Uncategorized – The Math Dragons

A Little Discrete Math

Recently I ran across a fun little problem about numbers at Ben Orlin’s blog https://mathwithbaddrawings.com/2019/11/18/the-anti-calculator-puzzle/. My rephrasing of it is as follows:

Consider every positive integer $n$ which can be represented in base $k$ as a $d$ -digit number. In other terms, we have

$\displaystyle n=[a_{n-1}a_{n-2}...a_{0}]_k=\sum\limits_{m=0}^{n-1} a_m k^m$ , (1)

where $1 \leq a_{n-1} < k$ and $0 \leq a_{m} < k$ for $m < n-1$ . We can then define the function f(n) over all d-digit numbers as

$\displaystyle f(n)=\frac{\sum\limits_{m=0}^{n-1} a_m k^m}{\sum\limits_{m=0}^{n-1} a_m}$ , (2)

i.e. the original number $n$ divided by the sum of its digits. The goal of the problem is to find the maxima and minima of this function over all $d$ -digit numbers.

Finding the maximum value of $f(n)$ is relatively simple; a little numerical experimentation and thinking shows that the sum of digits is always larger than or equal to the first digit, and thus heuristically we can say that the denominator can drag the whole sum down to be less than or equal to $k^{n-1}$ . More rigorously, we can rearrange the expression $k^{n-1}-f(n)$ into the form

$\displaystyle k^{n-1} - f(n)=\frac{\sum\limits_{m=0}^{n-2} a_m (k^{n-1}-k^m)}{\sum\limits_{m=0}^{n-1} a_m}$ . (3)

(Note that in the numerator of expression 3, the value of $a_{n-1}$ fell out of the sum. This is going to be a recurring theme of this discussion later on.)

Because every term $a_m (k^{n-1}-k^m)$ in expression 3 is either positive or zero for $m < n-1$ , and only zero if $a_m$ is zero, we can see that the maximum value of $f(n)$ is in fact $k^{n-1}$ , which is only realized when all digits of $n$ but the first are zero.

That was the easy part. Finding the minimum (and there is only one, as it turns out) is rather more subtle. While it is tempting to assume that the correct answer will be numbers of the form $[1ee...e]_{k}$ where $e=k-1$ , this can quickly be proven false using four-digit numbers in base ten. In base ten, $f(1009)=100.9$ , $f(1099)\approx 57.84$ , and $f(1999)\approx 71.39$ . Clearly we must look more deeply.

First, I will describe the solution: over all $d$ -digit natural numbers, function $f(n)$ as defined in expression (2) has a minimum only when $n$ 's base- $k$ representation is of the form

$\displaystyle n=[100...0ee...e]_k$ , (4)

where the number of digits that are equal to $e$ is given by the smallest integer $c$ such that

$\displaystyle c k^c > \sum\limits_{m=0}^{d-2} k^m$ . (5)

Note that for small $n$ , despite my representation in expression (4), there may actually be no or one zeros present in the true minimum. By the same token, there may be only one or two digits of $e$ . However, inspection of expression (5) shown that the number of zeros present is non-decreasing in the number of digits and eventually you will get an arbitrarily large number of zeros, albeit with a far vaster number of $e$ ‘s.

To prove (4) and (5), we will focus all of our attention on what I mentally label “the components of the discrete gradient” of $f(n)$ . That is, we consider the change induced in $f(n)$ by either increasing or decreasing a single digit. Obviously this would produce problems if we step outside of the valid range of digits (i.e. $1 \leq a_{n-1} < k$ and $0 \leq a_{m} < k$ for $m < n-1$ ), so we simply forbid taking such steps. All this said, the change in $f(n)$ given a single-digit change at position $l$ of sign $s=\pm 1$ can be calculated to be

$\displaystyle f(n+s k^l)-f(n)=s\frac{\sum\limits_{m=0, m\neq l}^{n-1} a_m (k^{l}-k^m)}{(\sum\limits_{m=0}^{n-1} a_m)(s + \sum\limits_{m=0}^{n-1} a_m)}$ . (6)

The most important thing to notice about the numerator on the right side of expression (6) is that it does not depend on the value of $a_l$ , which means that the sign of $f(n+s k^l)-f(n)$ is constant as we range $a_l$ from 0 to e. Thus either $f(n)$ is constant as we change $a_l$ (if $f(n+s k^l)-f(n)=0$ ) or the extremes values occur at the boundary points: $1$ or $e$ for the first digit or $0$ or $0$ or $e$ for the rest of the digits.

We can go a step further and notice that in the special case $l=n-1$ and $s=1$ , the right side of (5) must be negative or zero, because then all $k^m < k^{n-1}$ . From expression 3, we know that the true minimum must have at least one non-zero digit after the first (because all $n$ which end in anything besides n-1 zeros produce a $f(n)$ that is less than $k^{n-1}$ ). Thus the first digit of the true minimum must be one, because if it were greater than one, we could simply decrease the first digit and thus decrease $f(n)$ . Put another way, $f(n)$ is slanted towards smaller $a_{n-1}$ , and so all minima must slide there.

I then claim that

$\sum\limits_{m=0, m\neq l}^{n-1} a_m (k^{l}-k^m) \neq 0$ for any $l < n-1$ . (7)

I prove this below, but first observe that expression (7) combined with expression (6) gets us to the form described in equation (4), although not yet expression (5). As noted above, if $f(n+s k^l)-f(n)$ is not 0 for a particular digit (given the rest of the number), then $f(n)$ will change monotonically in that digit, thus producing extrema only at the boundaries. Thus besides the first digit, any minimum can only contain the digits $0$ or $e$ . We can get further by noting that exchanging digits with different values changes the numerator of $f(n)$ in expression 2 but does not change the denominator (the sum of digits) at all. Thus for numbers represented by the same sets of digits, the smaller ones will have the smaller values for $f(n)$ . Thus we need consider only numbers of the form of expression (4) as candidates for the true minimum. Finding the correct number of zeros will be the bulk of the solution once I have proved (7).

Expression (7) can be proved true by contradiction. If we assume there is a zero solution to (7), we can rearrange it to get the expression

$\sum\limits_{m=0}^{n-1} a_m k^m = k^{l}\sum\limits_{m=0}^{n-1} a_m$ for any $l < n-1$ . (8)

But the right-hand side of (8) is an integer which, in base $k$ ends in $l$ zeros. This proves that for any solution to formula 8, $a_m = 0$ for $m < l$ . But this in turn implies that we can subtract coefficients of $a_m$ on the right size of (8) from the coefficients of $a_m$ on the left, resulting in

$\sum\limits_{m=l+1}^{n-1} a_m (k^m-k^l)=0$ for any $l < n-1$ . (9)

But the left-hand side of formula 9 is positive (because $a_{n-1}>0$ and $k^m > k^l$ for every term). Thus there is no valid solution for equation (8) and equation (7) is proved.

The number of zeros and $e$ ‘s in the true global minimum can again fortunately be found from a detailed inspection of the consequences of expression (6). In particular, we consider all $d$ -digit numbers of the form shown in expression (4). In particular we will look at how $f(n)$ changes as you change $n$ from $[100...00ee...e]_k$ (with $r$ $e$ ‘s) into $[100...0eee...e]_k$ (with $r+1$ $e$ ‘s). We will find that for this case, the numerator of expression (6), whose sign is the sign of the change in $f(n)$ for this transition, starts out as negative when all the digits are zero, but increases monotonically as the number of $e$ ‘s are increased. The digit for which the numerator of expression (6) becomes positive is the last digit for which you can add another $e$ and expect $f(n)$ to decrease, and is thus the number of digits of $e$ which produces a minimum in $f(n)$ , now known to be unique.

To calculate the sign of the change in $f(n)$ when transitioning between having $r$ digits set to $e$ and having $r+1$ digits set to $e$ , we extract the numerator from expression (6) and rewrite it in our case as

$\displaystyle \sum\limits_{m=0, m\neq r}^{n-1} a_m (k^{r}-k^m)=-(k^{n-1}-k^r) + r(k-1)k^r - (k-1)\sum\limits_{m=0}^{r-1} k^m$ . (10)

But $(k-1)\sum_{m=0}^{r-1}$ can be expanded, and most of the terms cancelled, to get $k^r-1$, and so we can transform expression (10) into

$\displaystyle \sum\limits_{m=0, m\neq r}^{n-1} a_m (k^{r}-k^m)=-(k^{n-1}-1) + r(k-1)k^r$ . (11)

From this, we can see that the right side of expression (11) starts at negative numbers for $r=0$ and increases monotonically in $r$ . If we then use polynomial factorization to factorize $(k-1)$ out of both terms on the right side of expression (11), we then see that $f(n)$ does indeed start increasing and continues increasing once expression (5) is satisfied. q.e.d.

Why Math? Stay on top of today’s world

June 13, 2017June 13, 2017eldermathdragonLeave a comment

This post is part 3 of a series on reasons people give for why we need to improve math education. The first part of the series is here, where I beg you all to think about why we do what we do. The rest of the parts are:

Learning math gets you a better job
We need more people trained in science, technology, engineering and mathematics (STEM) in order to compete in today’s world
People need better math skills just to stay afloat in today’s world
Learning math teaches you how to think
Math is an art like music or painting; it makes us human
Math is the ultimate form of Beauty and Truth

We are floating in a sea of numbers.

Not literally. But when’s the last time you saw a number? Well, according to the digital clock on my computer, it is currently 9:59 on 6/13/2017. While I was coming home today, the speed limit was 70 mph. At work, I work on the first floor of building #6, while the rest of the company works on the fourth and sixth floors of building #14.

It gets better. Go read the news. I go to my news site and see that one particular candidate for a public office just won an election 55.9% to 44.1%, with 99% of the vote counting locations reporting. The article includes a handy map which allows you to see how the election results turned out in each county. Lots more numbers, and these ones definitely changed the world.

Oh, but there is so much more. On a pretty good hunch, I went to the health section of the news. I was not surprised to almost instantly discover an article which claimed a relationship between a mother’s health and the health of the child (no, I’m not giving details; they’re off topic), citing a study saying that if a mother had/did (something) during pregnancy, then the child would have a 44% higher chance of having (an unusual outcome). I have no idea if the study is correct or not, but there they are: more numbers! And important ones, at that. If people believe those numbers, it will affect an awful lot of children going forward.

And on and on and on.

There are numbers at the office and numbers at the store,

numbers in the news and oh! so many more!

There are labels on my cereal and bills in the mail,

warning me that to pay, I simply must not fail.

30 percent extra! 50 percent off!

How much really is it, that extra stroganoff?

Taxes and studies and numbers obscene,

I can’t help but wonder, what’s it all mean?

Nowadays, we all really need numerical fluency. Number still aren’t as common as words, but they sure do pack a punch.

Why math? We need more STEM students to compete in today’s world

August 16, 2016June 13, 2017eldermathdragonLeave a comment

This post is the second part of a continuing series about why we put our poor kids through all this crazy math education to begin with. The other sections are:

Learning math gets you a better job
We need more people trained in science, technology, engineering and mathematics (STEM) in order to compete in today’s world
People need better math skills just to stay afloat in today’s world
Learning math teaches you how to think
Math is an art like music or painting; it makes us human
Math is the ultimate form of Beauty and Truth

We need more STEM students to compete in today’s world

This one is true, without question. More and more, the jobs worth doing, and the ones that actually change the world, are in STEM fields.

I’m not going to talk for hours about the big picture and whose competing with whom, and why. That’s all besides the point. Here, we talk about individuals, families and teachers and how these ideas affect them.

So, given that its a good thing to have more folks in STEM fields, what does this mean for individuals?

Probably the most important thing I can say about this is just how important it is to spread good education around. Not everybody is going to be a scientist or engineer. The world would be a sadder place if we didn’t have any music or art. The important point is that we don’t know where we are going to find the folks who are able to be the next generation of scientists, engineers, programmers or mathematicians.

The more people who have a chance to learn, and the more who have a chance to learn well, the more people we will find who are willing, able, and enjoy working on these important fields.

The next most important thing? The math is taught properly. If math is taught properly, in context, then far more children will discover they are good at it than every have before. With a better understanding of math, what it is and what it is good for, comes a far better chance of actually choosing a STEM field when they get older.

How not to test math

July 30, 2016July 30, 2016eldermathdragonLeave a comment

This post is by James, but Jason and I spent a lot of time talking about it. It’s important enough we needed lots of eyes.

I’m guessing that more than a few of the people who come to our site (or will come to our site, at any rate) think that we’re some kind of…what, institutional shills? Voices for Big Education, trying to force unneeded change down people’s throats? Heaven knows they’re out there.

Maybe I just want to think that’s how things are. It would actually be a compliment, in a strange kind of way. Proving we aren’t is partly why we mostly spend time on elementary examples and practical problems—it’s hard to claim propaganda when most readers can test the answers and ideas for themselves. We think that’s the best way to show that math education really can be improved.

On the other hand, the people who criticize changes in math education sometimes have a point. Sometimes, the changing standards really are used as an excuse for selling new textbooks, rather than making the textbooks better. And making tests, and testing systems, can be a great way to make money whenever state education rules change. They are a less effective way to make money (right now) if you actually have to improve them first.

And the only way to fight this kind of cheating, without giving up entirely, is to hold the folks who make the tests accountable. That’s today’s story. Before we go on, though, I’d like to say that while I am showing an example of a bad test, I also saw examples of good tests made by the same folks. There was a 4th-6th grade science test seemed particularly good, and the high-school level math test was also better than average. But somebody really messed up on the fifth-grade math test, and that’s bad timing if you want the kids to get to the good tests.

In this post, we are digging into specifics. That’s why I am writing specifically about the Utah state SAGE testing for math, grade 5, based on the sample problems taken from the official SAGE test portal. The science test I liked, as well as the high-school one that was decent, are also available there for free. I actually took this test myself, and then reviewed my answers with my brother, before writing this post.

The problems

The test starts with this:
Utah Sage 1

That’s an OK start, I guess. I mean, it’s just a calculation problem, but it is the kind of problem that kids should be able to solve after grade 5. And we don’t want to start out too hard, do we?

SAGE problem 2

Problem 2A is also OK, but not great. Once again, there is no real reason why we are adding six and eight, and then multiplying by three, but kids should know how to translate the words into the symbols, like the question asks. The user interface isn’t great, but it isn’t terrible in this particular problem.

Problem 2B, on the other hand, is, well, bad. What you are supposed to do is to rewrite (6+8)\times 3$ as $(6\times 3)+(8\times 3)$ , a perfectly good thing to do. The problem is that the instructions are not clear. It confused me (briefly), and I study advanced mathematics in may spare time! It will definitely confuse students, not because they don’t understand math, but because there are lots of ways to create an expression in part B which equals 52. It’s only when they notice that they only have the numbers 3, 6 and 8 available that they might realize that they have to expand the sum. This problem would have been far improved by giving better directions. Something like “Expand the product in Part A to get an equivalent sum in Part B,” would have been much nicer.

SAGE problem 3

Y’know what’s funny about this problem? I had completely forgotten what a rhombus was. (I think I mixed it up with a trapezoid, which also has four sides, but is defined very differently.) All these years using all kinds of fancy math, and I got tripped up, here, on the meaning of basic words. Luckily for students, the test includes a dictionary, so this is just me being kind of rusty.

That said, this is testing important skills in classification, geometry and logic. The truly important ideas—relationships between different type of objects—actually is pretty important in real math. So this is a decent question.

SAGE problem 4

Finally, a word problem! It’s a pretty simple word problem, but at least its something. It’s also a multi-step problem, which is also an improvement. (You have to figure out how much money he actually spent, and then figure out how much more his estimate was than the actual amount he spent. You then have to turn it into an equation.)

Unfortunately, there’s a bit of a problem here, and it’s a pretty serious one.

Why are we turning this relationship into an equation?

I mean, seriously, why? It makes sense to say that the manager in the problem estimated way too high—ten times too high, in fact. (Or he overestimated the price by $4,500.) He obviously made an mistake in the multiplication. But—the equation doesn’t tell us anything about the situation that words wouldn’t, and just as well. The point of math is to communicate. This question is actually asking the student to make the result harder to understand for no reason at all. It looks like a word problem, but when you dig down to the real problem, the most important part is just math for math’s sake. A potentially useful skill, but it could be taught and tested in a much less—random—way.

If you want me to tell you when equations actually are useful, I will gladly tell you. It’s just, this isn’t one of them. The math they are testing is a poor fit for the simple word problem they created.

SAGE problem 5

This question is pure mechanics. Anyone who wants to roll back math education standards should love it. I’m going to get more into this type of problem later, but for now I’ll say we probably want a few on the test, but it’s just OK, not good or great.

SAGE problem 6

This is basically a good math problem, although a vague one. I’m actually glad that there is no single correct answer to this problem. (That’s true to how math works in the real world.) I just wish the question made it clearer to the student that they are just supposed to invent any two fractions which work. The problem should say that there is no one right answer, just right answers.

My far bigger complaint is about the way the problem is displayed. It wasn’t clear to me, on seeing the problem, that I was supposed to put one fraction in each box. When I first answered it, I put an addition problem in each box (providing two different answers to the question.) I suspect I would have gotten no credit for this problem, although I technically got it correct twice. This entire problem could have been solved by labeling the boxes more clearly, perhaps saying “Fraction 1” and “Fraction 2” to the left of the fraction boxes.

In fact, why bother with all the buttons and numbers on the bottom? Why force the students to use the fancy interface, when all they are really doing is typing in four numbers? Why not just have something like:

Find two fractions with different denominators which sum to $\frac{7}{12}$

$\frac{\Box}{\Box}+\frac{\Box}{\Box}=\frac{7}{12}$ ?

This is simpler and clearer and has a more natural way to enter the answers.

SAGE problem 7

Reading geometric shapes is a useful skill in science/engineering/biology, etc. (Think of reading the plans for building a house. Or digging for oil. Or building a space probe.) So is being able to parse the logic in the true or false questions they ask the students. So on the whole, this question exercises good math skills that the students need to have.

On the other hand, we’re seven problems in and we still haven’t gotten a decent word problem in any of these questions. This kind of geometry is perfect for word problems—the world is full of these kinds of strange shapes, and it’s amazing when knowing things about them would be useful. It’s an OK problem, but it could have been so much more. More on this at the end. SAGE problem 8.PNG

This problem is bad. I do mean, very bad. It’s easily the worst sample problem on this whole test and shows off the worst of what has gone wrong with it. It’s not me, or even mostly me. When I showed Jason the first draft of my discussion of this problem, he thought I was being too nice. And he was right!

So what’s wrong with it? How may I count the ways?

The “word problem” it’s attached to is probably the laziest excuse I have ever seen for a word problem. You could replace the first sentence with “Draw two rectangles,” and absolutely nothing else would change. There is no translating the word problem into math, or back again, the most important part of word problems. And the word problem as stated makes no sense. Who keeps a $4\times 1/2$ inch picture of anything anywhere? It’s too narrow! And why are we drawing it on a graph, anyway? It’s a transparent attempt to say “Yeah, we got a word problem,” without doing any of the actual work to have a word problem.
The way the user uses the machine to enter the answers is not how people normally use computers nowadays. It violates all the rules a student would have learned about doing graphics on a computer if they had used any other editing program anywhere. Jason says that his students found it very difficult to use. I certainly did.
I have no idea how the computer grades this problem. As a professional programmer, my head hurts just thinking about all the correct answers a computer would mark wrong on this problem, because computers just aren’t that smart.
What, exactly, is this problem supposed to be testing? The ability to draw rectangles? According to Jason, 5th grade is a bit late for that. It doesn’t match any of the known Utah Core standards. It doesn’t help students understand what a grid is supposed to stand for. What is the point, exactly?

Overall, a failure at about every level.

SAGE problem 9

This is again an OK problem on its own, and actually does stick to the standard. I’ll discuss my main concern later as part of the wrap-up. Jason’s main concerns was two-fold: first, the problem uses only sevens, which can confuse the poor kids. This would have been a better problem with different numbers. Secondly, it’s not free-form enough for his taste. He thinks the problem would be better being split into two parts. In the first part, the kid tries to write down the expression given by the words, using their own symbols. Once they’ve done that, they can say which of the given expressions match their expression. In his experience, kids do a lot better job of learning the notation if they do it in that order.

Conclusions

Well, I was going to go on, but there’s no point. These work fine as a decent sample.

Many of you are probably wondering what was so bad about this test. After all, I said that many of the problems were OK. And that’s just the problem:

This test is mediocre at best. All together, it is actually pretty bad.

Some of the problems are OK on their own. All of them (except problem 8) could have a place inside a good test. But taken together, they are just traditional math problems! Pointless, mechanical math problems. You need to test mechanics on a math test, but word problems do a perfectly good job of testing mechanics.

And there is the problem right there. There are no good word problems on this test. None! None of these problems have an interesting motivation or story-line. For instance, problem 1 could be replaced with the question:

“If a rectanglular kitchen has one side that is 68ft long and the other is 90ft long, how many 1ft by 1ft tiles will you need to cover the whole floor?”

The math problem the student has to solve exactly the same as in the original version. Translating the problem into math, and then back out, may be harder than the original problem, but it is necessary for students to have this skill anyway. The entire test could have been brought up a notch just by choosing better word problems. Not for every problem should be a word problem (mechanics really is important) but it should have happened much more often than it did.

And that’s not even as good as it could get. It could get much better. Go check out the fifth grade science SAGE practice test—that test is a whole lot better. On that test, you get to do virtual experiments with rocks, comparing their hardness, how they react with acid, and similar things. You use the result of one problem to feed into the next problem. The whole thing just feels more-real.

You can do exactly the same thing with math. With math, you can set up interesting scenarios which need math to solve. You can have one problem feed into the next. You can ask the students what the answers to specific problems mean. That’s how science and engineering and business actually use math in the real world! And this test did nothing to live up to this kind of standard. Argh! Frustrating! The high school level SAGE test was a lot better-it used more word problems for one-but even it could be seriously improved by being treated more like science—more like math is actually used.

And remember: math was made to be used.

Why math? Getting a better job

July 5, 2016June 13, 2017eldermathdragonLeave a comment

This is part of a series on “Why math?”

Why math?
- Learning math gets you a better job
- We need more people trained in science, technology, engineering and mathematics (STEM) in order to compete in today’s world
- People need better math skills just to stay afloat in today’s world
- Learning math teaches you how to think
- Math is an art like music or painting; it makes us human
- Math is the ultimate form of Beauty and Truth

Getting a better job

First of all, let’s get a few things out of the way:

Can learning math help your kids get a better job? Yes!
Can learning math make your kids be better at their jobs? Also yes!
Can learning math help you even if your job doesn’t need it? Yes, but it’s complicated. That’s for other answers in this series of blog posts.And most importantly:
Is getting a better job important enough to justify what we put our kids through to get it? My answer to that is a solid “Maybe”. The details are in the rest of this post.

But we’d better get a few things clear before we talk more about this. Probably the most important thing we have to understand about all this is that we are not talking about teaching kids to be computers. Computers are everywhere, and are incredibly cheap nowadays. I, personally, carry around an absurdly powerful computer everywhere I go. (It can also be used to make phone calls!) Businesses have more computers than they know what to do with. Nobody gets paid to be a computer any more.

No, the math we need to teach our kids is the kind that lets them use computers, and use them well. Doing that requires really understanding the math the computer is doing. If you tell the computer to do the wrong thing, it will happily do the wrong thing a million times in one second! Brilliant, you just got the right answer to the wrong question and can now happily do the wrong thing for the next six months—this doesn’t go over well anywhere I would want to work.

So with all of that said, lets get down to the really important question: Is getting a better job a good enough reason to make our math education worth it?

I’d say that on the average, yes, if we improve how math is taught. If we can manage to not drive students away by giving them only the boring bits, then there are going to be many more students who will discover that, hey, this math thing is kind of cool! Those students can get a better, more useful job, and actually like it while they do it. Even the ones who don’t actually like it are more likely to understand it, and that alone will help them in whatever other job they find they enjoy.

Even with all of that, though, it still seems like we need to consider what to do with those kids who, for whatever reason, really won’t get a better job because of the way we teach math. Even there, I believe that some of the other reasons I listed will be good enough reasons to teach our children math. Stay tuned.

Why math? Why do we do this? Oh, why?

June 22, 2016June 13, 2017eldermathdragonLeave a comment

“Why do I have to learn this? When will I ever use this?” — practically everybody at some point

This is a really good question. In fact, I’d have to say that it’s probably one of the most important questions anybody could ask. It seems to me, in fact, that it’s hard to justify our current system of forcing kids, with the full weight of the law behind us, to attend classes, if we can’t even say why we are doing it.

And don’t even get me started on testing. If we are going to do as much testing as we currently are, we had really be certain we are both teaching and testing the right things. Tests are hard on students, and they take them away from classes where they might be learning other things. They are also hard to make fair. I’ve always had an easy time with tests, and it feels incredibly unfair that some of my brothers and sisters have a harder time with tests and get worse grades and fewer opportunities even though they also know the material well enough for the real world.

Eesh. No wonder the question of how we teach math can make people so angry. And because we can’t all teach our children ourselves, we have to turn it over to the school. And because of that, we care deeply about how well the school’s live up to the job. On the other hand, it feels like other people are always the ones creating the lessons and the ideas, and we don’t have any control. This is especially true in math, where the way ideas are taught have changed so much over the past fifty years or so.

So what is driving the importance of math education? Why are we fighting so hard over what is going to be taught? What is the point of all of this? Until we can answer this question, it’s going to be very hard to get everybody working together on this important problem.

There are lots of answers to this question. Sadly, not everybody really agrees. It does’t take long to discover that when you go looking. Here are at least some of the ideas I’ve seen:

Learning math gets you a better job
We need more people trained in science, technology, engineering and mathematics (STEM) in order to compete in today’s world
People need better math skills just to stay afloat in today’s world
Learning math teaches you how to think
Math is an art like music or painting; it makes us human
Math is the ultimate form of Beauty and Truth

Each of these ideas is a very big idea on their own! They are far too big for this single blog post, in fact. So for the next few posts, I’m going to be covering them each of them more closely, trying to discover why people think that way and how well this matches with what happens in the real world. If I have any ideas that I’ve missed or comments on this, please let me know!

My Journey Into Math

June 17, 2016July 15, 2016senovfacLeave a comment

I’ll just come out and say it I guess; I’m a math lover. I can’t say that it has always been this way. If you were to talk to me when I was in junior high or high school I would probably have told you that I wasn’t good at it, and that it was hard. I would have also told you that it was useful (with my dad being a physicist it’s hard to not know it has some good uses). Yet, you would never hear me say that I loved it.

When I looked at what I was learning in class and seeing my dad take the math and use it in ways that I didn’t understand, as he would talk about how it was useful, I would experience a disconnect. If it is so useful, why can’t I find a way to make it useful? Why do some people just “get it” and others not? Why does it seem like such a drudge to me when I know that it can solve amazingly difficult problems?

I gave up on my dream of becoming an engineer my first year of college when I took a college algebra class and basically bombed the final. (I got an A in that class, somehow, which means everyone probably did poorly. Not a great indication of how well it was understood, and also taught, by people who obviously needed a little help.)

When I chose to start my major with education I was not too worried about taking my math classes, after all, I thought that I knew all of the necessary steps, and it would just be a brush up of my knowledge. The day I walked into the classroom for Basic Concepts of Math, I had no idea that my perspective of math was soon to change. The change started small. At first I was amused by the tasks given to prove why basic math concepts worked. Then, when it seemed like we were just about to enter into a rut, my instructor gave us a task that changed my life.

I am awed somewhat by the subtlety that my instructor used to build me up to the point so that, when that lesson came, I would be able to change my perspective. She had built up a classroom culture and structure in such a way that when the lesson happened, I was able to take the most of it and run with it.

The first task given was to write a formula to find the area of a square, then a rectangle, and finally, a simple parallelogram. The instructor told us that we couldn’t look up any of the formulas and that if we remembered it, we would have to justify it. For the square and rectangle it took only a few minutes to write the formula and justify it – way too easy. However, when it came to the parallelogram I was stumped. So I drew a picture, and just stared at it. After all, pictures are for dumb people, or people who can’t follow the formulas or steps well enough and therefore need remediation. Yet, there I was trying to figure it out, something so elementary that I shouldn’t have been struggling with.

It was a common practice of my instructor to let us talk and share our ideas with a neighbor as a way of processing our thoughts. Although I didn’t share, I did listen to a couple of my classmates. They were also working on it. I heard one of them say that the formula was the same as that of a rectangle, just length of the bottom multiplied by the height. So, I drew that on my picture. In that one spectacular moment as I looked at the picture, my mind stretched in a way that I hadn’t done for years, and in a way that I had rarely connected to math. I had a creative moment where I could see in my model “cutting” off the ends of the parallelogram and moving one side to the other to make a rectangle, where the formula matched what my classmate had said. I had realized that I could cut apart a shape and move it around in anyway I found useful!

At this moment I excitedly started sharing my picture with my classmates and seeking to justify my reasoning and seeking to actually answer their questions. My mind was flush with the joy and fun that I had just figured out. When we moved on to looking at the area of a trapezoid I was more than ready for the challenge and excited to get started. I was also surprised by the different ways that my classmates had solved it. It didn’t take long before I was recording what they were doing and drawing pictures to model and express what they had done, to explain it to a confused classmate or just to make sure to have it clear myself!

How I wish that I could take you through that wonderful process. It wasn’t always that way. I still had times where it was less than exciting. There were times where I would struggle harder than I would like to admit for concepts about basic math. More importantly, however, I found myself enjoying the challenge more, getting into the reasoning and process, and enjoying listening to my peers as we discussed the ideas.

In essence, I was surprised by how much about basic math I didn’t know how to explain (but could “use”) and how much more the math that I learned before made more sense. I learned that math could be a way to creatively express ideas and logic. I learned that it was more than just simply the steps behind the computation. I learned that math was actually creative, logical, applied, relevant, complicated yet simple, flexible and had some give in how it could be thought of, and a way of easily expressing my ideas.

The interesting part is that as I continue to deepen my understanding of math I find that I crave that deeper knowledge; and just being given a basic “here’s how” as being rather unfulfilling. Now, I must admit that my understanding of math still has many holes, especially with some algebra. But, I’ve gotten really good at the basics and I’m growing in my understanding of other math ideas.

Without actually having an opportunity to go deeper, I would never have learned to love math. I found that for me to really remember the “steps” of math, I actually needed the why also. I have found that I don’t forget the “steps” anymore, because I have a deeper understanding than I’ve ever had before. I can also generalize ideas about mathematical principles, without doing computation, and make judgments based on those. This has helped me be more confident and fluent when I work with any mathematical idea I come across.

This is why I am for reforming how we teach math. I have been on the side where it failed, yet I have also started down on the side where it succeeded. The side that failed was traditional math teaching. The side that helped me succeed is the one that is “confusing,” “illogical,” and “teaching to the lowest common denominator.” (Which, in many times in math, a lowest common denominator actually makes things clear and can be useful.)

I’ve had people tell me that we shouldn’t try to go deeper first because the kids aren’t ready for it, much like what happened to me, go deeper later. I can’t agree with that. I’ve seen kids go deep and make amazing, real growth that they are completely ready for. I’ve also seen some kids who are on the edge do amazingly well, until someone tells them that isn’t how you do math because getting the right answer quickly is the only important part. It isn’t by the way, but I’ve seen this happen.

So, the next time you hear someone complain about the way math is being taught, remember my experience. Think that there may be may be more to the story than what the infuriated person is saying.

If nothing else, remember that we want more than simple, easy “solutions.” We want what works best because it really works, even if that means a little more struggle and challenge.

Norbert and Errol Answer: Is Addition Really Simple?

May 27, 2016senovfacLeave a comment

Norbert: Yes, it is actually pretty simple. Though, there is some complexity that we don’t always think about.

Errol: Kids have to face these complexities and make sense of them. So, if you are dying to refresh your addition, please watch our newest video in the tutorial section. Enjoy!

The dreaded question: “Will this be on the test?”

May 19, 2016May 19, 2016eldermathdragonLeave a comment

“Will this be on the test?”

I doubt there’s a teacher alive who hasn’t heard this one. I doubt there’s a student alive who hasn’t asked it themselves or been very interested in the answer.

Teachers really don’t know how to answer this question. If you want to see some of the more creative answers I’ve seen, you can always look at this post at math with bad drawings. In fact, you should read it no matter what, because it’s great. It also made me ask a question.

Why don’t teachers like this question?

Because they really, really don’t like it. I mean, it drives teacher’s bonkers to have to answer this question. The answers in the post I linked to (written by an actual teacher) should make that clear. If you really don’t believe it, just ask a few teachers. You’ll see.

Before we continue, let’s get one thing out of the way: It’s obvious why students ask the question. It makes sense to us. After all, if it’s not on the test, that means that forgetting it won’t get us in any trouble. And studying is hard and takes time which we could spend doing things which we care about more. It’s just figuring out how to best use our time, under the assumption that our goal is to pass tests.

So why don’t teacher’s like it? I suspect that there is no one single reason, but easily the biggest one is that most teachers don’t want to teach their students how to pass tests. Oh, they want the students to be able to pass tests—they are trying to teach them after all. But the thing that teachers really dream about is that their students will someday, as adults, be using what they learned to make the world a better place. They want the students to be expanding our knowledge of the universe, curing cancer, solving social problems, or protecting the innocent.

Being able to take a test has very little to do with most of these. The real question isn’t whether they can take a test; it’s whether they’ve learned. Teacher’s try, and how do they try, to make tests which measure how much a student has learned. When all is said and done, though, the world’s best tests really don’t reach that goal. There is practically no problem in the real world which is solved by sitting in a room answering questions that somebody else already knows how to do. And it’s even rarer to have a time limit defined in hours rather than days. Or weeks.

And this is the heart of the problem. The students are trying to solve one problem—what do I need to do in order to get out of this class with good grades? What do I need to do in order for my parents to not get mad at me? How to I get this requirement taken care of so that I can go work on things I actually care about?

And no wonder teachers can’t get behind that. They love what they do, and they love the knowledge they are trying to give, hopefully to help a student for a lifetime. A student asking this question means that they aren’t looking past the end of the semester, and probably will neglect and forget all that information the teacher just taught them. To the teacher, this means that the teacher just wasted their time teaching, and the student wasted their time sitting in a classroom. And people hate being told that all their hard work is being wasted.

I don’t know what the right answer to this problem is. After all, it’s important for both students and teachers to know how well both are actually doing, so we need to test somehow. This all leads in to the eternal debate about the right way to test, which is waaaaay bigger than this post.

My first thought is, can we teach the students to look at the problem this way? I’m not sure how. After all, “looking at problems this way” probably isn’t going to be on the test…

Seeing part of the mind of God

May 4, 2016eldermathdragonLeave a comment

I’ve been busy recently working on an introduction for some videos we are putting together for the website. We hope to re-use it a lot when we start getting really in-depth into some of our tutorials, and so we are spending some real time making it look as good as it needs to be. We are also going to use these tutorials to introduce you to the actual math dragons. We’ve named them Norbert and Errol. The names seemed—appropriate.

All of this has made me think more, though, about something that’s very important to me, personally.

Why am I doing this? Why? It takes hours to do this kind of thing, and it can be very slow at times. I’m having to learn new things and practice with new tools all the time. I’m not an animator.

Why am I dropping all this time into trying to explain to people what math is? Why does it matter?

Well, part of it really is all the reasons we give on the introduction page. Both me and Jason really believe that math education is important for our children’s future. We both feel like math education can be improved, and that we have the skills to help people understand why. There’s so much that can be improved, and the world has finally gotten to a place where we can spread the message far and wide.

But that’s only part of it. There’s another part. There are ways I’ve always envied Jason. He likes teaching. It’s a big part of who he is, and he does a good job of it. We enjoy talking about ways he can help his class through some of this stuff, which can be really difficult. I enjoy the challenge, even while knowing he’s better at it than I am.

I’ve have spent some time teaching. I’ve tutored on and off throughout my life, never for money. I’ve had mixed success. With my experience, I can at least consistently get the students to the right answer, but sometimes the students got rather fed up because I kept asking the very simple “Why?” questions which they weren’t really used to. They just wanted to get the right answers so they could play basketball. And who can blame them?

In all of this, though, there’s been something very special that has always been treasured. It’s that moment when a child, or an adult, or even myself, finally understands. There’s the point where people get it. Something that was difficult before is—well, still difficult, because math isn’t easy. But it makes sense. They now know how to use it. This is a beautiful moment, like the moment you make it to the end of a hike and see the sun rising over the valley. Or the moment when you realize that this is finally it, I am really, truly going to graduate. (I’d add a marriage comparison here, but I’m still single.)

I believe so much that more people can get this kind of moment in mathematics that do right now. I believe that this clarity is a moment worth striving for in its own right. In our own little way, we see a very small part of the mind of God. And the best part is, it’s still useful for all those other things we keep talking about. In fact, it turns the STEM fields— science, technology, engineering and math—from a high-paying chore to a challenging puzzle. It’s still not easy, but it is worth it.

And I want so much to give more children and adults the chance to experience that for themselves. I see so many people who could get it, but have decided they just don’t know how. Combine this with the practical downsides of innumeracy, and you have a cause I can fully get behind.

Names and programs and politicians and school boards may come and go, but this part of math, I will push for regardless.