# #WeHateMath Storytelling in Math Education

Recently my math class was studying basic probability and I had one student who was really having a hard time wrapping her head around the topic. One evening I got an email from them, stating that they didn’t understand one of the practice problems:

Given that P(E) = 1/4 find the odds in favor of E.

I replied:

P(E) means the probability of E happening. 1/4 means that there are four possible outcomes and that E (the one we want) is only one of them.

We figure our odds for something by comparing the number of possible things that we want (in this case just one) versus the number of possible things we don’t (in this case the other three).

So the odds are 1 to 3 that our event E will happen because out of four total, only one of them is E.

I hope this helps.

They responded that it didn’t help and that they had read the chapter multiple times and just weren’t getting it.

I gave this some more thought and decided on a different tactic:

Let me give it one more shot:

We’re having a dog race — four beagles: Evelyn, Sam, Henry and Charlie.

All four dogs are evenly matched – same age, physical condition, everything – so each of them is equally likely to win.

However, Evelyn is your favorite and you’d like to know the probability that she’ll win the race.  There is a total of four dogs so she has one chance in four to win.  In math-speak, this is known as :

P(E) = 1/4

What about the odds? Well, It’s Evelyn against the other three dogs so we say that the odds for Evelyn winning are 1 to 3 or 1:3

Once I had expressed it in the form of a story, it clicked. They replied that they now understood what the problem was about and had a better sense of the material.

In my experience, the mere mention of ‘story problems’ (sometimes known as ‘word problems’) is enough to send my students into fits. (Google ‘story problems suck’. This is not an isolated issue.)

But stories are the primary way we humans communicate with each other. I use stories in my classes all the time. Sometimes they’re personal anecdotes (usually of the ‘See what I did there? Don’t do that’ variety), historical references or just something I made up like the example above.

I think that the issue with traditional story problems is that they’re not very interactive. Most seem to be little more than a written statement of the math problem we’re trying to solve.

So maybe we need better stories.

One promising project is Oppia. This is a Google project described as a “Tool for creating interactive educational content”. In other words, you can create a story that guides your student through a topic, periodically asking them to apply what they’ve learned to solve a problem within the context of the story.

You can set up your own copy of Oppia on your own computer to test it, contribute to or browse through the lessons at the official hosted server or just use their test server to see how it works. I played with one of the tutorials and it was very engaging. So head on over there and give it a try, let me know what you think.

# #WeHateMath A Better Way to Teach Math?

Ever since I started teaching math, I’ve been trying to figure out why so many students (and non-students) seem to have problems with it and what I could do about that. With that in mind, I’ve been trying different techniques to see which ones are effective.

The first one was what I call the “Age of Aquarius” technique. It usually has me saying things like “Math is the secret language of the Universe” and “We’re all made of math” in an attempt to engage student imaginations. Unfortunately these work best with an audience that is already receptive and math classes are generally full of people who want to be anywhere else. As a result, saying these things makes them think that either you are high or you’re trying to make your job sound more interesting than it is.

I call the second method “Eat Your Vegetables”. This consists of explaining how learning math is good for you, usually by citing the Stanford Medical School study that showed improved brain function from the very act of learning math. In other words, even if you never use this stuff in your real life, it’s the mental equivalent of CrossFit. This also fell flat, being a bit too abstract for a group that just wanted to get through the class with a minimum of effort.

I decided to tackle this from the other end. I wanted to find the source of this distaste for math. I’ve always felt that hate is a fear-based emotion so if I can lower the general anxiety level in the room, I should get better responses.

One technique I used was to use tools to handle the mechanics of problem-solving. For example, whenever convenient I encouraged the use of calculators once the math portion of the problem had been processed. (If you understand the math well enough to explain it to a machine, this is a good thing.) For statistics, I showed how to use spreadsheets to quickly and easily observe the effects of different sample data on results as well as how to create different types of charts to visualize the numbers. For probability, I brought in decks of cards. I used a drawing program to sketch out problems and scenarios.

I also tried to minimize the use of jargon during class. While it’s useful to have a common vocabulary, I wanted students to pay more attention to the relationships and patterns than to worry about coming up with the correct terminology. We introduced terms as needed but it was okay if they had to use a few more words to explain what they were talking about or if they had to draw a picture or diagram.

This was the first term I used these techniques so it’s early days. But my last class session was today and overall I feel quite positive. Several students have remarked that this was the best math class they’ve ever had and my retention rate was pretty good. For math classes, it’s not unusual for half or more of registered students to withdraw before end of term. Out of an initial class of twenty one, I lost just seven and of those, four withdrew before the start of the term.

All in all, a good start. I’m teaching two sections of the same class next term so I’ll continue to refine my techniques and report my data here.

# #WeHateMath One Man’s Meat is Another Man’s Poisson

Recently I got to thinking about my grad school course on system performance. The instructor had a day job at Bell Labs where he did the math that he tried to teach to us. It was all about things like queuing theory, think time and all sorts of statistical and stochastic analysis of information systems. (It was pretty overwhelming at the time but I’d like to take the class again.)

Anyway…one of the concepts that I took away from the class was about Poisson distributions. It got me wondering about how I would explain this concept to my College Mathematics students.

Let me explain.

College Mathematics at my school is a 100-level class for non-technical students. Most of my class are in programs like Medical Assisting, Graphic Design and Business Administration. This is the course description:

This course develops problem-solving and decision-making strategies using mathematical tools from arithmetic, algebra, geometry, and statistics. Topics include consumer mathematics, key concepts in statistics and probability, sets of numbers, and geometry. Upon successful completion of this course, students will be able to apply mathematical tools and methods to solve real-world problems.

Basically, it’s a functional numeracy class. Don’t get me wrong — it’s fun to teach and I enjoy the challenge of presenting math concepts in an interesting and accessible way. One of the ways I do this is start each class session with a ‘Math Minute’ where I run through a quick math concept to give my students a little something to ponder.

The Poisson distribution was developed by Siméon Denis Poisson in 1837. According to StatTrek:

A Poisson distribution is the probability distribution that results from a Poisson experiment.

Okay, a Poisson something-or-other comes from a Poisson something-else. Could you vague that up for me?:

A Poisson experiment is a statistical experiment that has the following properties:

• The experiment results in outcomes that can be classified as successes or failures.
• The average number of successes (μ) that occurs in a specified region is known.
• The probability that a success will occur is proportional to the size of the region.
• The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

That’s a little better. Let’s look at the classic example of a Poisson distribution: bus arrivals.

On your route, a bus comes by, on average, every ten minutes. On this particular day, you arrive at the bus stop and there is no bus to be seen. How long, on average, do you think you’ll have to wait for the next bus?

At this point, your audience is expected to say “Why, five minutes, of course.” The standard answer is actually ten minutes. The problem with this example is that it doesn’t give the right picture so the actual answer doesn’t fit our mathematical intuition. That’s why at this point the explainer hauls out the equation:

Poisson Formula: Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

And everyone’s eyes glaze over.

I’m not complaining but I’m looking for a way to explain this to a non-technical crowd. So the bus stop example can be much more effective if we just tweak a few parameters. A Poisson distribution is generated out of experimental data so let’s adjust our mental picture.

Instead of being the person waiting for the bus, let’s instead picture ourselves sitting on a park bench across the street from the bus stop. In one hand we have a stopwatch and the other a spreadsheet. We can start our experiment any time but let’s say that we begin when a bus arrives. As we observe busses and passengers, we record two things:

1. The time the bus arrives
2. The time that each person arrives at the bus stop.

We need to gather enough data for a useful answer, so we’re going to be here for a while. It could be an hour, two hours, twelve, whatever. At some point we get bored and go home to crunch our data. Lo and behold, we discover that the bus interarrival time does in fact average ten minutes but the average wait time is greater than five minutes.

Why is that? Well, from our perspective as an outside observer, this phenomenon makes a lot more sense.  Let’s look at our bus arrivals over a timeline:

bus 1  –  5 mins. – bus 2  – 15 mins.- bus 3 – 10 mins.bus 420 mins.bus 5 and so on…

As an individual waiting at the bus stop we didn’t have this perspective. But when we look at it from outside, we have a better picture of what’s going on. More passengers arrive during the longer intervals than during the short ones. As a result, the wait times for these passengers are going to skew our average wait time, giving us our previously unintuitive result of greater than five minutes.

Uncertainty is part of our lives, so it makes sense that it would show up in math as well. Probability (and statistics) don’t eliminate uncertainty but rathergive us a useful way to acknowledge  and honor uncertainty.

# #WeHateMath Math without Calculations?

I ran across a couple of articles that discuss something that I’ve been pondering (and talking about on this blog) for a while now. That is, teaching mathematics without requiring that students do the final calculations by hand.

Here’s the basic idea. Computation (formulating solutions to problems) is more important now than ever. However, since we have calculators, computers and even Web sites to crunch the numbers for us, doing the calculations by hand is out-of-date and should be de-emphasized.

This idea was recently promoted by Conrad Wolfram, the head of Wolfram Research, which produces the software package Mathematica and the math engine Wolfram Alpha. Here’s an interview with Wolfram where he describes his idea in a bit more detail.

I’m not saying this is a bad idea, but I can see that there are any number of ways for someone to take this in a negative way. It takes a while to think this through for those of us who (like me) were raised and educated in the traditional math curriculum.

I teach undergraduate math, so I’m always on the look-out for ways to improve my classroom content. That means that sometimes I use my classroom as a lab for a little empirical research.

For example, this term I’m teaching College Mathematics. This is a 100-level class aimed at non-technical students (we’re a career college) and my students are in a mix of majors like Medical Assisting, Graphic Design and Business Administration. The course description is as follows:

This course develops problem-solving and decision-making strategies using mathematical tools from arithmetic, algebra, geometry, and statistics. Topics include consumer mathematics, key concepts in statistics and probability, sets of numbers, and geometry. Upon successful completion of this course, students will be able to apply mathematical tools and methods to solve real-world problems.

So it’s essentially a course in functional numeracy. For most of these students, this is the only math class in their program.

I decided to test the theory that the thing most people who claim to hate math actually hate arithmetic. My personal opinion is that arithmetic is unnatural and mechanical (since the learning strategy consists of memorization) and that math places more emphasis on creativity, intuition and critical thinking. With this in mind, on my first day of class I did the following:

1. Explain the difference between math and arithmetic.
2. Set a policy of ‘no arithmetic by hand, unless it’s absolutely convenient’. (For example, I’m not going to grab a calculator to multiply 9 by 5.)

For each session, I start with a “Math Minute” where I present a short puzzle or thought experiment to get students thinking and discussing some math concept. For the rest of the period I discuss this week’s subject (I can’t change the text or the lesson plan). However, I keep the conversation focussed on concepts rather than calculation. When we work through problems, we spend most of the time thinking through the set-up and then use a calculator (or spreadsheet or Wolfram Alpha) to get the answer. (This is only for problems where the calculation isn’t obvious. See 2 above.)

As you can see, I haven’t made any big changes (evolutionary, not revolutionary). I still have to pay attention to different student learning styles, encourage group participation, share problem-solving tricks and all of the other classroom techniques I’ve been using for years now.

As expected, there was some resistance. I was taking an unusual approach by de-emphasizing those parts of a math class that are traditionally the focus. But overall the response has been positive.

Now I don’t think that all classroom math should be abstracted to machines. In my class, we still use decks of cards and dice to talk about probability, count floor tiles to think about surface area and lots and lots of whiteboard work complete with diagrams. The point is to get students to connect with math and I’m just shifting that connection to the problem set-up process and I’m a fan of whatever works.

# #WeHateMath Why I Hate Arithmetic (And Why You Should Too)

I’m Tom and I hate arithmetic. (“Hi, Tom!”)

Let’s face it. Everybody hates arithmetic. It’s boring, mechanical, unnatural and an unfit occupation for a respectable human. We hate arithmetic so much that we’ve spent the past 40,000 years inventing things to do arithmetic for us.

Consider Stonehenge.

• Up to 165 stones, about half of which were 18 feet long and weighed 25 tons.

• There is no building stone in the area, so the stones had to be dragged from hundreds of miles away.

• The entire construction took 30 million hours (about 1500 years).

All of this work and when it comes right down to it, it’s basically a big calculator. We hate arithmetic so much we will go to ridiculous lengths to avoid doing it. Worse yet, if you openly admit that you hate arithmetic, you’re shamed and ridiculed.

Math is not arithmetic but the way we teach both subjects doesn’t make this clear. Arithmetic is mainly memorization which is expected for something so mechanical.

“Here’s the times table. We’ll quiz you on it later.”

“Here’s the transitive property. Memorize it.”

The basic strategy for surviving arithmetic class is memorize, memorize, memorize. Then we get to math class and NOBODY TELLS THAT THIS ISN’T ARITHMETIC ANY MORE. Instead we’re given more things to memorize without any human context. Math is about creativity, imagination, intuition and seeing the patterns that go on all around us and trying to understand them. There’s none of this in a typical math class, much less a typical math textbook.

Let me put it another way. Let’s say that you like stories.  You like to tell them, you like to create your own, you enjoy listening to others tell them to you. So you start school and you’re all excited because you’re finally going to learn how to write these great stories that you’ve been imagining.

You start out and learn about all of the letters. Some of the letters have special names (‘vowels’, ‘consonants’) and you learn them. You learn how the letters go together in words (spelling) and placing words in sentences (grammar). You memorize the structure of a sentence and all of the different parts (‘subject’, ‘verb’, ‘adjective’,’adverb’, ‘gerund’ [yeah, that’s a real thing])

Are we ready to write stories yet?

Nope, still got more stuff to memorize. Subjunctive, declarative, indicative, imperative, the difference between passive and active voice, between first person singular and third person plural, proper use of semi-colons, dangling participles, prepositional phrases….

Okay, I’m officially exhausted and we’ve completely given up on our original goal to be able to write our stories to share with others.

A few years back I was teaching a Pre-Algebra class for some adult learners and I had one older gentleman who was really struggling with the material. One day I had him come up to the board, work a problem and explain how he solved it. He got the right answer, but he did it in a way that I had never seen before. That’s when I asked him what he did outside of class.

It turned out he was a carpenter. He used math all the time, because you need math to create good-looking functional pieces. When he was learning carpentry, he picked up the tools and techniques he needed in service to this. He knew what a correctly proportioned bookshelf that wouldn’t fall over should look like and feel like and the tools of math were just things he picked up on the road to that goal.

Math is about exploring the world around us and trying to understand why things are the way they are. Why do certain combinations of notes sound pleasant and others don’t? Why is the sky blue? How do I throw a stone to make sure it goes as far as possible? Numbers, proportions, sequences, patterns are around us all the time. We can see them and we’re compelled by our very nature to try and understand them. This is what math is about.

# Project Euler #5 – A Fool Remains

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

He who asks is a fool for five minutes, but he who does not ask remains a fool forever.

Chinese Proverb

This is an example of why students hate word problems. It’s also an example of why it’s important to practice with word problems.

You see, it’s not just about asking a question. It’s about how the question is framed. Question framing can make a big difference in how easy it is to find the answer. This is related to (but not the same as) the loaded question, which is one of the classic logical fallacies.

So when I come across a problem like this, I try to rephrase the question in such a way that my path to the answer becomes more clear. In this case, the question makes more sense to me when phrased like this:

What numbers can be factored into the first 20 integers? Find the smallest one.

Since I’m lazy, I asked WolframAlpha but it didn’t understand the question. I also considered that multiplying all of the numbers from 1 to 20 together (otherwise known as 20 factorial or 20!) would answer the first part of the question but not the second. (It’s 2432902008176640000, by the way.) (Yeah, probably too big.)

Now, of course, I have a computer that has no problem with mindless arithmetic so I could totally brute-force this by starting at 2520 (since the answer certainly can’t be smaller than that), factoring each number, examining the factors to make sure all required integers are present and so on. But I think there’s a lazier (and hence smarter) way of doing this.

20 – gives me 2, 4, 5 and 10

18 – gives me 2, 3, 6, and 9

16 – gives me 2, 4 and 8

14 – gives me 2 and 7

15 – gives me 3 and 5

9 – gives me 3

And so on and so forth. Let’s try that technique with a number we already know, 2520 and the factors from 1 to 10.

10 – gives me 2 and 5

9 – gives me 3

8 – gives me 4

7 – a prime but needs to be in there because we don’t get it anywhere else

6 – got it (2 from 10 and 3 from 9)

5 – got it (from 10)

4 – got it (from 10 via 2 and from 8)

3 – got it (from 9)

2 – got it (from 10 and 8)

1 – duh

So the factors that I care about are 10, 9, 8 and 7. If we multiply these together we get 5040. Not the answer we’re looking for but certainly better than 10! (otherwise known as 3,628,800). My keen eye, however, observes that 5040 is exactly twice 2520, so I have an extra 2 somewhere in my numbers that I don’t need. (I’m going for smallest, after all.) Well, I already get a 2 from the 10 so I don’t need 8. But now I don’t have a 4 (almost sounds like we’re playing Fish) so I’ll substitute to get the factors 10, 9, 7 and 4. Multiplying these together gives us 2520 which is the correct answer.

This is a problem-solving technique I like to use when I want to experiment with solutions. Find a similar, but smaller problem (preferably one where you already know the answer or can easily get to it) and use it as a test case.  (Make sure your logical assumptions are sound, though.)

I think I’m ready to tackle the original problem now. I’m going to use a table so that I can organize my factors more efficiently

 Factor Gives me 20 2, 4, 5, 10 19 19 18 3, 6, 9 17 17 16 2, 4, 8 15 3, 5 14 2, 7 13 13 12 2, 3, 4, 6 11 11 10 2, 5 9 3 8 2,4 7 7 6 2,3 5 5 4 2 3 3 2 2 1 duh

Since we don’t get 19,17, 13 and 11 from anywhere else, they have to be part of our final answer. Let’s look at that table again and mark off the numbers we need and don’t need.

 Factor Gives me 20 2, 4, 5, 10 19 19 18 3, 6, 9 17 17 16 2, 4, 8 15 3, 5 14 2, 7 13 13 12 2, 3, 4, 6 11 11 10 2, 5 9 3 8 2,4 7 7 6 2,3 5 5 4 2 3 3 2 2 1 duh

So it looks like we just need 19, 18,17,13,11,10,8 and 7. Multiplying these together gives us 465,585,120. A quick calculation at WolframAlpha confirms that the first 20 integers are indeed factors of this number. This seems like a large number but not when compared to 20!, which is  2.43290200817664 × 10^18 or a bit over 2 quintillion.

So that’s definitely an answer, but is it the answer? Based on our experience with the smaller version of this problem, let’s try to substitute 4 for 8 as one of our factors. This gives us 232,792,560 which is not only smaller but is also correct. But can we keep going down? A bit of experimentation shows that with further substitutions cause factors to drop out so that’s no go. So we end up with 19,18,17,14,11,10,4 and 7.

So we looked at a couple of problem-solving techniques that we can add to our intellectual toolkits:

• If possible, rephrase the question so that it suggests a path to the answer

• Use a smaller version of the problem to quickly test your potential solutions.

• As always, use a computer for the boring stuff.

# Why We Hate Math II

I know that there are people who do not love their fellow man, and I hate people like that!
Tom Lehrer (1928 – )

This is the second in a planned ongoing series of posts where I try to figure out what problem(s) people have with math.
Today I’d like to talk about an essay in Harper’s by Nicholson Baker called “Wrong Answer: The Case Against Algebra 2”. Baker is an award-winning writer of both fiction and non-fiction whose most recent work is the novel “The Traveling Sprinkler”.
Just to digress a bit, I would note that he’s not making the case against math instruction altogether but for some reason is focused on Algebra 2. Just so we’re clear, Algebra 1 is basic algebra and covers topics that include (but are not limited to) inequalities, factoring polynomials, exponents and logarithms. Algebra 2 (sometimes known as Advanced Algebra) includes (but is not limited to) functions, quadratic equations, geometric equations and some trigonometry.

So let’s look at the arguments.

Baker begins by reminiscing about the good old days (you know, the 1500’s) when textbooks were written in Latin with notations that were made up out of whole cloth by their authors. He notes that we’re still teaching the same set of rules even today.
I’m not sure if he thinks this is a good thing or a bad thing. It’s not as if math techniques go out of style. If a rule doesn’t work, we fix it or discard it. (We’ll talk more about this in a bit.)
Then he gets quite poetic:

That’s what’s so amazing and mysterious about the mathematical universe. It doesn’t go out of date. It’s bigger than history. It offers seemingly superhuman powers of interlinkage. It’s true.

Okay, two paragraphs in and he’s gone off the rails. Math is possibly the most pragmatic skill set that a human being can possess. When humans wanted to stop living in caves and instead build houses and cities, they needed to have math to make sure their homes wouldn’t fall down and kill them. When you have cities, you have lots of people working together and that takes organizing, which needs math to assess wealth, track profits and loss and yes, to collect taxes.

You know who doesn’t need math? Australian aborigines. Their language has a word for ‘one’ and that’s about it. If you ask a man with four children how many he has, he’ll say, “Many.” There’s no talk about distances between places. Instead they map out their world with songs. There’s nothing wrong with this. It works for them. Humans are very pragmatic, when it comes right down to the day-to-day business of getting on with things.

I have a serious problem right off the bat with Baker presenting math as this mystical, seemingly otherworldly, mysterious thing that is beyond the human experience. Nothing could be further from the truth. Math exists because some human somewhere had a job to do (build a house, run a business, run a government) and needed an intellectual toolkit in order to do it.

All right, two paragraphs into a seven page article and I’m already annoyed. But let’s be fair and let Baker state his case. Moving on…..
For Exhibit A, Baker cites a poll taken on a Web site where 86% of respondents report hating math in general and algebra in particular. Other opinions on that same site:

Wow. Just….wow.
Out of respect for Mr. Baker, I will refrain from making snarky comments about the use of public Internet polls in scholarly argument. In addition, I’ll assume that he’s just warming up at this point and has not yet presented his case.
On the next page Baker approaches the vicinity of a point. He describes a math textbook:

The textbook’s cover is black, with a nice illustration of a looming robotic gecko. The gecko robot has green compound eyes and is held together with shiny chrome screws. It has a gold jaw and splayed gold toenails. Perhaps you like the idea of robotic geckos, and you might expect, reasonably, that there would be something about the mathematics either of geckos or robots somewhere in this book. There isn’t.

Zing! Take that, math lovers! No robot geckos for you! (Seriously, this sounds like a very cool book cover.) Just for sake of comparison, I looked up Baker’s “The Travelling Sprinkler” on Amazon.com and not only does the book cover NOT have a picture of sprinkler on it but the book itself isn’t about sprinklers! Oh, the cruel bait-and-switch scams inflicted on the reading public by the publishing industry!

Fine, I’m being childish. I mean, if you want to have the discussion about how messed up our public K-12 education system is and how California and Texas unduly influence our national textbook purchasing system, fine, we can have that discussion. But that’s beside the current point.
Baker continues:

So no lizards, no geckos, no robots. Here’s what you actually learn about rational functions in Chapter 8 of Pearson’s Algebra 2 Common Core:

“A rational function is a function that you can write in the form f(x)=P(x)/Q(x) where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x)=0.”

So now we begin to approach agreement. Assuming Baker is being honest in presenting this quote out of context, I agree that many math textbooks are poorly written for their target audience. Let’s also ignore the fact that if you got through Algebra I, then you already know what real numbers are and what a polynomial function is.
I’m not a professional author/writer/novelist like Mr. Baker but I did publish a textbook back in the day. Since it was my first book, I made the classic rookie mistake of letting my students (the intended audience) give me feedback on the book content. As a result, one of the things I did was, whenever I hit a spot where I thought things had gotten a bit complex, I paused and spent a few sentences explaining in clear language what had just happened. For the example given above, I might have added something like the following:

Remember, P(x) and Q(x) are just numbers. Since real numbers are simply those numbers that we can represent with a fraction, it makes sense the P(x)/Q(x) can represent all real numbers. The only exception is when Q(x) is zero, since dividing by zero is undefined.

I have no problem with complexity, as long as the author takes a moment to make sure everyone is on the same page. The more advanced students can ignore the aside and move ahead while everyone else can take a moment to let their brains cool. Like any area of study, math has a specialized vocabulary. Using the formal terminology lets students get used to the jargon.

Baker continues his dissection of the textbook:

“If a is a real number for which the denominator of a rational function f(x) is zero, then a is not in the domain of f(x). The graph of f(x) is not continuous at x=a and the function has a point of discontinuity.”

See how my little added explanatory text becomes useful? A function f(x) is continuous if, for every x, there is a result that we can put onto a graph. We therefore can draw an unbroken line through those points on the graph. However, since dividing by zero is undefined, if some x causes us to divide by zero, then we can’t plot that point and our line is broken, or discontinuous.
Baker goes on to cite a story problem about calculating free throw percentages in basketball. He’s doing well until this:

How very odd, you think: I don’t have to know any algebra at all in order to figure out that the answer to this free-throw question is 6. All I need is arithmetic and a little trial and error. But that’s not what the textbook wants. [emphasis mine]

What?!? The textbook doesn’t want you to just guess, it would actually prefer that you apply what you just learned to work out the answer in a precise, provable and most importantly, repeatable way?

If you were having an appendectomy, which technique would you prefer your surgeon use?

Just because the example problem is trivial, it doesn’t mean that the solution methodology is unimportant. Also, while I’ll agree that many story problems in math textbooks are typically awful, I’ve at least had some experience trying to come up with back-of-the-chapter exercises to let students apply what they’ve learned. It’s hard and a typical math textbook requires many hundreds more of these than my own book did.

Besides, this gets to the whole point of math. There are numbers all around us, every day — time, temperature, wind speed, velocity, mass, acceleration, force, luminance, what have you. Math gives us the tools to put these numbers in context and thus gain a better understanding of how the world around us works.
You may argue that not everyone has to have a deep understanding of math and with that I’ll agree. However, if nothing else, it’s extremely useful to know enough math to know when someone is lying to you.

This is my fundamental problem with the whole ‘math is icky’ crowd. By deliberately avoiding an entire field of knowledge, they leave themselves vulnerable to hucksters and charlatans. Look at a typical daily newspaper (for those of you born after 1980, go to a typical news Web site). How many of the stories (including the editorials) involve numbers? How many of them are using your fear of math against you in order to push a political or ideological agenda? If someone tries to sell me on their philosophy using numbers that are meaningless without proper context, they are knowingly committing fraud. But if you’re the type of person whose eyes glaze over when anything numerical comes up in the conversation, you’re the perfect mark for these con-men. This isn’t even a new thing. Darrel Huff’s seminal (and very readable) book “How to Lie With Statistics” was first published in 1954.

I’m not even going to go into the Stanford University study showing that the very act of learning math improves brain function.

Moving on….

After speaking with educators (and pausing to lambaste the Common Core educational standards as “micro-managerial, misbegotten” and “joy-stunting”) Baker proposes a new approach to math instruction:

We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mind-stretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the infinitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.

As long as we have public schools that rely on local property values for their funding, this will never happen. Oh, perhaps in those special ‘charter schools’ where the administration can pick and choose what students are enrolled and they have additional corporate funding and sponsorship as well as an actively engaged group of parents, but otherwise, no. Plus, this entire essay so far was arguing against the forcible teaching of advanced algebra and now he proposes the forcible teaching of a melange of algebra, geometry, trigonometry, calculus, statistics and probability? Seriously?!?
Well, I’m exhausted. I’m just going to sum up the rest at this point. Baker does a decent job of putting education reform into an historical perspective except for this one sneaky sentence:

By 1950, at a time when only a quarter of American high school students were taking algebra, the nation’s technological prowess was the envy of the planet.

I call this sneaky because he completely buries the lede by not pointing out what else was happening in America during that time:

• The top marginal income tax rate was 90%
• Organized labor membership was at its peak
• There was a thriving middle class
• We had a strong manufacturing base, which along with a strong labor movement, meant that you could get a job with decent pay and benefits and support a family without a college degree.
• The rest of the world was still rebuilding their infrastructure from the damage of the second World War.

Compare that to our situation today. It’s okay, I’ll be here when you get back.

Listen, nobody is denying that primary education is important and that it has problems. But there is no one thing you can point to and say, “If was just fix this, we’re good”. Yes, math is not taught consistently well, math textbooks aren’t consistently well-written and more math instruction is not the silver bullet that will fix our education system. It has ugly, messy, multifaceted problems that will most likely require ugly, messy, multi-faceted solutions.
I’d like to believe that Mr. Baker is making an honest effort to suggest improvements because that would make his arguments simply wrongheaded, rather than disingenuous. I could publish a similar essay arguing against forcing students to write research papers and that advanced writing classes are the reason that American students are dropping out of school. I would be no less wrong.