First Day Jitters

Today is the first day of classes for me.

I’ve been teaching in one form or another for 20 plus years. I’ve survived the Great Calculus War, and even spent time in the Technology Underground engaging in acts of passive resistance against the ancienne regime. And yet the start of every term is fraught with anxiety for me: What am I going to do this semester?

I don’t know how common this problem is among my colleagues. I mention it because it underscores what I believe to be an important fact of life: Never be too comfortable. This doesn’t mean you should go out and buy ill-fitting shoes and eat kale 24/7. Rather, it means that a little discomfort is good, because the general human reaction is to try and change things to become more comfortable.

So I’m anxious about what I’m going to teach this semester. That’s good, because it forces me to look a what I’m teaching and ask myself the all-important question: Why would anyone want to listen to me talk about this subject? (I don’t have an answer for this term yet…which is why the anxiety persists)

On the other side, learning mathematics is all about anxiety too. The critical question is not whether you suffer from math anxiety; it’s what you do about it. Remember that the human reaction is to change to become more comfortable in whatever situation you’re in. For all too many people, the reaction to math anxiety is to avoid math.

And that’s fine, if you live in a stone age society where mathematics can be left to specialists. (Go ahead, ask me about paleomathematics…) But in the modern world, it’s not practical to avoid mathematics. It’s possible, in the same way that it’s possible to avoid reading. But you won’t get very far, and you condemn yourself to being a second or third class citizen.

Instead, the way to fight math anxiety is to accept the discomfort…and push through it anyway. The most important lesson we can learn in life is that we can survive a little discomfort, and when we get through to the other side, we are better for it. And soon enough, you’ll find yourself addicted and actively seek the discomfort because you know you can get through it.

Psychologists no doubt have a litany of strategies for dealing with anxiety disorders. But here’s my suggestion for dealing with math anxiety: Do a little math, every day. The good thing about math is that it’s something you can do in the clamor of your own mind as you go through daily life.

Count things: that’s the beginning of mathematics. Don’t look at the line at Starbucks; count how many people are there. If you do this often enough, you’ll start to think about more efficient ways to count: you’ll find yourself counting by twos, threes, and fives. You’ll also develop what many have called number sense: the ability to estimate quantities with reasonable accuracy.

Once you’ve gotten into the habit of counting so that it’s second nature…in other words, once you’re comfortable with it…introduce a little anxiety and start to do arithmetic on a regular basis. There’s eight people ahead of you in line at Starbucks; how long is it going to take before you get your coffee? How much is Starbucks making off the people in line? What’s the average wait time? Soon enough, you’ll be running through calculations like “If those eight people are like me and ordering a $2.35 coffee then that’s 8 \times 2.35 in revenue for ten minutes…”

And again, every time you get comfortable…move it up a notch.

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Linearity and the Cashier’s Method

Linearity is an important concept in life and mathematics. It’s closely tied to another concept, proportionality, and in fact it’s often confused with that concept.

Two quantities are proportional if k times one quantity corresponds to k times the other.

Below are boards and their prices.  The first picture shows a 12-foot board costing $5.26, while the second shows that a 16-foot board costing $7.16.

Now 16 feet is 1 and 1/2 times as much as 12 feet, so we might ask:  Is $7.16 is 1 and 1/2 times as much as $5.26?  There are several ways we could answer this question.  Just for fun, and to build up your mathematical abilities, let’s do it without a calculator.

Here’s one way:  1 and 1/2 times as much as $5.26 is 1 times $5.26 (which would be $5.26) and 1/2 times $5.26.  There are many ways to get half of $5.26, but here’s how a cashier might do it:  half of $5 is $2.50; half of 26 cents is 13 cents.  So half of $5.26 is $2.50 and 13 cents, or $2.63.

So 1 and 1/2 times $5.26 is $5.26 + $2.63.  Again, a cashier might do it this way:  That’s $5 and $2, for $7; and 26 and 63 cents, for 89 cents.  So 1 and 1/2 times $5.26 would be $7.89.

What does this mean?  It means that if the cost of wood were proportional, then the 16-foot section would cost $7.89.  Instead, it costs $7.16, a little less.  While we might expect some deviation from an exact proportionality, the difference (76 cents, or about 10%) is rather sizable, and so we might conclude the cost of wood is not proportional.

However, it might still be linear.  In many ways, linear is an easier concept:  an extra k corresponds to an extra m.  So here, the extra 4 feet cost an extra $7.16 – 5.26.  Again, think of it like a cashier:  You have $7.16 on the table, and want to remove $5.26.  I see $7.16 as a $5, two $1s, and 16 cents.  So I remove a $5, then make change for one of the $1s to remove 26 cents.  This leaves me with a $1, 74 cents, and 16 cents:  $1.90.

To check for linearity, we need to look at another value.  In this case, an 8-foot board (not shown) will cost $3.23.

If the cost is linear, then an 8 + 4 = 12 foot board would cost $3.23 + $1.90 = $5.13.  Here the difference between what the board should cost and what it does cost is only a few cents, so we might reasonably conclude that the cost of lumber is linear.

Of course, only politicians and pundits base their conclusions on one piece of evidence.  What about a 16-foot board?  If the cost is linear, then a 12 + 4 = 16 foot board should cost $7.16 + $1.90 = $9.06, and we find it costs $9.19, which is reasonably close.

One final note:  Even if the cost of lumber is linear, we can’t expect that it will always be linear.  Since the 8-foot board costs $3.23, we’d expect a 4-foot board to cost $3.23 – $1.90 (subtract $2, return 10 cents) $1.33, and a 0-foot board would cost $1.33 – $1.90 = $ -0.57.  But if you go to the checkout with ten 0-foot boards, the cashier is not going to hand you a five spot.

And remember, lumber doesn’t grow on trees.  (Yes, I meant it…wood grows on trees, but no tree in the forest grows 16-foot boards)  A 200-foot board might cost significantly more than linearity would suggest.

 

Obesity, Poverty, and National Security

According to the internet, if you ate only ramen, you’d save thousands of dollars each year in food.

That sounds great, except there’s a problem: ramen lacks a wide range of essential nutrients and vitamins. You’d lose your teeth to scurvy, a lack of vitamin D would cause your bones to become brittle and easily broken, you’d suffer nightblindness from a lack of vitamin A, and you’d be tired all the time from a lack of iron and the B vitamins. In short, all the money you saved on food, and much, much, more, would be spent on increased medical care.

The problem is that eating healthy is costly. And this leads to a national security crisis.

If you want the short version, I’ve summarized the key points in a ten-minute video:

A little more mathematics:

Food buyers face what mathematicians call a constrained optimization problem: they have to meet certain caloric and nutritional goals (the constraints), which defines a feasible region. Generally speaking, any point in the feasible region defines a solution to the problem; what you want to do is to find the optimal solution.

The optimal solution is generally determined by the objective function. For example, if you lived off x packages of ramen and y eggs, the important objective function might be the total cost of your meals. At 15 cents a pack of ramen and 20 cents an egg, the objective function has the form L = 0.15x + 0.20y, and we might want to minimize the value of the objective function.

In the following, I’ll assume you want to minimize the value of the objective function; the arguments are similar if you’re trying to maximize the value (for example, if you’re designing a set of roads, you might want to maximize the traffic flow through a town center).

There’s a theorem in mathematics that says the optimal solution will be found on the boundary of the feasible region. The intuition behind this theorem is the following: Imagine any point inside the feasible region. If you change any one of the coordinates while leaving the others the same, the value of the objective function will generally change. The general idea is to move in the direction that decreases the objective function, and continue moving in that direction until you hit the boundary of the feasible region.

At this point, you can’t move any further in that direction. But you can try one of the other directions. Repeating this process allows us to find the optimal solution.

We can go further. Suppose our objective function is linear (like the cost function). Then the same analysis tells us the optimal solution will be found at a vertex of the feasible region. This suggests an elegant way to solve linear optimization problems:

  • Graph the feasible region and locate all the vertices. Generally speaking, the constraints are themselves linear functions, so (in our ramen and egg example) the feasible region will be a polygon.
  • Evaluate the objective function at each vertex,
  • Choose the vertex that minimizes the value of the objective function.

Easy, huh? Except…

  • If you have n commodities, you have to work in \mathbb{R}^{n}.
  • This means the feasible region will be some sort of higher solid.
  • This also means that finding the vertices of the feasible region will require solving systems of n equations in n unknowns.

In 1945 ,George Stigler did such an analysis to find a minimal cost diet that met caloric and nutritional requirements. To make the problem tractable, he focused on a diet consisting of just seven food items: wheat flour; evaporated milk; cabbage; spinach; dried navy beans; pancake flour; and pork liver.

“Thereafter the procedure is experimental because there does not appear to be any direct method of finding the minimum of a linear function subject to linear conditions.” The problem is that with seven items, you’re working with hyperplanes in \mathbb{R}^{7}, and the constraints will give you hundreds of vertices to check.

Note the date: 1945. What Stigler didn’t know is that there was a method for finding the minimum value easily. But that’s a story for another post…

The Most Important Letter

A question came up on Quora about what letter’s removal would have the greatest impact on the English language.  The obvious answer is “E”, since it’s by far the most common letter in English.

But let’s consider that.  Can you writ a comprhnsibl sntnc that dosnt us ths lttr?  Ys, you can!  So its not clear that “E” is all that important.

So let’s do some mathematics.  The key question is:  How much information does a given letter provide?    Consider the following:  I’m thinking of a color.  You know the color is either red, green, blue, or fuchsia.  (I have no idea what color fuchsia is…I just like the word)  Your goal is to determine the color I’m thinking of by asking a sequence of Yes/No questions.

One way you could do this is by asking “Are you thinking of red or green?”  If the answer is “Yes”, then  you might ask “Are you thinking of red?”  If the answer is “Yes”, then you know the color is red; if the answer is “No,” then you know the color is green (since I answered “Yes” to the first question).  On the other hand, if I answered “No” to the first question, then you know I was thinking of blue or fuchsia, so you might ask “Are you thinking of blue?”  A “Yes” tells you I’m thinking blue; a “No” tells you I’m thinking fuchsia.

Now reverse it.  If you know I’m thinking of the color red, then you have the answer to two Yes/No questions.  We say that “red” has an information content of two bits.

So far so good.  But suppose I’m somewhat dull and can’t think of any color other than red. In that case, you already know what color I’m thinking of, and don’t need to ask any questions.  In this situation, “red” has an information content of zero bits.

As an intermediate case, suppose that half the time I think of “red,” one-fourth the time I think of “blue”, and one-eighth the time I think of “green” and one-eighth the time I think of “fuchsia.”  Then you might ask a different sequence of questions:

  • Are you thinking of red?  (Half the time, I’ll answer  “Yes”, so the answer “red”gives you the answer to one question:  it’s 1 bit of information)
  • If the answer is “No,” then “Are you thinking of blue?”  Half the time this question is asked (remember it will only be asked if the answer to the first question is “No”), the answer will be “Yes,” so the answer “blue” gives you the answer to two questions:  it’s 2 bits of information.
  • If the answer is “No,” then the final question “Are you thinking of green?”  Again, half the time this question is asked, the answer will be “Yes,” which tells you that “green” is worth 3 bits; meanwhile, the answer “No” means I’m thinking of fuchsia, so “fuchsia” is also worth 3 bits.

It might seem difficult to determine the information content of an answer, because you have to come up with the questions.  But a little theory goes a long way.  The best question we could ask are those where half the answers are “Yes” and the other half are “No.”  What this means is that if n is the answer to the question p_{n} of the time, then the information content of the answer n will be -\log_{2} p_{n}.  Thus, if “red” is the color half the time, then “red” has an information content of -\log_{2} (1/2) = 1 bit.

So what does this mean?  “E” makes up about 12.7% of the letters in an English text.  But this means that knowing a letter is “E” answers very few questions.  So the letter E contains about 3 bits of information.  In contrast, “Z” only makes up 0.07% of the letters in an English text, so knowing a letter is “Z” answers many questions.  So the letter Z contains about 10.4 bits of information (the maximum).

At first glance, this suggests that “Z” may be the most important letter in the English language:  losing the letter “Z” will lose the most information.  However, there’s a secondary consideration:  “Z” doesn’t often appear in a text.  So every “Z” you drop from a text loses a lot of information…but you don’t drop that many.

And here’s where the greater prevalence of “E” comes in.  While the letter “E” only gives you about 3 bits of information, it’s common enough that dropping the letter “E” from a text will lose you more information overall.  For example, suppose you had a 10,000 character message.  Of these 10,000 characters, you might expect to find 7 Zs, and losing them would lose you about 77 bits of information.  In contrast, there would be almost 1300 Es, and losing them would lose about 3800 bits of information.

 

Exact is Not Accurate

Numbers.

Over the next few years, you’ll be certain to see a barrage of numbers thrown at you.  While researching the latest atrocity promoted by the administration, I came across the following tidbit:  The average tuition for private schools is $10,003.

Now, if I want to include this in a blog, vlog, Facebook post, or public speech, I have a conundrum.  Compare the two sentences:

  • The average tuition for private schools is $10,003.
  • The average tuition for private schools is about ten thousand dollars.

The first sounds like I know what I’m talking about:  that I’ve done some high-level research and wrestled a number to the ground.  The second sounds like I spent thirty seconds on  Google.  (Actually, the first number was based on thirty seconds on Google)

The difference is that I sound more convincing with the exact figure.  In fact, there’s a story (which might  or might not be true) that when the first surveyors found the height of Mount Everest, they came to a value of 29,000 feet…but they published it as 29,002, because that sounded more accurate.

The problem is the exact figure might not be accurate.  Consider the two statements:

  • The population of the United States is 324,595,182.
  • The population of the United States is 325 million.

The first gives an exact number, and sounds very accurate.  But it is almost certainly false.  In particular, even if the population of the US was 324,595,182 at some point, it is almost certainly not 324,595,182 right now.  On the other hand, it’s still about 325 million, and will be so for awhile.  (I talk about this in my FOCUS article).

There’s a concept in the sciences called significant figures.  The gist of it is this:  When I give you a number, I am giving you a guarantee that the non-zero digits of the number are correct.  (The zeroes are a little more complicated:  if you want a crash course on signficant figures, here’s the video I have my students watch)

  • If I claim 324,595,182, then I’m guaranteeing each and every digit is exact…and if the population is 324,595,183, then I’ve fed you misinformation.
  • If I claim 324 million, then I’m guaranteeing that the population is somewhere between 323,500,000 and 324,499,999 (since anything in this range would round to 324 million).

What’s the big deal?  One problem with statistics is that people don’t believe them.  You’ve heard the quote:  “There are three types of lies:  Lies, damned lies, and statistics.” I suspect part of the reason is that if someone says “The average tuition at private schools is $10,003,” they can respond with “But at our school, it’s $7500, so how do you get an average of $10,003?”   This generally leads to a discussion of how to calculate averages, and often degenerates into accusations of skewed samples.

On the other hand, if you say “The average tuition at private schools is around $10,000,” then to the person who says “But we only pay $7500,” the response is “Which is around $10,000.”  By avoiding the mechanics of computing the number, we focus on the value itself.

Math for Democracy

I surrender.

I’ve been trying to keep this blog politics free, or at least minimize the politics:  when I talked about the March Across the Hudson, I focused on estimating the crowd size and not on the reasons behind it.

I’m still going to minimize the politics.  But it’s clear that we’re heading towards a major crisis.  I’m not talking about the person in the White House, or Russian interference, or anything that minor.  I’m talking about the denial of basic fact-finding.

You’ve heard the term “fake news.”  The problem is that most Americans get their information from one or two sources, which they don’t verify.  If those sources are unreliable, then they’re going to get a warped view of the world.  So I have a new mantra:

Five minutes a day.

Take five minutes a day to track down a fact.  You might start with the news story, but don’t end with it.  Who did they interview?  If they’re reporting on a piece of research, track down the original article and check out the legitimacy of the publisher.  If they’re reporting on an incident, go to the local newspapers and see what their coverage is.  If they’re talking about waste in government spending, go to USAspending.gov and see how your money is spent.

So let’s talk about that.  One of the promises of the new administration is to drastically curtail the U.S. Department of Education, returning control of schools to the states. Sounds good, right?  But go to USAspending.gov to see how the Department of Education actually spends your money.  Note that I’m giving you the source, so you should feel free to check my claims.   (A guaranteed way to identify something as “not a fact” is that lack of a source:  If there’s no source, it’s not a fact.  Keep in mind this does not work in reverse:  you can cite a source and still spew non-facts)

Most federal agencies suck in a lot of taxpayer dollars…and then shovel them back to the states in the form of grants.  Find the government department you’re interested in, then download the grants database: this tells you who they’ve given money to, and how much.  You can import it into Excel, or download it as a CSV and  use your own spreadsheet software.  Then the fun begins…

You can sort the grants by any category you want.  The cost of the elected President’s recent trips to Mar-a-Lago have been in the news:  current estimates for the three weekend trips (out of five weekends in office) are around $12 million, so here’s a few grants made by the Department of Education that are around this much.  I’ve deliberately chosen programs that benefits states where Trump support was very strong:  yes, New York, California, and other states get money from the Department of Education, so of course we’re concerned…the point is that states that supported Trump need to be even more concerned, because here are some of the things they’re going to lose:

  • Nevada: $9,928,139 for Vocational Rehabilitation training. Nevada has received almost $200 million in grants from the Department of Education since January 1, 2016.
  • Kansas: $10,669,790 for Department for Children and Families for Vocational Rehabilitation training. Kansas has received more than $210 million in grants since January 1, 2016.
  • Texas: $11,187,178 to Bexar County Texas for “Impact Aid.” The army base Fort Sam Houston occupies a good part of Bexar County, and this land can’t be taxed, impacting the county’s ability to pay for schools. That’s money local taxpayers don’t have to pay.   The Department of Education has given more than $600 million in Impact Aid grants since October 2016, reducing tax burdens around the country.

Now for some math.  On a dollar basis, California, Texas, and New York have received the most from the Department of Education.  But they’re also the biggest states in the country.  An easily googlable fact is the population of these states; if you divide how much each states gets by its population, you obtain a per capita figure.

These are interesting.  A few more states that stand to lose big if Trump eliminates the Department of Education:

  • Alabama: $15,912,537 for preschool programs. Alabama received more than $400 million in grants. On a per person basis, that’s 26% more than Connecticut gets.
  • Louisiana: $9,177,379 for preschool education programs. Louisiana has received nearly $500 million in grants. On a per-person basis, that 37% more than California receives.
  • West Virginia: $9,828,491 for vocational and rehabilitation services. West Virginia has received more than $160 million from the Department of Education. On a per person basis, that’s 50% more than Massachusetts.

 

Lies, Damned Lies, and Statistics

We all know the quote:  “There are three types of lies: lies, damned lies, and statistics.”

But like many things that are short enough to tweet, this statement is misleading.

Statistics don’t lie.  People do, generally by omitting key pieces of information.   Any statistic worth repeating should include two other numbers.   If these are missing, the whole truth is being kept from you. 

The two numbers to look for are:

  • The sample size.  This is the number of cases examined.  If you base a conclusion on one example, you’re a politician or a pundit, relying on anecdotal evidence and shouting instead of facts and logic.  While a large sample won’t guarantee reliability,  a small sample will almost always be untrustworthy. 
  • The p-value.   This is a little more complicated,  but roughly speaking, it measures how convinced you should be.   A  small p-value (0.05 or less) means the evidence is very convincing. 

I’ll talk more about these later.  Until then,  remember: if someone doesn’t give you these values,  they’re not telling you the whole truth. 

Underreporting of Deaths

The White House released a listing of 78 terrorist attacks it claims were underreported by western media. Both the BBC and the New York Times have responded by posting links to the numerous stories they ran on these incidents, debunking the belief that these incidents weren’t reported in detail.

We can go further. The vast majority (56) of the terrorist attacks resulted in one or fewer deaths. Of these, only 19 people actually died; the remaining victims were wounded. The articles run by the New York Times on these 19 deaths had an average length of 705 words.

Of course, this number alone doesn’t tell us much. To be meaningful, we need some basis for comparison. One possibility is the average word length of articles on single murders. Unfortunately, there’s no shortage of such articles:

  • On February 6, 2017, a Virginia woman shoots her 6-year-old daughter.
  • On February 3, 2017, a 12-year-old shoots a store clerk in Arkansas.
  • On February 2, 2017, a 14-year-old girl shoots her brother over a video game in Toledo.
  • On February 2, 2017, two men shoot another man during a Craigslist robbery.
  • On January 9, 2017, a Florida police officer is killed.
  • On December 24, 2016, a man in Arkansas shoots at a car for tailgeting, killing a toddler.
  • On December 1, 2016, Joe McKnight is killed in what appears to be an incident of road rage.
  • On August 3, 2016, the body of Karina Vetrano is found in a Queens park.

I’m still collecting data, because it seems there’s a journal article here, but the preliminary data is too interesting to ignore.

These eight articles have an average length of 386 words. Actually, this figure is probably higher than the average for single murders: Joe McKnight was a NFL football player, and Karina Vetrano had considerably more coverage because of its local nature.

What this suggests is that if you’re killed by a terrorist attack, your death is likely to receive twice as much coverage as it would if you were merely killed as part of an ordinary crime. A similar analysis of stories from the BBC suggests that terrorist attacks get four times as much coverage as ordinary crimes.

Manufacturing and Mining

One of the oft-quoted statistics is that if it were an independent country, California’s economy would be among the ten largest.  In 2015, it was sixth, just behind the UK and just ahead of France.  Texas and New York are also major players:  Texas’s economy is slightly larger than Canada’s (in 10th place), while New York’s is just behind Canada.

However, there’s an important factor:  California (where I was born) is also more populous than New York (where I live) and Texas (where I have relatives).  Thus, while China’s economy is larger than New York’s, China has more people; as a result, the standard of living in China is lower.  From the ground, the important question isn’t “How much does my country make?” but rather “How much do make?”

For that, you want to look at the per capita figures:  that’s the total GDP divided by the total population.  I won’t do the comparisons for other countries, but only for California, New York, and Texas.  Under this comparison (again for 2015), New York comes out 2nd, California is 10th, and Texas is 13th.  Put another way:  If these states had the same populations, New York’s economy would be about 20% larger than California’s. (I’m using data from Wikipedia, if you want to play with the numbers yourself).

Now, unless you’ve been living under a rock, you know that the United States has a new President who’s rather controversial.   However, one of his campaign promises is that he’ll bring manufacturing and mining back to the US, and much of his appeal is in the so-called “Rust Belt,” where over the past twenty years millions of jobs have been lost.

Part of the argument is that the various free trade agreements made over the past forty years have destroyed manufacturing and mining, by making it easier to ship jobs.  That’s probably true, though almost every economist who’s studied trade has concluded that it also generates quite a lot of jobs here.  Again, the problem from the ground is that the jobs it generates are very different from the jobs that disappear:  It’s little consolation to a steelworker than the financial services industry is booming.

I’m going to take a look at one very specific industry, that seemed to support the new President very strongly:  coal.  In fact, one of the very first things the new government has done is ease regulations on coal mining companies regarding what they can dump into streams; this is being hailed as a way to re-open mines and get more miners to work.

Sounds good, right?  Except there’s a problem:  Productivity.

  • In 1985, the US coal industry produced about 900 million tons of coal, and employed about 180,000 workers.
  • In 2015, the US coal industry produced about 900 million tons of coal, and employed about 65,000 workers.

What should be clear from these figures is that coal mining jobs haven’t disappeared because they’ve been shipped overseas:  we’re producing as much coal as we did thirty years ago.  But we’re using one-third as many workers.  So what does this mean?  Reopening mines and restarting coal production will produce a bump in employment.  But the vast majority of coal jobs are never coming back.  

This problem is true across the spectrum:  technology is being used to get more done with fewer people.  We can bring back coal mining…but not the jobs.  We can bring back auto production…but not the jobs.  We can bring back textile manufacturing…but not the jobs.  The vast majority of manufacturing jobs are never coming back.

So what can be done?  We can revive the industry…but the jobs won’t be there.  We can take consolation in that other industries are booming…but that doesn’t help the displaced workers.

The only viable solution is education and retraining.  Rather than waste time, money, and effort trying to rebuild an industry that won’t employ many workers, it would be far more useful to spend that time, money, and effort to retraining our workers so they can build their own industries.

 

Problem Solving

In case you’ve been living under a rock for the past few years, you’ll know there’s something called “common core mathematics”, and that many states had adopted it while others have rejected it to produce their own state standards.  I won’t talk about the standards here (other than to say the biggest difference between Common Core and Your State Name Here Standards is the name).  Instead, I’ll talk about something that is part of the standards:  Problem Solving.

 

Unfortunately, mathematicians (and by extension, math educators) are terrible at producing names.   Chemists talk about adiabetic processes; biologists talk about glycophosphlipids; geologists talk about regoliths, and so on.  Meanwhile, mathematicians talk about sets, rings, fields, continuity, surfaces.  The difference is that mathematicians use these words in very specific ways.  One way to tell a mathematician is to see them cringe every time someone refers to a group of people…

So what about problem solving?  To ordinary people, this is a problem:  “Find 2398174 \times 139871.  But in the context of mathematics education, this is not a problem.  There’s no commonly accepted word for it, which is too bad (how you speak influences how you think:  see Neither Borrower Nor Lender Be),  so I propose the name “Task.”  Finding this product is a task:  You know how to do it, and it’s a question of following a set of steps to get the answer.

 

If 2398174 \times 139871 is not a problem, then what is?  It might come as a surprise, but this can be a problem: Find 4 \times 3.  It all depends on context.  If you know the multiplication fact 4 \times 3 = 12, then this is not a problem:  it is a task (specifically, the task of recalling what 4 \times 3 is equal to).  On the other hand, if you don’t know what 4 \times 3 is equal to, then this is a real problem.

So how can you solve this problem?  One way is to wait patiently until someone whispers in your ear “4 \times 3 = 12.”  But this relies on waiting for someone to give you the answer, and (from a broader societal perspective) programs you to believe what you are told instead of thinking for yourself; I’ve noted elsewhere that one of the reasons higher mathematics is important for a free society is that it develops the habit of questioning what you are told.

Instead of waiting for someone to give you the answer, you can try to solve the problem.  In this case, the problem solving might go something like this:

  • We’ve defined a \times b to be the sum of a bs, so 4 \times 3 is the sum of four 3s.
  • This means 4 \times 3 = 3 + 3 + 3 + 3.
  • But I know how to add:  3 + 3 + 3 + 3 = 12.
  • So 4 \times 3 = 12.

Ideally, the last thought is “Cool!  I can figure out mathematics on my own and not need to wait until someone tells me what to do!”

What’s important to understand is this:  Problem solving is a skill,and like all skills, it gets better the more you practice.  But you only get one chance to solve a problem.  That is to say, once you’ve solved the problem, then no variation on the same problem gives you a chance to problem solve, and you will never again have the chance to solve the problem.

(Admittedly, it’s possible to find a new solution to a problem.  But basic arithmetic has been around for thousands of years, so finding a new solution to the problem of multiplication is very difficult…indeed, finding a new solution to any problem is something that could earn you an advanced degree in mathematics)

Thus, once you’ve solved the problem of finding 4 \times 3 = 12, then 9 \times 7 is not a problem:  you’ve figured out how to solve it (in this case, as the sum of nine 7s).  In fact, once you’ve figured out 4 \times 3 = 12 this way, then you know how to solve 2398174 \times 139871and this question is not a problem anymore!

What does this mean?  Consider a traditional elementary school math lesson which shows students how to multiply two numbers using the standard algorithm.  The instant students are shown how to multiply two numbers using the standard algorithm, they lose forever the opportunity to solve the problem of multiplication.  They will never get another opportunity to solve the problem of multiplication.

This requires a substantial shift in viewpoint in how we teach students mathematics.  Traditionally, students have been programmed to apply certain Standard Algorithms for basic arithmetic operations.  This is ideal…if we want to create students who are programmed.  However, those who can only follow a program will be doomed when they confront something outside of their programming:  such students will never progress beyond their teachers.

Instead, we need to create students who are able to solve problems.   The only way to do that is to give them the opportunity to solve problems…which means holding off introduction of the standard arithmetic algorithms for as long as possible.

You might be concerned that this means students won’t be able to multiply 43 \times 15 without a calculator.  And that’s a legitimate concern.  However, let’s consider:

  • A student who hasn’t learned the standard algorithm for multiplying two 2-digit numbers can’t multiply 43 \times 15 without a calculator:  they have no way to even begin to answer this question.
  • A student who has solved the problem of multiplication can find 43 \times 15 without a calculator:  they can add forty-three 15s together.

The problem of 43 \times 15 exists independent of the student’s knowledge of multiplication.  The difference is the student who’s done problem solving will be able to solve the problem; the student who’s only learned how to apply algorithms will only be able to solve the problem they have an algorithm for.