in a "high tech" way, by having an e-textbook with a slider. Adjusting the slider changes the level of abstraction and granularity that it teaches the concepts at.
Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Or whatever, you get the point. People can't relate at all to math when all the examples use generic terms without context. At that point it just becomes rote repetition. Obviously this commentary applies moreso to earlier math education.
Couldn't disagree more.
The whole point is that you don't care about the meaning of the numbers. They are just numbers. A big part of math is learning to strip numbers of their context and look at them bare.
Besides, variables with several letters in them are extremely ugly. They look like products. (They should be banished from programming languages also, but I realize that this is a controversial idea.)
I would think that by now that shouldn't even be up for discussion - much less for programming where your objectives, besides achieving a calculation or a side-effect, are also that your code be intelligible to other human beings. Even in assembly you have symbols for move, jump, etc.
And no, different people have different ways of thinking, not everyone has the same combination of nature and nurture to be completely comfortable with pure abstract mathematical notation - and for most part they don't need to. When you assign fancy greek letters as symbols you're also labelling them, so why not label them with something that a person learning might recognise.
Outside of teaching people that aren't familiar with it, yeah, sure, use all the numerical elegance you can muster (and I would still argue that even there it's still detrimental if the objective is sharing knowledge as widely as possible).
Or do people here really think that nobody has ever tried to make a math textbook with lengthier names and descriptions? It isn't used since it doesn't make things better, it makes them worse.
We can either believe that there's just a big percentage of people that will never be able to grasp beyond arithmetic (mathematically dumb) or that perhaps we aren't teaching things in the best way possible?
When I say using descriptive/relatable labels, is in the context of a relatable problem. I also think this is common, in the very first levels of introducing maths for small children it's a normal procedure. At some point between that and more advanced topics it seems to jump off and the conclusion seems to be "Many students fail to master requisite concepts before advancing to more complex ideas, leaving them ill-prepared to succeed in higher level Science, Technology, Engineering, and Mathematics (STEM) coursework.". Of course it's the students (usually kids) that fail, not the very smart lecturers and teachers. At every point in time probably people think of the way they're doing something as the most sensible or best way of doing it.
I think there was quite a leap when programming let go of pure mathematical notation to more regular language likeness - it seems many at the time voiced their opposition to that too.
Not long ago I spent some hours chasing around a problem where had the authors I encountered tried to explain it using more descriptive problems I would have grasped it in minutes, because it was really basic. But yeah, maybe it sounds really smart when it's shrouded in jargon, symbols and meanings that only someone immersed in that field of study would understand.
The problem here is that she don't know what speed is. She knows a lot of formulas, but not what speed actually is. That is a typical student, if you add speed in a formula you will make them confused and the problem will get harder.
The word problem was basically the same as in the video. If you're moving 20 miles/hour how long will it take to go 40 miles. (honestly, this was 20 years ago ... the problem probably said "20 MPH" ... it wasn't a very good textbook)
I read the problem to her, similarly to how the guy in the video was trying to explain. "If you are going 20 miles... per hour..."
She was in drivers ed at the time, so I tried talking about her speedometer and "MPH and how that stands for Miles Per Hour"...
Eventually I wrote it down for her as a fraction "20 miles / 1 hour" and not only did she figure it out but she also said "Oh my god, that's what MPH means? I thought that was just a number I had to match with the speed limit signs!"
One of the toughest parts of teaching is figuring out where you need to start (for each student). The thing that gets a student stuck might not even be related to the material... they may have an incorrect preconceived notion that you're not aware of so they think they know what MPH is, but it's actually something completely different, so you are not effectively communicating with each other because all the words you are using have different meanings.
What I am saying is
First, if student don't even understand what is carAccerleration or carSpeed is, then how do you expect them to understand the X and Y term?
Second, to explain an example, you may not use carSpeed or anything related to it at all, but why use X and Y while there are so many better names out there. It really depends on the context of the example trying to show.
So my point is using carSpeed or carAccerlation as programming naming technique bad? Yes, it's bad. But using greek terms to show a student who are really bad at understanding things comes even worse. How we can explain them what is alpha, what is beta, etc
I am not an official teacher at any school, but I volunteered to an educational organization to teach students about science experiments. The odd is there is a huge gap in knowledge in ONE class, students range from 6th grade to 9th grade. Student of 6th grade in class don't even know what is physics and chemistry (all stuffs they know is adding and substracting). My job is explaining to them what electromagnetic field is and how the tesla coil works. That's come to my point, you can't just throw to them a bunch of equations and tell them it is it. I had to solve this by breaking the term down into smaller pieces and didn't talk about any equations at all. After that, could I expect them know how to solve physics problems? Of course, it's not. But at least they had a push, and if you want someone to move forward, a small push is what they need (Newton's first law related). That is the 'A-hah' moment.
One more point, if you have read 'A Brief History of Time' of Stephen Hawking, you'll see he barely used any complicated terms or equations in his book (he did use, but not much), he just explain things and break it down into small pieces so even non-physiscal peoples will understand.
And in conclude, if it's for generalize things, then an equation with greek terms is needed. However, to explain things to student who are bad at math and really need help, there is so many better ways to name terms in math examples instead of using greek letters.
You can apply context to any generic abstraction/formula, otherwise the math is useless, by definition.
Teaching people effectively requires context.
I hope I don't have to read your code with single letter variables everywhere then! There's a reason that's considered a bad practice. Math gets a pass, why?
I'm not sure that this kind of snark is helpful if you want to teach something.
My point is precisely to make things as easy as possible, not harder! Variables are just slots. How would you rewrite the following using "meaningful" variable names?
The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ?
Any "meaningful" names that you could assign to the terms X, a, b, c would depend heavily in the context. You would convert this equation into an unrecognizable monstrosity. But this equation appears in many different, completely unrelated contexts! The main point of math, I'd say, is that no matter what is the context, the equation is always one and the same and it is solved in exactly the same way. Thus it pays off to have variable names as meaningless as possible.
You would convert this equation into an unrecognizable monostrosity."
For any beginner this is a unrecognisable monstrosity.
Having real names for the variables can give them something to hold on. Something they can relate to. So it makes sense to their brain.
And later on, you can abstract it - and they are fine with it and see the benefits.
(I still want programming variable names to be as descriptive as possible)
You (and probably everyone else around here) are used to abstraction - a beginner to math is not!
(and sadly, most people are even after 12 years of schoolmath still beginners in math)
And the focus on abstraction is a huge wall, making everything harder than it needs to be.
" Thus it pays off to have variable names as meaningless as possible. "
I agree that this is very helpful to achieve, when you are dealing with plain math only. But I think this thinking actually lead to lots of real world harm - when variables are not meaningless factors in math - but representing real persons. Which is forgotten quite often, if some buerocrat for example just sees them as a "meaningless variable" and treats them like this. But in reality this can mean, someones existence just got crushed.
Can you rewrite this same equation using "real names" so that it becomes less monstrous? I don't think that's possible, but maybe you have something in mind.
I recognize it exactly as this: as some abstract equations - but nothing more. (am I supposed to recognize them?)
There is of course no real point in adding names later on. At least not without context.
I would start with a real world example.
And then - yes have the equations with long names.
And then after that - have the equation as abstract plain math - in the shape you posted it. But not starting with it.
And like I said: nothing wrong with using x, y and z - when teaching und using abstract math.
My point was and is, that this should come later. After the real world examples.
I don't think so, as you can embed allmost any math concept in a real world context. Then the numbers get meaning - then the brain accepts dealing with it easier.
The first time I had real fun with math, was when I dived into computergraphic and had to come up with equations for various vector related stuff.
Suddenly there was meaning to it - the equations provided real benefit. With great visual results. So I had fun improving my equations with pen and paper. I never had that doing abstract math. Only slight relief when some equation finally solved, but it remakned abstract and meaningless.
So this is where I would start: a real world problem - and then to solve it, show how abstraction helps. And all the various needed math concepts along the way.
But starting with abstractions or focusing too much on them, just creates more mental blocks. I certainly had some with math - and I liked science from being very young. For common persons it is much worse. Just have a look how the common people discuss numbers related to covid for example ..
Not without adding a lot of complexity. What I meant is that the concepts taught in maths very rarely map directly to any real world problems. GPU programming is a good example, it uses a lot of vector related maths but the leap from not knowing about vectors to GPU programming is huge and not a good way to learn the subject, since you would have to think both about the complexity of GPU's and graphics and about the vector abstraction at the same time. The reason you had fun doing GPU programming is because you already knew about the interface of vectors so you had no problems working with them.
I had to relearn them pretty much completely, as I forgot all the abstract lessons before. But I had fun doing so. Because my brain saw real world use.
My suggestion is to first teach the generic concept wrapped in context that's more relatable to the student. Similar to code, it's easier to understand the purpose of a variable when you understand the surrounding context as well.
For your formula above, you could define the axes as something meaningful. Say, e.g. instead of A, B and C you have Supply, Demand and Price.
If I read a formula such as:
Demand = 100 + 50Price
Supply = 50 + 20Price
It's easier for somebody to relate to utility of the math than something like:
A = 100 + 50C
B = 50 + 20C
Of course, the math is exactly the same, but you give the student something to relate to.
Now, Supply/Demand is not something the average person can relate to. Obviously context has something of a cultural and familiarity bias. But there are examples that are likely to be close to universal. I use economics here, just because it's a discipline I'm familiar with :)
I don't advocate teaching strictly in contextual examples, but to use them first to provide ELI5 level introduction to concepts. The utility behind the math becomes more clear.
Of course, this is just my take/experience. I spent the majority of time in college skipping class and self teaching, and I discovered that the fastest approach to learning something new is to start at highest level of abstraction and work down. Usually I would learn concepts from resources other than the textbook, as they conveyed information more effectively.
Trying to learn concepts "bottom up" rather than "top down" works, but is less time efficient.
No, you give the student an extra problem they need to solve: Figure out what the hell supply and demand has to do with price. Understanding that isn't expected of high schoolers and teaching it isn't a part of mathematics.
History shows there are no intuitive equations outside of basic counting and summing. Even the concept of rate of change weren't explored before the 17th century, before then people had very ridiculous views on what equations physics uses, that is what you get if you try to invoke students intuition here.
Are more people going to relate more to a problem framed that way, or framed as what A has to do with B?
If you're comparing the two, it's self evident that the real world example is easier for the average person to understand. People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this domain is not the most widely understood... just off the cuff to make a point.
The solution to both problems is the same, does not require specialized domain knowledge at all, so I don't get your point.
There are plenty of more culturally widely understood domains we could use, rather than economics. Physics is of course the easy go to. Not sure what you mean by ridiculous views on physics, clarify?
Kids dont know basic physics naturally. They need to learn that and it takes quite a lot of effort.
Using physics examples kids who struggle with C = 40*B is not making it simple. It is complicating whole thing by making claims the kids don't believe yet.
Have you ever tried to argue with an average person? Those deductions aren't simple to make at all.
> Are more people going to relate more to a problem framed that way, or framed as what A has to do with B?
All students have already seen countless of examples of what A and B can be. So when they see an equation with A and B they can relate it to apples, to money, to distances, to mass, to temperature etc, at once. If you use special names for the variables most students assumes that the equation only works for those variables and either memorises it or tries hard to understand what makes those input kinds special.
> Physics is of course the easy go to.
If you use physics then your class is no longer a math class but a physics class. It takes years for students to build intuition for even basic physics.
> Not sure what you mean by ridiculous views on physics, clarify?
Before newton people didn't understand rate of change or movement properly, instead they came up with shit explanations like this:
> Based on Aristotelian physics, Scholastic physics described things as moving according to their essential nature. Celestial objects were described as moving in circles, because perfect circular motion was considered an innate property of objects that existed in the uncorrupted realm of the celestial spheres. The theory of impetus, the ancestor to the concepts of inertia and momentum, was developed along similar lines by medieval philosophers such as John Philoponus and Jean Buridan. Motions below the lunar sphere were seen as imperfect, and thus could not be expected to exhibit consistent motion. More idealized motion in the "sublunary" realm could only be achieved through artifice, and prior to the 17th century, many did not view artificial experiments as a valid means of learning about the natural world. Physical explanations in the sublunary realm revolved around tendencies. Stones contained the element earth, and earthly objects tended to move in a straight line toward the centre of the earth (and the universe in the Aristotelian geocentric view) unless otherwise prevented from doing so.
That was what the best of humanity could come up with over 2 millennia. You can't expect an average modern high schooler to do better than that without first being taught physics.
Demand = 100 + 50*Price
Supply = 50 + 20*Price
Now, for this:
A = 100 + 50*C
B = 50 + 20*C
It's easier to do abstract reasoning (spotting deep connections between patterns) when you abstract things. I'm pretty sure this has some solid evidence behind it.
It's really a question of whether you want to teach solving problems in general, or teach how to reliably solve a specific problem. If it's the latter, sure, in other subjects (e.g. accounting, science) they use specific terms a lot more, because the problem is more important than the problem solving.
As a teaching tool, teachers start with "3 apples" rather than "3x" problems. This is usually closer to K than 12 though.
Not sure what you mean. Providing context does not change underlying math or mechanics of solving the problem at all. Just the same as if you asked somebody to solve FizzBuzz or BeepBoop.
What it does is make the rote exercise of solving more meaningful and relatable. Things that are relatable tend to be more easily internalized.
Talking about early childhood apples etc is missing the point. I'm talking about generic methodology to learn a new concept in a timeboxed way, not a multi-year development. Though thinking behind the two is the same.
And before you say "just learn those subjects first then?", often it is easier to first learn the abstract math and then learn those concepts than the other way around, since making sense of those equations is hard even when you do know the math and close to impossible when you don't.
Take the math for Physically Based Rendering (PBR), for example the Cook-Torrance reflectance equation.
For me at least, it's 1000x easier to understand with descriptive variable names than with L, p, w, kd, c, ks, D, F, G, n, L, dw
No idea where else this equation is useful because it used single letter variables.
Single-letter variables are extremely superior if you're working out math on paper or at a board. It is simply not possible to work through a long and complex calculation or derivation with variable names that are not compact.
You also end up concentrating more on structure than on identity. You recognize idioms (like the dot products in your example) and it almost doesn't matter what the things being dotted together are; the physical form of the problem guides you to what they must be, even if you lose track. There is only one way it can go.
The flip side is that if someone else does the derivation, and you just need to use the result, it's basically completely incomprehensible to you. You haven't put in the time to understand the problem yourself (if you had, you'd "just" have derived or at least worked through everything yourself), so the extra affordance of long, descriptive variable names often makes a tremendous difference to final understanding.
This, incidentally, is a major point of difference between programming code and mathematics. For any good mathematics problem, there is only one right answer, and any attempt to solve that problem will reach it. (Possibly in a different guise, but the solutions will certainly be isomorphic.) Programming is so often business-driven or real-world-constraint-driven that this is not true. Two attempts to solve the problem may diverge radically. And no one will ever derive business logic from ZFC set theory! So for mathematicians, the fundamental understanding of the problem domain necessary to do good mathematics enables the use of terse, fluid notation. Since programming just doesn't work that way, its verbose, rigid notation is much more useful.
A lot of people are bad at math, but decent programmers. Yet, programming is basically just composing expressions the same as you would in math? Why the disparity?
Then the more you abstract things the less variable names matter. If you create a general purpose list calling its content type "item" or "x" are both equally readable since "item" doesn't say anything that "x" doesn't.
Now math, the entire subject of math is just a bunch of abstractions. Now there are a lot of concrete applications of math, but math itself isn't concrete.
The notation did not developed because people were bored. The notation developed and sticked because it made expressions easier to deal with.
School math was hell because my brain does not memorize useless uninteresting stuff.
A student starts in some state and that state may not have a direct transition to “understanding abstraction”.
A teacher needs to understand the state the student is in and find a sequence of transitions that lead to the desired state.
It’s difficult for teachers to do this if their expertise is limited to knowing what the final state should look like, but not being familiar with (or having forgotten) the state space that leads there.
What do you do when you need more than 26 variables in your program?
My experience is that best way depends on the student, the thing you want to teach and probably even on the time of the day.
The best teachers are able to adapt to that.
Teaching someone is more like a swiss cheese. Different students have different problems at different levels, and different holes at different levels, that they need to fill to progress more effectively.
A slider (or any completely linear progression of learning) is not, in general, going to be helpful to anyone.
Traditional textbooks can be of variable quality, but it's a benefit that you can flick through to any page. If a teacher can identify what page might be most useful to you, all the better. A point on a slider is unlikely to give you that.
> Why use X and Y when you could carSpeed and carAcceleration?
One of the purposes of math, one that makes it useful to teach to everyone universally, is that it teaches 'abstraction'.
Math questions very often are about car speed and acceleration, or some other contextual example (at least in Western education systems). Using v and a to represent those makes it easier to use algebraic tools that you learn that apply generally, rather than re-learn tools for each specific situation.
The point is you start with context specific examples, and high level naming. Teach concepts in plain english. After which you can re-cover the same material in progressively more generic and low level forms.
It's much much easier to understand the generic form of something when you actually understand contexts where it's useful, and see practical examples.
I think you misunderstood my slider comment. You have a textbook, but slider adjusts the level of abstraction that it teaches you the concept at. In what way is this incompatible with flicking to a page?
The TLDR; version is Explain like I'm 5, then ELI10, ELI15... and so on. There's a tendency in the hard sciences to look down on presenting concepts in a simplified form, so I understand many here will disagree with this approach. You have to struggle through the seminal 1000 page textbooks from the 70's to truly learn!
Every school already does that. For the first years math is almost only questions like: "You have 5 apples and you get 3 more Apples from Emmy, how many Apples do you have?", with associated pictures.
Then after many years of using apples and then learning numbers you start to show them that you can have variables like X and Y. Then when they already understand how variables work there are no issues using them. If they still have problems it is because they never properly understood variables, which is strange since we already spent like 10 years teaching math up to variables. You might argue that 10 years isn't enough time to learn variables, but many kids learn them way faster than that and we can't slow them down even further just so a few more kids don't feel left out.
It's not faster to code using one letter variable names, for obvious reasons. Because you're bounded by time to understand and make decisions, and not by how long it takes to press a few keys.
I don't advocate teaching more slowly, but teaching more effectively. Teaching "top down" is more effective, and not widely used, from what I've seen.
In your example you describe a multi-year period where basic examples are given in early childhood, and then made more generic over time. That's similar to what I'm describing, but not really the same.
I'm describing the same methodology but done repeatedly with any new domain being learned.
When you learn a new concept in computer vision, this methodology can still apply. Sure, you can throw the formula to do some image transformation at somebody and have them go at it. But showing it being used, explaining the logic behind it in natural language and so forth will convey the idea much more effectively.
The "bad" math education that turns so many people off tends to be more of a "bottom up" approach, solve 10 generic and meaningless problems using this formula.
I used to think this too, even when I started teaching. But I think if you look at a (modern?) textbook you'll see that those problems are not actually generic.
They are often designed problems that are trying to help you gain insights as you solve them. Each problem is probably demonstrating a different interesting pattern, if you are looking for that. Admittedly, students are not taught what to look for in those problems and if the text book as "answers to odd problems" in the back of the book, then there are probably 2x the number of problems necessary.
I teach HS Computer Science and I have to teach about division with ints and doubles. It would be nice if I could just explain the rules and give 2 or 3 examples and be done with it... but it turns out I have to have them work through lots of "generic and meaningless problems"
10 / 5
5 / 10
10 / 10
10 / 3
3 / 10
10 / 5.0
10.0 / 5
10.0 / 5.0
10 / 3.0
10.0 / 3
10.0 / 3.0
10 / 10.0
10.0 / 10
10.0 / 10.0
Each of those problems demonstrates a different "edge case". Towards the end there are some repetitive ideas I could leave off (once they know that double/int, int/double, double/double all evaluate to a double it may be overkill to keep repeating those patterns, but...)
If I don't have them work through each of those problems, it is likely that at least one student will pick up the wrong pattern and run into problems in their labs when they encounter those patterns.
That being said, I think the biggest problem with math education (in the US) is that many Elementary teachers would probably rank their Mathematics skill as one of their lowest skills. I personally know an elementary school teacher (with a master's degree) that took her last Math class her Junior year in high school. I met her in her college Math credit course which was a Computer Science survey course where we listened to talks by researchers and wrote 1 page papers about what they were researching.. I know she's a great teacher, but she just doesn't have the math background to teach more than "these are the steps to do math, follow them to get the answer."
Absolutely yes. I'm a secondary level math/physics teacher, and I've come to the conclusion that the problem truly stems from elementary level teachers who don't understand the math and just teach rote steps. Then, when things diverge from those rote steps (usually around fractions, negatives and variables, from what I've seen), that's where you really start to lose students.
The teachers themselves have no underlying grasp of the material and what it actually means, so of course they can't really impart that to the student (especially multiplying/dividing fractions and doing operations with negatives). Then it just compounds in middle school and by the time they get to high school it's a huge mess.
The first thing I do, and recommend anyone do, when learning new algorithms is to immediately convert the one letter var gibberish into a real program with proper variable names. Suddenly what looked like a huge, meaningless expression becomes so easy to understand!
The lack of focus on readability is rampant through math and math adjacent fields.
Another commenter mentioned the Cook-Terrance Reflectance Model. When I google this, I get a large expression with one letter variables. So I'm supposed to store in active memory which letter maps to which real world concept?
There's a reason one letter variables is considered a terrible coding practice. Why rely on human memory caching, when you can actually be descriptive?
Unlike a math textbook, you don't show how combinations of symbols are manipulated over time in a codebase as part of a coherent narrative that builds toward a math conclusion.
You also don't generally try to teach something new in the code itself, as opposed to in a team wiki or cited reference.
Feel free to clarify!
The slider is not useful - first because the goal of differentiating material by ability is already met in schools, and second because a student can't easily identify where on the slider they need to be, so it creates a further challenge for them to figure out their own curriculum. We only ask this of students when they get to Masters level, or beyond, with good reason.
If I were teaching you math, I would identify that your concept of mathematical algebra is poor, given that you haven't identified why variables x and y are useful in math as an abstraction (and why the justification for this is clearly different from computer science).
You're missing the point. I'm talking about a strategy for learning any new concept. Just because you're taught addition at age 5, doesn't mean learning calculus at age 15 is now easy. Is it a necessary abstraction to understand how to do higher math? Of course! But math education tends to focus on bottom up approach with rote exercises rather than teaching from an intuitive perspective first.
A good example is learning integrals. Another commenter mentioned they had been solving integrals for a year before understanding at all what their potential use was. Here's a formula, solve 20 problems. That's rote learning. That's not useful at all.
It doesn't matter the concept or discipline. If I give you simple explanations first, it will be faster for you to learn, versus starting with more complex ones. Otherwise there wouldn't be an art to teaching at all.
Your last comment is funny, because it implies a misunderstanding of what I'm saying. If you name X or Y, Speed or Acceleration, the math is exactly the same. There is no change in abstraction or takeaway in the education.
And it's different from comp sci how? I can change variable names arbitrarily in a program and behavior remains the same. I can explain a data structure to you via an easy to parse drawing and high level explanation, or I can show you a formula and say go at it. Same concepts, different teaching methods.
High school level math education is notoriously hated, for good reason. The fact that materials and practices are not entirely overhauled to make it easier for the average student, is just amazing to me. The focus by and large is not in making things relatable or intuitive at all. But there will always be people like you that defend a broken system to the death!
I think school curriculum is just fine, and already covers the points that you are trying to make, but better.
A much bigger problem is that there are few teachers competent to teach math in this way, because there is a shortage of math teachers, because teachers are significantly underpaid for the high-level skills you are touching upon.
The math curriculum, as it is in most countries, is fine. The "Common Core" maths principles begin with "1. Make sense of problems".
Algebra is done with letter variables with good reasons: the process of solving a problem involves translating reality into a mathematical model, doing the mathematical work abstractly in a standardised way, then translate back into reality.
A main purpose of algebra is that you don't have to hold your model of reality in your head while you do the mathematical work, which would quickly lead to cognitive overload on harder problems. [In computer code, the opposite is true - since you have a computer to do the mathematical work, you want to be always focussed on the model]
Concerning the final note: that is already the case. And then one abstracts away from that for two reasons: 1. This works just as well for boats, trucks or non-vehicles, so you use v and a. 2. Have you ever tried to write and read a formula like MassOfTheCar * VelocityOfTheCar * VelocityOfTheCar /2 =....? Notation needs to be actually usable in practice, not just pretty for single line examples.
I'm describing a "top down" approach, as opposed to "bottom up". Or outside in, rather than inside out.
> Notation needs to be actually usable in practice, not just pretty for single line examples.
Yes, math was practiced long before computers were a thing. There's a reason it's considered a bad practice to use single letter variables in code.
I can assure you it's much easier to read complicated formulas in well written code than in conventional math notation!
If you were to use all one letter variable names and cram all computations into a single large expression, that would be considered by most to be poor code. There's a reason why, and it's been proven out in industry.
Math is no different, other than cultural biases and historical reasoning.
I could not disagree more. The long names obstruct the shapes of the expressions, making it difficult to recognize common concepts visually. There's a higher need for intermediate variables (to avoid expressions becoming physically too long), and the intermediate variable names often need to be very long to properly describe their meaning (and may represent highly abstract concepts). Terser code can avoid trying to name these things that cannot be meaningfully named.
> If you were to use all one letter variable names and cram all computations into a single large expression, that would be considered by most to be poor code. There's a reason why, and it's been proven out in industry.
I agree that it's not a good idea to put everything in a single expression. But it's not a good idea to put every single mathematical expression in its own variable either, and in my experience the sweet spot lies much closer to the terse code.
I've worked with math-heavy code in both terse and verbose styles, and refuse to work on verbose math code again.
What is the strong reason?
- It easier to see the 'essence' of an equation/problem when all irrelevant detail is omitted.
- Many concepts are so general that you'd need a small essay to have a 'descriptive' name, rather than just, say, x.
- Algebraic manipulation often leads to non-intuitive concepts in intermediate calculations, e.g. meter-dollars. It's handy to be able to just call such a thing 'a' (or whatever) if it helps make calculations easier.
Definitely no beyond elementary school level and very basic concepts.
If is much easier to do abstra manipulation and calculations with letters then words. And beyond very very basics, you will do a lot of that.
Extremely bad idea, and I learned it the hard way. A lot (most?) textbooks have more details than you need to know. It's better to find out the most important material in the book, focus on that first, and then go to the next tier of important material in the book, and so on. At some level (perhaps after the first), feel free to try other books without completing the current one.
Unfortunately, it's not easier to know which material in the textbook is important and which are merely detailed examples on your own. You need someone with mastery to tell you that.
> If you face difficulties, do not give up but instead go back twenty times if that should prove necessary and only then allow yourself to investigate another mathematician's solution
Partially agree. Pick an N that is large enough, but not too large. Another mistake I would make is refusing to move on until I've solved the problem. A more practical approach is to give it N times over M hours/days, and then move on and/or look up the solution. You'll learn more this way.
Without some guidance on which book to choose, (1) could condemn a reader to mathematical purgatory. In any case, I’d also say that taking multiple math courses in high school helped me to understand that math is most often about relationships and that there were multiple ways to describe the same relationships. This was in spite of no one bothering to really teach that idea. Previously I was mostly focused on axioms and rules (probably because of how I was taught), which, while useful, missed some important stuff. I also prefer to read through multiple texts because it gives me a break, let’s me see new connections, and if I come back to the text after working on something else and still know what’s up I know I really grok the material. (It’s insanely easy for me to memorize and apply rules; that’s not indicative of understanding.)
(3) is good reading practice in general, but also, it’s a skill most people haven’t developed. It’s very easy to say, “Pfft, I know all this,” and just skip something without actually really grokking the information; it’s too easy to over-estimate understanding. For most people I’d say at least work through any proofs and the most difficult questions in each section. This is actually something I think we should be learning in middle school at the latest.
The advice implies a completely linear process of learning. I think that works for some portion of learners and isn't useful at all for another portion.
Edit: never abandon a book you have chosen without working all the way through it
I've read a number of math books on my own. There are plenty of bad books out there. Some will have you wasting an awful long time if you never abandon them (since they are filled with extremely difficult problems or simply don't explain themselves well). This is advice for someone who, at minimum, has a teacher feeding them good books.
Abel is famous because of his algebra results. He had awful good abstract thinking and was indeed very good at math. That does not only everything he ever said is correct. In particular, it does not imply general teaching skills.
Being good at proving theorems does not imply being good at teaching students.
Both means that if the book you picked assumes you already knew something you don't really, you are stucked. If your book happen too explain something later or is not perfectly written, you are set up to fail.
Very often, later concept let's you figure out the first one. Very often, another book or source adds details to have been missing. Very often, you need to do something else for a while and when you come back to chapter, it suddenly clicks.
See the MAA's Instructional Practices Guide:
Here's info on research on the effectiveness of inquiry learning in math:
It wasn't until undergrad that we had this fantastic calculus lecturer - everything just made sense. He started at the very, very beginning; adding together numbers, and worked his way up to pre-calc, in around 2 weeks. He never skipped any steps, and was very good at explaining the implications of various proofs, and pointing out the typical / common errors and fallacies made by students.
Of course, if you had a good grasp of math leading up to calculus 1, this was all boring and old news. No big deal, those students would skip classes, it was after all in the very beginning of the semester. But for me, it was simply life changing.
And parallel to that, people started posting vids and websites explaining math. I suddenly had multiple different sources of people trying to explain math their way - which also turned out to be invaluable. Suddenly this guy Salman Khan started posting vids on youtube, and I discovered Pauls Math notes.
Well, the rest is history for me - I graduated with good grades in all math courses, and a love for math.
If anything, it thought me that there's no single way of teaching math. Again, when I grew up, it was one book and one teacher.
Paul Halmos taught for two years in the 1970s at the University of California, Santa Barbara, where I was an undergraduate majoring in linguistics. I was also interested in math, and I ended up taking three courses from him: an introduction to mathematical logic, with about twenty students; a seminar on set theory and the foundations of arithmetic, with four or five students; and a one-on-one tutorial on point-set topology.
For the latter two courses, he taught based on his “do mathematics” principle. He would provide us with definitions of basic concepts and then have us work through a series of not-so-difficult problems that led eventually to proofs of the major theorems. He let us do most of the work ourselves, though if we got stuck he would provide pointers to help us keep moving ahead. His method was inspired by that of Robert Lee Moore , though Halmos was not as hard-assed about it as Moore reportedly was . In the seminar, he encouraged the students to work together.
More than forty years later, I remember many details of the two problem-based courses, while the first course, in which he lectured on mathematical logic, has mostly slipped from my memory.
I found that it significantly expands my repertoir, or grammar and vocabulary so to speak instead of staring at the lecturer trying to explain their thought process.
In fact I only understood why integrals were useful, when I started learning for the final exam.
This stuff can be really cool, but if even your teacher does as if it is the most boring and useless thing in the world, if even your teacher is unable to explain beforehand why you want to know this, how would you?
We took a triangular ruler outside, leveled the base and lined it up by eye with the top of the building. Then we measured the distance to the point where we were standing and the distance from the ground to the ruler.
I really wish education reduced the amount of rote learning involved and focused more on letting students explore and make mistakes with the educator as a guide.
Then one day we all went outside with our solar cookers and hot dogs and had a cookout. IIRC, there was a lot of variation among the cookers. Different reflectors were used (piece of aluminum sheet, bent and held by end brackets. Mylar film laminated to a wooden parabola "bed". One kid got adventurous and designed his with a vertical dog holder, and a circular parabola that was actually several different profiles stacked to "smear" the focal point along the vertical axis. I don't remember if it worked well or not)
I think we were meant to learn some sort of process engineering in addition to our respective subjects? Or just "teamwork", I guess.
In my experience, students are expected to memorize all sorts of equations without ever really being expected to understand why you're calculating the integral in the first place (even at decent universities). Obviously the risk with that is that you won't recognize any problem can be solved with integrals, no matter how concrete. Once I left the fancy tricks behind I felt like a fog lifted and it was actually really fun to see how integrals fit into both applied and abstract examples.
Going back to that simple explanation once people are a bit more experienced can make things click for a lot of people. Remaining people what they are dealing with and why can also help.
My class consisted of hour after hour of solving integrals of different types using all the different shortcuts, and the exams were largely the same. I feel like that time and effort could have been much better spent.
Aren't people who try to do this a running joke?
'Adam has five apples and seven buckets and must drive to Rapid City...'
People hate that stuff.
You can do a ton of cool stuff with maths, including calculating the movements for figures in a computer game, extrapolating the distance to the moon, etc.
And before someone claims I am beeing unrealistic: in my youth I worked with teens from socially complicated backgrounds and helped them to learn for their maths exams, exactly like this. These were kids that in bad moments would pull a knife on you or throw furniture out of the window. This was literally the only way to get them to start caring about the result of a mathematical formula and then they demanded to know more..
Well, I don't think there's a royal road to teaching mathematics.
First ingredient if you want to be a good mathematics teacher: preparation. When you go to class, just make sure you went through all the examples, all the exercises and problems, and you know them in and out. Be prepared.
When they train you how to teach, they never tell you that. It is just assumed. They tell you about the attention curve (which supposedly looks like a bathtub: high at the beginning and at the end, low in the middle), about body position, voice volume, and other stuff. But for some reason they forget to tell you that first and foremost you need to prepare.
Second ingredient: feedback. Ask a colleague to come and observe you, and, if possible, go with the same colleague and observe another teacher whom you want to learn from. They (the colleague) can point at what you did differently from that teacher, what you did well, what you should work on next. If that colleague is good, they will give you a list of 20 things to work on, but also tell you to focus on the first one or two. Are you stopping to look at the kids and see if they understood? Are you making eye contact with a single kid, rather than cycle through all the classroom? Is your voice loud enough (pro tip: it never is, speak louder). Did you leave the kids enough time to think on their own? Did you give them too many hints? Are you biased in always letting only the best kids in class answer out loud? Are you pointing with your finger when you want to emphasize something (it's natural, but don't do it, it intimidates the kids).
But all these things are second order things. First thing: prepare for the math you are going to teach. It's much more important to not make any mistakes, such as a sign mistake in the middle of a calculation, than to make eye contact. Mistakes erode the kids' trust in you. Avoid them. Prepare.
Once you got to a point where you consistently deliver a flawless lecture, then you can think of becoming the next 3Blue1Brown. But you need to walk before you run.
I wonder if that's really the best way to teach, like a classical music performance where you can't play a single wrong note. This almost conveys that the teacher has some special intuition that the students will never get. It's also far removed from how anybody actually uses math. We try things, go down blind alleys, look stuff up, check our work to see that it's believable, etc.
Now I do believe you need to be solid on the concepts, so that when you do make a mistake, you can say: "See, this is a mistake because it violated X."
If you make mistakes regularly on curriculum content, on content that students are expected to perform in a test without mistake, students will quickly lose confidence that you can teach them anything.
What is a mistake and what is new material often looks the same, to a student: strange processes that they haven't seen before. They will have to overcome your mistakes in order to figure out what they are supposed to be learning.
> Mistakes erode the kids' trust in you.
In addition to this, I often find myself trying to digest concepts broadly because my mind is lackluster with the mechanical side of calculations. When a professor makes a mistake, I occasionally have to revert an already imperfect record (with or without notes) and adjust my conceptual framework.
Just an anecdote about why I personally suffer from poor teacher preparedness as a student.
> When they train you to teach...
I started a job as a professor this year and never got any of that basic training. Before this I was a postdoc reading papers and writing code all day. In job interviews my lack of teaching experience wasn't seen as an issue. The faculty would say, "oh teaching isn't so hard, you'll figure it out quick enough." So far I have been muddling along ok, but I have this anxious feeling that there are a lot of aspects to this that I don't know that I don't know, and don't know where to get started on finding out about them.
I was extraordinarily fortunate to have just such a colleague as I mentioned. I consider her to be a genius when it comes to pedagogy. We went together and observed a great Professor teaching. She pointed to me lots of things, but one stuck to my mind: "Did you notice that he was wearing a tie? He doesn't wear a tie when he's in his office, or on the hallways of the Department, but he wears a tie when he teaches". That's respect for your students and for the action of teaching.
My Dad was a teacher. One day a former student of his told me this: "I remember the day he came and taught us X"(where X was a great poet - my Dad was teaching literature, not math). "He was wearing a black suite and white shirt and tie. That's how much he cared about teaching about X." I was impressed. I realized that kids care. And they notice. They notice if you put the effort, and they appreciate it.
Good luck in your career. I don't know of any book. I'm not a teacher or a Professor anymore. I work in the industry, but I do take interns every year, and in a sense that's a teaching (and research) experience. I enjoy it a lot, my next intern will join exactly one month from today. And just like I was preparing for my kids in the past, I'm preparing a lot for this intern. I hope he'll enjoy the experience.
Personally I found the book To Know as We are Known very inspirational:
First, the class of sequences under consideration is not clearly defined. (I am not going to invest time solving a problem, if I am not sure about the definitions involved, constant coefficient/variable coefficient etc).
Second, a question is asked "Is a product of two such sequences also of the same form?" and the meaning of product is also not clearly defined.
Third, the author links to an article whose PDF on the publishers website is unreadable, and does not seem to answer the question from what I can tell. The only purpose linking to this article serves is intimidation.
There is a happy ending to my rant. I googled the topic alluded to by the author and found a clear post explaining the solution. In this stackexchange post the question and solution are both clearly stated.
Actually, make the test be a minority of the overall grade - i.e. 80% of grade from homework.
It was boring at times, but after three years I had a rock-solid base on how to turn one expression into any other. The applications of those skills came later once I got into Physics and Stats.
I don’t know if it worked out better that way, but I think there’s kind of a chicken-and-egg situation. To really understand the applications, you need the math background. To appreciate the math background, you need the applications.
Just like you cannot learn to program by reading books, people have to really dive into problems and rediscover certain facts by themselves.
The end result is that basically everyone who learns math creates a lot of errors in their understanding of it, and then those errors gets hammered out after countless exams as they progress through their education. Without the pressure of education most people just continue to accumulate errors in their understanding until they no longer can progress since the backlog of stuff is too long and they don't know where to start.
Of course, I'm not going to suggest that learning math solely from a book is optimal (neither is learning programming from a book), but it certainly isn't impossible.
My solution: short sessions of a few pages at a time, with me sitting by his side, ready to intervene when he gets sidetracked.
I tell stories with mathematical concepts - anything involving money (selling and buying things) is popular around here. And the Boolean Logic Baby Gate was a big hit (you have to put in just the right filter function if you want to let your friends in but keep the disruptive babies out).
I sing Baa Baa Black Sheep but with lots of different numbers of wool distributed in different ways (this also keeps the 1yo entertained).
Guess My Secret Number gets lots of requests: "If you have my secret number and you add four more, you get seven. What's my secret number?" (We take turns guessing and giving clues.)
Another good game is Tiny Polka Dot - we've come up with so many of our own variations on the games.
In terms of actual books, we really enjoy Life of Fred.
Even recommending which books to read -- a very personalized topic -- is something a good tutor can do.
It you want something that starts with the absolute fundamentals, What Is Mathematics? is a perennial classic. I think that one starts with addition and takes you all the way through advanced calculus.
If you want a comprehensive survey of nearly all major branches in mathematics written by some of the most prominent Soviet mathematicians, you’ll want Mathematics: Its Content, Methods and Meaning.
If you want more of a reference-style encyclopedia, The Princeton Companion to Mathematics is a good one.
If you want to build a solid foundation for math tailored to computer science, look no further than Concrete Mathematics, coauthored by Donald Knuth himself.
Every time I've recommended Calculus Made Easy  it's been a huge success. The writing is lively and full of motivating examples, and it's an enjoyable read.
> You can teach people math by teaching them how to read, and then suggesting a good math book.
It didn't work for all. The correct way to teach maths is to remove NUMBERS!
1. Don't teach that math is a "ladder". Trigonometry is not "harder" than geometry, although many concepts in trigonometry are expressed as geometry. Calculus is not "harder" than algebra, although many concepts in calculus are indeed expressed as algebra. Teaching that math is a ladder allows people to believe that once they've covered the "basics" of a field of math (as if you could ever learn all of "algebra"), they shouldn't have to think about the basics again. Try telling a pianist that once they've learned a set of scales they don't need to study them any more, or try telling Richard Feynman that once he understood the standard atomic model there was no need to continue examining it in greater depth.
2. Don't teach that math was "discovered" in its current form. What most students think of when they think "math" is in fact western notation for patterns observed in reality, handed down by a bored teacher as universal law. A sheet full of squiggles isn't "math", it's just a set of the most-predictable and best-notated patterns made up by past mathematicians. "Math" is the process of discovering and refining those squiggles. The relationship between the length of the two perpendicular sides of a right triangle and the hypotenuse exists outside of human knowledge, but the Pythagorean Theorem is a only a certain method for expressing that relationship. The numbers 1 through 9 are one possible code for counting, but the quantities I through IIIIIIIIII and onward exist outside of the notation of Arabic numerals.
3. Don't immediately move on from concepts once a student has mastered them. Imagine if every time you figured out how to do your job correctly, your boss moved you to a completely new task requiring a completely new set of skills, giving you no time to enjoy applying your prior mastery. If you've just learned how to factor quadratic equations, why move on? Why not explore programs that factor equations? Why not dig up old exams and show how factoring would have solved earlier problems faster? This is always met with cries of "but they're supposed to use what they've learned to go on to even harder problems!" Sure, I agree, but how can you possibly enjoy any task if the only thing you can really expect is that even if you master your task, you'll be struggling again within the week? Would you join a rec basketball team if every time you started hitting three-pointers consistently, they moved the rim higher and farther away without giving you any time to feel what it's like to be good at basketball?
4. Don't teach that you need to know math because your grades have to be high. For nearly every profession available, grades are immediately forgotten as soon as you start receiving wages. Teach that you need to know math because the world is constructed and controlled by mathematicians and those acting on the advice of mathematicians. If you don't know math you'll be taken advantage of by those who do, whether it's in advertising, gambling, banking, medicine, insurance, politics, entertainment, engineering, programming, or any of the many other fields driven entirely by math.
Of course, not teaching those four lessons becomes difficult, because those lessons work in opposition to the central tenets of mandatory state education, which necessarily operates as a giant brainwashing factory working to justify its own existence. Modern ediucators spend most of their time trying as hard as they can to teach people that knowledge is best bestowed by authority and then proven through certificates, that various fields of inquiry exist separately from one another (for instance, that physics and biology can or should be studied independently of history, mathematics, grammar, semantics, or art), and that failure to live up to expectations must necessarily lead to shame and corrective action.
Don't send your kids to school if you can avoid it.
I'm homeschooling my kid and have been using sample problems from both Russian and Singapore math. But I only have a single source for Russian math problems, which is a pdf of a paper by a Russian educator who taught in the US and then wrote a paper comparing the two educational systems.
It's interesting because he talks about how algebra is a big scary subject that is introduced late in the American system, while in Russia they start doing algebraic-type problems early in primary but they are solved with pictures. So when algebraic notation is introduced later it's just a next step to the types of problems that students have already been solving.
If you look at another of my response comments to my parent comment, I mention that I don't really know if the system could be replaced with something better, but I don't think standardized education is good for anyone when compared to non-standardized education.
While true, I just don't think a one-on-one education, however skilled the tutor might be, could possibly match the social development that comes as consequence of classroom education.
I admit that their ability to not embarrass themselves among their peers is greater than those who are homeschooled or from experimental private schools, and their traditionally-schooled biting, ironic sarcasm is unmatched among the more naive non-schooled population. Traditional schoolchildren are also much better at taking instructions, falling in line, acquiescing to the wishes of their superiors, and bearing assaults on decency and freedom without complaint.
I made it through 12 years of public school and am extraordinarily "well-socialized", and I am disgusted by what it took to make me that way. It's only now that I've been out of school for about a decade that I've finally started to turn into a human being.
I think we could get into an endless debate here; I would say that the classroom model is a symptom of a post-industrial society where everything must be compartmentalized, sorted, and certified by a governing authority, which thus destroys the human connection with all of human endeavor; you, and I apologize for putting words in your mouth, would likely say that we never could have gotten to the point of a post-industrial society where you and I could debate the merits of industrialized math education on the internet without the classroom. Assuming my projection of your argument is accurate, I'm not even sure you're wrong.
It's just that whatever they're teaching today is bad, and I think it's worse than not teaching it at all. People come out of school and college technically skilled but intellectually hobbled. I don't know if the system itself can fix that, or even if it should be fixed.
As a starting reference, I mention "the Seven Lesson Schoolteacher": http://www.uvm.edu/~rgriffin/GattoDumb.pdf
As a college instructor at a community college, you're teaching math to people who likely have spent their whole lives watching other kids easily ace exams that they struggled with, who have never spent any of their genuinely free time doing math, and who just want to be let out of school. They are products of the system.
 'How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes', Barton https://www.amazon.com/gp/product/B079K3HVMJ/ref=ppx_yo_dt_b...
A teacher can provide a little guidance, but mostly its up to the student.
In my opinion learning is not an obvious thing, it is not obvious why things need to be presented as problems for example, and many students rebel against them as it leads to an empty feeling every time they even attempt to solve a textbook math problem. Curiosity is innate, but learning has to be taught.
So the clash occurs because they begin by teaching math to people that do not know how to learn, or why to learn.
One, the first thing the professor does is put things in "human terms", the issue is plain that the mode of conveyance used in the textbook is wanting. And this is coming from "Author in Action" videos provided by Pearson, ostensibly from Sullivan himself. I think a portion of this issue is derived from being "too simple to state."
Two, calculation takes precendence over everything. I've seldom been offered the opportunity to actually apply any of this myself in real world practice. What's worse is that when professors have included "working" problems they're quickly glossed over, to me it seems like this is absurd - as a non-math major this is precisely why I'm in the class, for application. This branches into arguments about the math taught being entirely abstract and stripped from a model, which makes it infinitely more difficult for someone like me. What compounds it is none of the work asks things like "why do we use exponential functions in the calculation of half-life?" instead, you just translate to TI-84 and plug your answer in. I'd quite like to see more applied concepts, and less calculation. It'd also be nice to self-direct, but I'm sure that's in reality impracticable from an assesment standpoint. I noticed repeatedly that I quickly grasped certain concepts, but was forced to repeatedly compute similar problems, time that would've been better spent working on points where I was lacking.
> Okay, I’ll bite. You wrote, “What I do not always get is some totally obvious ideas. Does this make any sense? Here is an example.” How is what you say about recurrences an example of a totally obvious idea, or of not getting a totally obvious idea? I don’t understand.
IBL reminds me a lot of breakout sessions or whatever you want to call mini group projects that take place in the span of instructional block. Some people in those groups will already know the answer, some won't care, and some will count on the person who knows the answer to carry the group because that person isn't so great at teaching it themselves. Why would they ask questions about answers they already know? When I was still in school IBL style was used far more in sciences than something like history. But I loved history. Especially lecture based history classes. Part of the reasons for that many professors I had loved to talk about the subject they taught. That doesn't mean the professors didn't love the subject they taught, but talking about it didn't seem to be something the loved doing. I personally struggled with the IBL format and did well with the lecture based format.
I'm not an educator myself. I just bring that up because at least for me I did not find the technique helpful and got less out of it than I did with lecture based classes. Everyone is different. I imagine the IBL format does appeal to some more than others.
Teach how Humans found Maths. Context is important.
Teach what Abstraction really is. Start from a Lion Picture and draw it and abstract by step by step.
Teach use cases.
Once you done that, teach Algebraic thinking. Then the student can learn anything.
I learned that my students came to me with a wide variety of interesting beliefs about math, such as:
1. Math is subjective! The teacher solves the problem by knowing the "trick" that gets them the answer they want to see. They can't explain where the trick came from, or why your answer is better than theirs.
2. They were explicitly taught "test taking skills," such as the "guess and try" method. (This method was described in a hand-out for parents, that one of my kids brought home from school).
3. They will never use their college math after college. Even students who are destined to become engineers believe this. They heard it from somewhere, not from their math teachers, meaning that culture plays a role in math education.
In addition, I perceived that the curriculum itself reinforced a problem solving method that is unrealistic: 1) Identify the "form" of the problem, which is one of the "forms" studied in the most recent chapter. 2) Extract the parameters from the problem statement. 3) Perform the algorithm associated with that form, to produce an answer.
For instance, they only learned one function with an extremum: The parabola. So in the chapter on "maxima and minima," every single problem boiled down to finding the parabolic equation representing the problem statement, and then finding the vertex of the parabola. This bears virtually no relationship to any useful or even theoretical math topic.
I decided that with as much ** as had been handed to them, the least I could do was help them get good grades that were needed to get into some of their intended majors such as Business. But not wanting to completely abandon my integrity, I adopted a compromise: Instead of test taking skills, I would teach math learning skills. Imagine you are working on a homework problem, and have not been paying attention in class at all, but you have the textbook. How do you approach the problem? It taught them how to learn from the textbook, which is at least kind of analogous to how most of us do math today -- by looking things up. And I promised them that if they added just one more ingredient -- repetition -- they would get good grades on the exams.
I avoided pulling up facts from memory. Instead I would say "I know the answer, but let's find it in the textbook."
There were dozens of sections of the course, and one great big exam, that was graded by a team of graders. So I got to see how my kids scores stacked up against the "competition," and they actually did quite well. I also had kids from other sections dropping in to my lectures. So I can't claim to have some foolproof method, but it at least got me through my teaching stint with my conscience intact.
As a pet peeve, school math isn't even real. It doesn't represent how anybody does math: Mathematicians, STEMmies, or laypeople. Nobody does anything with math, without a computer in front of them. In my perfect world, I would place a much greater emphasis on computation when teaching math.
I was particularly proud of a guided discovery task I came up with for introducing some of the more complex laws of indices to my Year 11 class two years ago. The worksheet looked like this:
Nice, eh? Again, I ask the question: what could possibly go wrong?
Well, quite a lot, as it turns out.
When considering a guided discovery task, the question I should have asked myself is: what is the best that can happen? Take the laws of indices lesson. The best that can happen is that all students discover the laws of indices for themselves, leaving no gaps in their knowledge, nor developing any misconceptions, in a reasonable time frame. We can then proceed with the rest of the lesson, maybe moving on to application questions, or interleaving other topics into the examples (see Chapter 12), such as indices involving surds or fractions.
How often does that actually happen?
In my experience, literally never.
What actually happens is that one or two students discover exactly what I wanted them to discover. They are feeling great about themselves, and rightly so – as we have seen in Chapter 2, success is motivating. A handful of students have some kind of idea what is going on, but with an eclectic mix of gaps in their knowledge and newly formed misconceptions. Some of these students are aware they have gaps and misconceptions, others are blissfully ignorant. And the rest of the students do not have a flipping clue what is going on. They are feeling confused and pretty down about themselves when they see their fellow classmates have figured it out.
Any form of decent formative assessment strategy (Chapter 11) quickly reveals this disparity between levels of understanding, and as such I cannot move on with the lesson. So what do I inevitably end up doing? Teaching the laws of indices, of course! Maybe I will set those students who seem to have understood it off on the work I hoped everyone else would be moving on to – mind you, I would really like them to hear my explanation and do the worked examples, but how can I justify doing so when they have demonstrated their understanding? Hmmm…
Anyway, back to the rest of the class. By this stage, I am 30 minutes into a 50-minute lesson, rattling through a series of worked examples on the laws of indices far quicker and with much less care than I should. There is zero time for the students to practise their newly acquired skills and hence consolidate their knowledge, nor sufficient time for me to do any kind of application questions which would show them the full breadth of the topic.
But it is even worse than that. Even if I could somehow freeze time and spend those lost 30 minutes going through carefully structured and well-chosen worked examples (Chapters 6 and 7), I am not back at square one. I am behind square one, because my students are no longer coming at the topic with fresh eyes. Many of those who failed to ‘discover’ the key relationships have already decided that indices are difficult, and yet another area of maths that they don’t understand. It’s going to take more than my magically retrieved 30 minutes to turn that one around.
And so I wave goodbye to a group of confused students trundling out of the door, promising that we will pick this up again tomorrow, assuring them it will all be fine. I am already dreading the lesson, wanting to open up proceedings by saying ‘okay, everyone, forget what happened yesterday’. In the past, I blamed my students for this – if only more of them could have figured it out. Now, I know the blame rests squarely at my feet. I never gave them a chance.
 "How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes" https://amzn.eu/3BkjAsl
1. Find a teacher that is knowledgeable and passionate about math. Grad students might fit the bill. Engineering grad students also. Low level math (e.g. calculus, differential equations and below) can be taught by anyone in STEM, it doesn't need to be a trained math teacher. Engineers, physicists, or chemists often do a better job and passionate ones might be easier to find. Passion is important, because a good teacher will be a window onto another world, that you are invited to explore. A bad teacher is a window onto a brick wall. I used to tutor undergrads in math and the worst students were the math education people. I was often told "I hate math" by these students. I would even plead with them to find another subject, that it was unfair that someone who hates math becomes a math teacher, but they insisted that they love "teaching", they just hate math. Welcome to the US Public school system.
2. Include history in the math education. This depends on the personality of the student, but for me, I loved learning about the lives of the people who made mathematical discoveries, and the circumstances of those discoveries. To this end, I recommend books by George F. Simmons, for example:
3. Look at the Russians. The Soviet Union had an amazing pedagogical program in math education, with fantastic books, puzzles, newsletters, etc. Some of that survived the turn to capitalism, and today they still punch above their weight. Some resources:
4. Interdisciplinary approach. A great way of studying math is to present some problem in physics or engineering and develop the mathematical techniques to solve it. This is how much great math was discovered.
5. Ask questions. Rather than telling students techniques and then watching them use those techniques, you can create a dialogue where you start by asking specific questions and guiding the student to discover the math on their own. This of course requires a smart student and a smart teacher, but it's a very effective way to really learn a topic. This was famously done by the Texas Topology department in the mid 20th Century.
6. Teach concepts. If you cannot get the student to discover concepts, you can still emphasize the teaching of concepts. For example, there is no reason why a 5th grade math class can't begin to cover concepts such as simply connectedness, or Euler characteristic, or hamiltonian circuits. These can be done with simple pictures, yarn, paper and scissors. Then encourage the student to generalize to other shapes. Similarly in geometry classes, working with a ruler and compass creates a more memorable hands on experience for young students and can be a technique to introduce them to early theorem proving, for example bisecting a chord.