It's not all snake oil

If you are a teacher, what do you think about the following (real life) situation? 

Teacher: "It’s 1 o’clock. Today, school ends at 3 o’clock. How long do you have to wait?"
Student: "Two hours."
Teacher: "What’s 3 minus 1?"
Student: (After considerable struggle, cannot answer.)

—

Teacher: "What’s 4 divided by 2?"
Student:  Cannot do this, but announces the fact "Four times three is twelve."
Teacher: "Here’s four marbles for you to share among two students."
Student:  Without hesitation, separates the marbles so that he holds two in each hand.


The interaction described above is an abridged version of an account related by Stanislas Dehaene in his book "The Number Sense." (Links provided below.)

Stanislas Dehaene is a cognitive neuroscientist with a strong background in math and computer science. The student is M, is one of his patients. M is not a child, but is a retired and talented artist who has a brain lesion in the inferior parietal lobe of his brain.

Mr. M has some knowledge of numbers. He knows the history of certain numbers and can give lengthy lectures about the significance of some historical dates.  In his mind, he can move back and forth among the hours of the day. He is sporadically successful when dealing with numbers in other very concrete settings. He has some rote verbal memory and can recite such things as "Three times four is twelve," but he has no comprehension of what that means. He has absolutely no understanding of numbers when they are presented as abstract quantities. Despite considerable effort, Dehaene has been unable to teach M how to do even the most simple arithmetic computations in an abstract setting.

For a platonist like me, numbers exist independent of my own mind. They have meaning without any concrete context whatsoever. Five plus four is always nine, whether it is five apples plus four apples, or five cars plus four cars, or five miles plus four miles. The concrete setting — apples, cars, or miles — has nothing to do with what 5 + 4 is.  Five plus four is nine, and that’s it. It has meaning all by itself. It is one of the ultimate permanent properties of number.

But Mr. M shows us that, whether or not numbers are real things, the platonist’s concept of numbers as abstract entities is not independent of one’s brain. For Mr. M, such numbers do not exist.


Dehaene’s book is an account up to 2011 of what cognitive psychology and neuroscience has revealed about number sense and how we create it. Much of this is made possible by advances in brain imaging techniques that provide highly detailed pictures of the functioning of the brain (see the notes at the end of this post).

Thousands of studies have shown that the adult brain has an intricate network of neural circuits that react to numbers. The studies show that numerical undertakings activate localized regions throughout the cortex, and that these regions are found in the basically the same physical location in every brain.

One fairly specific region that reacts to numbers is located at the back of the brain in the groove between the two hemispheres. It is called the HIPS region (short for the horizontal part of the intraparietal sulcus). It seems to be almost exclusively devoted to number size and magnitude. In other words, it is one of the regions that reacts to cardinality — to numbers as quantity as opposed to numbers representing order or sequence.

The HIPS region doesn't appear to react to anything other than numbers. And it reacts no matter what mode is used to input them — spoken words, printed Arabic numerals, collections of dots. You cannot even sneak a number into the brain without activating that area. It reacts to numbers even if you are not consciously aware of them — the briefest subliminal exposure of an Arabic numeral hidden amongst a stream of other visual stimuli activates the HIPS region while the same stream without the numeral has no effect.

It's satisfying to discover that a certain part of the brain is tuned to a notion so fundamental as cardinality. But there are some surprises: the relationship between the activity in the HIPS region and cardinality is not so neatly packaged as one might hope.

To illustrate the complexity, take a few seconds to solve the following simple puzzles. In each puzzle, you have to place the correct symbol, either = , or < , or > ,  in the square to make the math correct. As usual, the symbol < means "less than," while > means "greater than."

The puzzles are not very challenging. But did you attend to the length of time it took you to solve them?  Here is what I think happened:

Puzzle A: You solved it instantaneously.
Puzzle B: You solved it quickly, but you performed a brief double check.
Puzzle C: You counted the dots in some manner (or you just skipped this puzzle altogether).
Puzzle D: You solved it instantaneously.
Puzzle E: You counted the dots.


The speed with which you solved these puzzles echoes the activity in the HIPS region, and the correlation is quite strong. The greater the activity in the HIPS region the quicker you will have solved the puzzle.

What is happening is this: when you are presented with two numbers that are small and different (puzzle A) or numbers are large and fairly far apart (puzzle D), there is strong activity in the HIPS region. For numbers that are close in magnitude, the activity in the HIPS region diminishes as the numbers get larger (puzzles C, and E, and even B to some extent).

What you have just experienced is three separate ways that we compare magnitudes. As adults, we typically do it by counting to obtain the exact magnitude, but that is not the only way.

We solve puzzle D without actually knowing the exact magnitudes. Cognitive neuroscientists call this the approximate number sense and describe the quantities as analogue magnitudes. They use a pan balance (an analog device) to model what is happening.  A pan balance can readily distinguish between, say, a collection of 60 one-gram tokens and a collection of 80 one-gram tokens without knowing the actual number of tokens in each collection. However, when comparing 60 tokens with 59 tokens, the pan balance cannot so easily distinguish the difference, just as the HIPS activation is weak when two large numbers are close in magnitude.

Puzzle A illustrates yet a third way that we compare magnitudes. Small numbers seem to have a special treatment by our brains. Although you may think that you just rapidly counted the dots, it is unlikely that you really did. Psychologists refer to the ability to accurately estimate small quantities without counting as subitizing.

Young children can subitize the difference between 1, 2, or 3 objects. Many animals also have this ability, and both children and some animals have an approximate number sense. What young children do not have, and what animals do not have, is the adult human’s precise sense of cardinality. Older children acquire it but animals never do. Humans can eventually learn to compare magnitudes of virtually any size — we learn how to count, and we learn what counting means. This is a number sense that we are not born with — it is a cultural artifact — we acquire it through education provided by our parents and teachers.

If we are given a bag of marbles and we count them 1, 2, and so on, until we reach the last marble with a count of 31, then we learn that 31 represents precisely the quantity of marbles in the bag. We learn that every bag of marbles has a precise quantity associated with it. And if for a second bag the last number reached turns out to be 34, we know that second bag has more marbles than the first one.

Tribes or groups of humans that do not count, or that only have a very rudimentary numbering system, cannot discern the difference between even moderate quantities when the quantities are close in magnitude. For such people, there is absolutely no difference between a bag of 31 marbles and a bag of 34 marbles.

Don’t you find this intriguing? One of the foundational aspects of mathematics, the notion of a precise cardinality, is a human creation.

Some additional notes

Brain imaging

In the past two decades new brain imaging techniques such as fMRI (functional magnetic resonance imaging) and MEG (Magnetoencephalography) have been developed that allow the brain to be studied in great detail. These techniques examine the functioning of the brain, not just the anatomy. With fMRI it is possible to scan activity across the entire cortex with a spatial granularity of 1 to 6 millimetres. With MEG we can monitor duration and response times with a resolution in the order of milliseconds. These techniques are noninvasive and hundreds of images can be gathered in a few minutes without danger to the people and patients who have graciously allowed researchers to test them.

The distance effect  

When numbers are large but close together, it is difficult to discern a difference between them. This is called the distance effect.  For puzzles A through E above, using dots enhances this effect, but brain scans and psychological studies show that the distance effect remains even when numbers are presented using Arabic numerals. It takes a bit longer to decide which of 18 or 19 is larger than it does to decide which of 4 or 5 is larger.

Ordinal numbers

As well as using numbers to describe quantity, we also use numbers to describe order. When we want to distinguish between the two different uses we typically refer to them as cardinal numbers and ordinal numbers.

As children learn to count, they are also acquiring an ordinal number sense. When a child counts "1, 2, 3, 4,"  she must understand that order is important, that 1 is always before 2 which is always before 3 which is always before 4.

Brain imaging studies have been used to explore how the brain reacts to ordinal numbers. Instead of asking "Which of these two numbers is the larger?" the question becomes "Are these three numbers in order?"  In one study, the ordering question was presented both symbolically using Arabic numerals, and nonsymbolically using dots.  With Arabic numerals, the brain regions that are are activated for the ordinality task are separate from the regions that are activated for the cardinality task. With nonsymbolic representations, there is considerable overlap.

Implications about education

Education strategies based on cognitive psychology are often viewed as some sort of snake oil. Indeed, I react somewhat negatively when I see phrases like "brain based teaching methods."

But current research, which combines "hard" neuroscience with "soft" psychology, seems to be very solid, There are already important conclusions about how we acquire number comprehension, and I expect that as the science progresses, we will have to rejig current theories about learning.

For example, we now know that children are born with neural circuits that can be adapted for number comprehension, and that they are ready for such accommodations quite early. The research show that infants have an approximate number sense and can subitize small numbers. And contrary to the widely accepted belief about "object impermanence," young children do not think that objects cease to exist when they are out of sight. In other words, children are ready to begin acquiring arithmetical concepts much earlier than prescribed by Piaget’s constructivism.

Another result, perhaps more disturbing, is that recent fMRI studies indicate that not all brains are tuned with the same acuity (thereby raising the spectre that there is such a thing as "math brain").  It is not known whether the differences are due to nurture (practice and training) or nature (biological). The current hypothesis is that it is due to both.

The new cognitive neuroscience does not have all the answers, and there is danger of being too optimistic about what it offers. But it tells us that we should paste handle with care labels on our lesson plans.

Links etc.

Stanislas Dehaene, The Number Sense [How the mind creates mathematics], revised and expanded edition, Oxford University Press, 2011. 

Here are two related online videos by Stanislas Dehaene. The first is an 8 minute interview (French with English subtitles). The second one is long and in French and covers the material in his book.

Where do mathematical intuitions come from?

La bosse des maths 

The following 30 minute video (in English) shows the detail possible with modern brain imaging techniques. Although it is about reading rather than math, it is what led me to buy Dehaene’s book about the number sense. 

2012 WISE Summit Focus

A letter to my cousin

Hi Barb,

Thanks for the link.

I have thought a lot about this controversy (the Canadian Math Wars). It’s a horrible state of affairs, and it should stop.

I'm afraid that I'm going to have to play "Blame the messenger," because articles like this one
pit the general public and some academics against the teachers and our education systems. I think the fault lies with the laziness of news columnists who reflexively go to the easiest sources. Whenever something is written about math education in Canada you can almost guarantee that the most prominent position in the article will be given to Robert Craigen or Anna Stokke deriding the current state of affairs. Controversy sells newspapers and TV programs.

In my opinion, the math wars stem from two things. First, new students entering university are ill-prepared for their math courses (upsetting the math profs because the students do not know the fundamentals). Second, teaching methods have changed since the students' parents went to school (upsetting the parents because they cannot help their children with their homework).

This gives rise to the sentiment that math education was better in the past, and that it is currently in decline. This is bolstered by the fact that Canada’s scores are slipping in international math tests such as PISA. [ PISA = Programme for International Student Assessment. ] The result is a desire to go "back-to-basics" and to confine teaching to the "traditional" ways.

Here is my take:

When I first started teaching at U of A forty years ago, I also found the students to be ill-prepared, so this is not a new perception.

It is true that changes have been made in the way math is being taught, but I don't think it's a bad thing. Just ask a few of your friends about how they did in math at school. I’ll wager almost every one of them will say "I was no good at it!" In other words, we were pretty bad at teaching math in the past, so changes in our teaching methods had to be made. And I cannot understand why someone who says they are bad at math would want things to revert to the way they were taught.

As far as PISA goes, I have doubts that the results are a reliable measure of math education. But even if they are, I don't think we have slipped that much. Canada is still among the better performing countries. Finland is perennially cited as one of the best, and in the 2012 PISA round of tests, Canada was merely one step below Finland. Incidentally, Pasi Sahlberg, a highly respected Finnish educator, recently tweeted "The world needs more Canada."

And by the way, newspaper columnists almost always use the term "discovery math" when talking about some of the more up-to-date teaching methods. It's a very loosey-goosey term that implies that current teaching methods forbid direct or explicit instruction thereby leaving students to flounder helplessly on their own.

I don't believe that this is really happening, especially the part about forbidding direct/explicit instruction. In the past three years I must have read hundreds of blog posts by teachers describing how they taught one or another topic in mathematics. Most of them presented thoughtful routes through the lesson. They all involved direct or explicit instruction at some point, 'though not the way you and I experienced it. 

I guess I can sum up my feelings this way:

I agree that there is room and need for improvement in our math education, and I have opinions about that, but "Forward To The Past" should not be an option.

Regards,

Ted

PS. Somewhat off-topic: A few years ago, the doll with your namesake caused quite a stir when it was programmed to say "Math is hard!" A school teacher told me Barbie should have said: "Math is hard, and teaching math is even harder."

PPS. You can read about my biases against PISA here and here.

Math and History - a comparison of two subjects

History
My grades were not good. They were "consolation" grades — the sort of grades that say: "You showed up for the exam and spelled your name correctly so I’m giving you a pass in the course."

I did not develop a sense of epoch. I had trouble grasping the significance of major historical events. Our history books and lessons were an unitemized list of facts, dates, and names. Wars and revolutions happened, and then they ended, and that was it. There was no connective tissue. There was no pattern. History was a confusing and uninviting subject.

Mathematics
In high school, I liked math and I was "good" at it. My grades were excellent.

My brush with Arithmetic in elementary school was disagreeable. However, in high school computational skills were not front and centre, and I got along extremely well without them.
My Algebra and Geometry and Trigonometry courses made sense. There were clear bonds between them. There was a sense of inevitability as we worked through the material.

 * * * 

Of course, the description of my History courses is not accurate, but that’s how I reacted to them. So I ask: 

What sucked the joy out of my History courses? 

I do not know the answer.

Now go back to the first two paragraphs of this post and replace the word History with the word Mathematics. Does this describe how some students feel about math?

So what sucked the joy out of their Mathematics courses? 

I don’t know the answer to that question either.

How did you learn your times tables?

When I was a kid, I liked playing ice hockey. I was actually not very good at it — no Connor McDavid here!  But I did acquire the basic skills. For example, I figured out how to lift the puck. (For you non-hockey players, that means shooting the puck in such a way that it flies off the ice into the air. It’s a essential skill if you want to be able to score goals.)

I practiced that skill a lot. Whatever I did, it worked. I could lift the puck consistently without thinking about it. I haven’t shot a puck for many decades, but whenever I imagine doing so, I swear can feel the memory in my triceps.

Of course, I really did not "figure out" how to lift the puck. I did not know the theory behind the lifting action. And to the extent that the skill was necessary, I didn’t need to understand the theory.

The lesson is this:

When learning something new that you will need for later use, master the mechanics first. You can learn why it works later.

* * * Warning: possible straw man ahead * * * 

It’s a useful lesson.  It helps me understand the approach to mathematics teaching advocated by the back-to-basics people: You can be successful by learning the how without understanding the why. Just learn the essential basic facts and algorithms. Don’t worry about why the puck flies into the air — just practice shooting enough so that you can lift it consistently and effortlessly.

Reasonable advice? Maybe. But, no matter how hard I practiced, I could not always "lift" the multiplication tables. As far as the basic multiplication facts are concerned, I do not have what some people call rote recall — I do not have the ability to rapidly and effortlessly retrieve all of the basic learned facts from memory.

A great chunk of my own elementary math education was founded on the contrary belief, that rote recall is, in fact, achievable by everyone — that all it takes is practice.  Accordingly, my classmates and I were regularly drilled and tested on the multiplication tables. I did not do well, and I argued with my teachers. Ultimately, I was punished for my inability to memorize the 12 x tables.

I don’t believe that my recall difficulties are exceptional. The more blogs I read, the more I suspect that there are many people who, no matter how much they practice, will never possess rote recall of the basic arithmetic facts. In that sense, those people can never know the basic facts.

So, it was with interest that I read that Nikki Morgan, the secretary of state for education in the UK, has decreed that:

"we are introducing a new check to ensure all pupils know their times tables by age 11"

An interesting post by @thatboycanteach asks what it means to "know" the times tables. Like me, he suffers from what might be described as rote recall deficiency. And like me, he survived (and even thrived) by using various work-arounds to compensate.

The UK times-table test will be computerized and time-restricted. It looks like it will be based on pure rote recall. For the flunkies, there will undoubtedly be some sort of penalty. It’s unlikely that they will be physically punished like I was, but even non-corporeal punishment can inflict great stress and harm and, in the end, may prevent them from learning mathematics.

What is of concern to me in Alberta is that, however sincere the back-to-basics people may be, they seem to be basing their reform efforts on the very thing that caused me difficulties, namely, the belief that all students can and must achieve rote recall, that this is the only way to know the basic facts.

That is the conclusion that I draw from reading their petitions and press releases. If I’m wrong, if I am raising a straw man, it is difficult to understand why they also want to banish the teaching of alternate approaches to the basic facts and algorithms that are needed so that people like me can compensate for our deficiencies.

Math fair workshop at Banff

The 14th annual math fair workshop at BIRS will take place over the weekend of May 6/7/8, 2016.
(BIRS = the Banff International Research Station.)

Right off, let me say that I have a pretty bad attitude about school science fairs.  You know — those competitions with poster sessions, baking soda volcanoes, and parent-created displays. The ones that end with an obligatory showcasing of a winner — a bright student who looks like he/she will go on to become the next Neil deGrasse Tyson, and who, for a short while, will be a poster-person for our education system.

OK, that's harsh, but it is still very much the norm to single out a winner.

How about having one that does not overly favour the highly talented? One that even a less confident student would enjoy and not end up feeling like a failure because he or she did not win a medal.

If you’re like me, you do not enjoy being tagged as a loser, and you would likely withdraw from a situation where that is liable to occur. Aviva Dunsiger touched upon this in her blog. Although her post is about phys-ed rather than mathematics, she paints a clear picture of the response to anticipated failure:

Yes, there were always strong athletes, but those that struggled (and I was one of them) wanted nothing to do with phys-ed. With my visual spatial difficulties, games like volleyball, basketball, and baseball were a tremendous struggle. I certainly never got picked for a team, and I couldn’t blame anyone. Why would I want to be physically active if I was only going to meet with failure?

[the emphasis is Aviva's]

Can we have a math fair where students can be mathematically active without the anticipation of failure?  One where students do not need a badge or ribbon to confirm that their efforts have paid off ?

Such math fairs do exist. They’re called SNAP math fairs because they are Student-centred, Non-competitive, All-inclusive, and Problem-based.

The fairs are built around math-based puzzles. The students first solve the puzzles^[1] and afterwards prepare the artwork and puzzle pieces that are required to display them.  

Visit such a fair and you will find students manning their puzzles. But, you will not see them exhibiting the solutions. Instead, they will invite you to try the puzzles yourself, and they will give you hints and help when you run into difficulty. The math fair is very interactive. It is much more than a poster-session.

* * *

Here are a couple of puzzles from past math fairs. The first one is for younger students to solve. 

Cats Pigs and Cows

A farmer has nine animal pens arranged in three rows of three. 

Each pen must contain a cat, a pig, or a cow. 

There is already a pig and a cat in two of the pens. 

The farmer wants you to fill the remaining pens so that no row or column contains two of the same animal.

The second puzzle is for older students.^[2]

The Sword of Knowledge

The dragon of ignorance has three heads and three tails. 

You can slay it with the sword of knowledge by chopping off all of its heads and all of its tails. 

With one stroke of the sword, you can chop off either one head, two heads, one tail, or two tails.

But the dragon is hard to slay !! 

• If you chop off one head, a new one grows in its place. 
• If you chop off one tail, two new tails replace it. 
• If you chop off two tails, one new head grows. 
• If you chop off two heads,  nothing grows.

Show how to slay the dragon of ignorance.

* * *

A SNAP math fair is remarkably adaptable to many different circumstances. If you are interested in learning about how you can incorporate a SNAP math fair into your own teaching environment, come to the BIRS workshop. You will meet teachers who have organized math fairs in their own schools. You will also meet a few mathematicians who have taught courses in which a math fair was key ingredient. 

As well, there will be math fair resources available, and the participants will be involved in puzzle-solving sessions.  

The BIRS workshop has room for about 20 participants, and it is oriented towards (but not limited to) K-9 teachers.

For more details about SNAP math fairs, visit the SNAP math fair site. And while you are there, take a look at the Gallery to see how students react.

For more information about the workshop, and who to contact, the link is here.

End notes

^[1] The solving part is a crucial element of the math fair.  Ideally, students solve the puzzle by themselves. They are surprisingly persistent.

^[2] I imagine the Sword of Knowledge puzzle would work with junior high or high school students. However, I once visited a SNAP math fair where two grade five students had solved it. Their teacher told me that they struggled with the problem but solved it after one of them grabbed a handful of pencils (tails) and erasers (heads).

How to divide by 2

Holy Moly! I thought I pretty well knew everything about dividing a number by two. I was about to hit the publish button, but I reckoned I should first do a quick scan of the web. Again, holy moly !!

This post was initiated by my watching a very skillful carpenter reface our fifteen year old kitchen cabinets. He was a "measure twice, cut once" sort of guy. In the course of his work, he did a lot of marking and checking of centre lines so that handles and panels could be precisely located.

Locating a centre line comes down to finding the midpoint of a measured length, in other words, dividing a number by 2. How he did this might surprise you, as did the advice I encountered on the web.

The measurements to be halved are usually mixed numbers. I happen to have a board that is 9⁷⁄₈ inches wide, and I asked my wife (who is not a mathematician) how she would find the centre line. This is how she explained it using her usual yardstick (which has a resolution of one-eighth of an inch and which dictated her approach).

The midpoint would be half of 9⁷⁄₈ .
Half of 9⁷⁄₈ is half of 9 plus half of ⁷⁄₈, which is 4¹⁄₂ inches plus 3¹⁄₂ eighths.
So measure 4¹⁄₂ inches and tack on an extra 3¹⁄₂ eighths to get the midpoint:

I imagine that a carpenter might do it in the same way, except that he or she would be using a tape measure with a higher resolution and would likely think of the midpoint as ⁷⁄₁₆ inches beyond the 4¹⁄₂ inch mark (rather than 3¹⁄₂ eighth-inches).

There is another way which comes to mind: 9⁷⁄₈ is the same as 10 − ¹⁄₈,  so half of 9⁷⁄₈ is the same as half of 10 minus half of ¹⁄₈,  which is 5 − ¹⁄₁₆ ,  in which case you would probably find the midpoint by locating the 5 inch mark and backing up ¹⁄₁₆ inch. I would have used this way myself, and I suspect some carpenters would do it this way as well.

What did the web say?

On the web almost every "explanation" of how to divide a mixed number by a whole number reduced the problem to dividing two fractions using the invert-and-multiply trick. According to these posts, dividing 9⁷⁄₈ by 2 should be done as follows:

1. Convert the mixed number to an improper fraction: 9⁷⁄₈ = ⁷⁹⁄₈.
2. Convert the 2 to an improper fraction: 2 = ²⁄₁.
3. Do the division  ⁷⁹⁄₈ ÷ ²⁄₁. (by which they meant:  ⁷⁹⁄₈ ÷ ²⁄₁   = ⁷⁹⁄₈ × ¹⁄₂ = . . . ).
4. Convert back to a mixed number: ⁷⁹⁄₁₆ = 4¹⁵⁄₁₆.

By the way,  my wife said she never really understood the invert-and-multiply thing. She said she would change the two fractions so that they had a common denominator and then divide the top numerator by the bottom one. For example, to find ²⁄₃  divided by ³⁄₅  she would reason as follows:

  ²⁄₃   divided by ³ ⁄₅  is the same as ¹⁰⁄₁₅  divided by ⁹⁄₁₅ ,  which is the same as 10 divided by 9, or ¹⁰⁄₉ .

What is somewhat astonishing is that I found two sites that described an algorithm designed precisely to solve our very specific problem, namely, how to divide a mixed number by 2. The algorithm differed according to whether the whole part of the mixed number was even or odd.

Here is how it applied to dividing 9⁷⁄₈ by 2:

1. Divide the whole part of the mixed number into half (ignore the remainder): 9 ÷ 2 ➞ 4. 
2. Add the numerator and denominator of the fraction: 7 + 8 = 15.
3. Double the denominator of the fraction: 2 x 8 = 16.
4. The answer is 4¹⁵⁄₁₆

Most of the web stuff mentioned above never really explained why the particular algorithm worked. And I did not encounter any post on the web that used the distributive law to divide a mixed number by a whole number, that is, no-one suggested doing what my wife and I did:

Which method did the cabinet installer use?

Answer: None of the above.

Instead, he used a self-centering tape measure. This is a tape measure that has two number lines on it. The top one in black shows standard feet and inches, and the other one directly below it in red shows the half measurements. Here is the board being measured by a self-centering tape:

It shows that the width of the board is  9⁷⁄₈  inches and that centre line of the board is at the 4¹⁵⁄₁₆  inch mark. The point on the centre line can then be immediately marked on the board without actually doing any computations.

* * * * *

As I was rewriting this post, I encountered a couple of tweets by John Golden (@mathhombre) and Denise Gaskins (@letsplaymath) that directed me to their posts* about Richard Skemp's work which seemed to be relevant to what's going on here.  (Thanks.)

Skemp observed that people lean in one of two opposite directions when they describe what it means to "understand" mathematics. He called the one way an instrumental understanding, and the other, a relational understanding. (A detailed summary can be found in the posts mentioned below.) He contended that the way you tilt affects both how you learn math and how you think it should be taught. A person with an instrumental viewpoint would tend to think of math and teach it as a collection of rules to be memorized and applied. A person with a relational viewpoint would likely think of and teach math as exploring the connections between various parts of the subject.

If I grasp Skemp correctly, the stuff from the web that I mentioned above falls very much to the instrumental side while the approach that uses the distributive law is more relational.

The carpenter’s use of a self-centering tape would also appear to reflect an instrumental view of mathematics. But not necessarily—it could simply be a tradesman using a tool that simultaneously decreases the chances of making errors and increases the speed of doing the work.

* * * * *

* The posts by John Golden and Denise Gaskins are here and here. Also, the posts about Skemp by David Wees and Gary Davis are definitely worth a read. 

Apples and oranges and the number line

You can add apples to apples and oranges to oranges, but you cannot add apples to oranges.

One of my teachers used this old chestnut to explain that we had to convert to the same units before adding similar quantities. It doesn’t make sense to add 2 and 6 to find the combined volume of 2 quarts and 6 gallons — before adding, you have to convert everything to quarts, or everything to gallons (or, perhaps, everything to litres).

Apple + apples = ?

Of course there are circumstances where it makes perfect sense to add apples to apples, but there are also a lot of situations where it doesn't.

What do you get when you add my PIN and my wife’s PIN? Or my brother's telephone number and my sister's telephone number,  or the grocery product numbers for tomatoes and cucumbers? It seldom makes sense to add numbers that are used as identification labels.

But even when the numbers are more than just ID numbers, it may still not make much sense to add them. Would you perform the following additions?

• Hours to travel by bus + hours to travel by car from Edmonton to Regina.
• Average pay in Alberta + average pay in Ontario.
• Johnnie's  Algebra grade + Tina's Algebra grade. 
• Life expectancy for a person born in 1990 + life expectancy for a person born in 2015.
• Current Montreal time + current Vancouver time. 
• Cruising altitude for flight 107 + cruising altitude for flight 108. 
• Sunrise time + sunset time for today in Saskatoon. 
• Number of voters in the 2011 election + number of voters in 2015 election. 

Apples − apples = oranges ?

The odd thing is that in each of the above cases, although it makes little sense to add the numbers, it does make sense to subtract them. Here's another example.

I’m at the corner of 105 avenue and 34 street. If I walk along the street from 105 avenue to 112 avenue, how far will I have to walk?  

You would probably answer "seven blocks." Here you are subtracting an avenue number from an avenue number and getting a distance expressed in blocks.

Even when the result of a subtraction uses the same language to express the units of the answer, the units are sometimes measuring something different. Consider the following:

The temperature in Calgary at 10 AM today.
The temperature in Calgary at 4 PM today.
The difference between these.

Although the three measurements are expressed as degrees Celsius, the first two have a different meaning than the last.

Apples + oranges = apples ?

And there are case where you can add apples to oranges. (There’s the useful observation that adding 2 apples to 3 oranges makes sense because it gives 5 things, but I’m not thinking of that.)

Dinner Time 

Q: What time is it?
A: 5 O'clock
Q: How long will it take to cook the roast?
A: 2 hours. 

Did you just add hours to O'clocks?

Number Lines

I imagine that most people, when asked what 5 + 2 means, conjure up what I would call an aggregate model, like a collection of 5 tokens together with a collection of 2 tokens.

We tend to think of numbers as describing an aggregate or assemblage — the number of marbles in a box, or the total weight of your luggage.

In the Dinner Time example, above, the numbers are not being used in this way. Instead, one number describes a position (time of day) and the other describes a movement (a duration of hours). If you were asked to explain to a child why 5 O'clock + 2 hours is 7 O'clock, perhaps you might draw something like one of these:

These are variations of the familiar number line¹, where numbers are depicted in the two different ways that numbers are often used:

1. as a position on the line, with positive numbers to the right of zero, negative numbers to the left, and
2. as a directed distance or movement (a vector), drawn as an arrow pointing left or right. The length of the arrow represents the magnitude of the number and the direction indicates the sign of the number (right = positive, left = negative ).

Addition

On the number line, addition is thought of as

current (or old) position + movement = new position,

as, for example, in

current bank balance + deposit = new balance.

With a number line, addition of 5 + 2 and 3 + (−7) is represented as:

For addition, the movement arrow is drawn with its tail at the old position. Its head indicates the new position.

Subtraction

By far the most common application of subtraction is to describe the movement that separates one position from another:

new position − old position = movement,

as in

Best before date − current date = days left before milk goes sour.

With a number line, here's one way to represent 2 − (−4) and 8 − 5:

As with addition, the arrow is drawn with its tail at the old position and its head at the new position.


Children typically learn subtraction as taking-away, which is demonstrated with an aggregate model by removing a certain number of tokens from a larger group and counting what's left. In fact, so strong is this interpretation of subtraction that the language often persists throughout the rest of one's life.

The take-away model of subtraction, can be described as

old position − movement = new position.

It is also possible to use a number line to model this interpretation of subtraction. Here's what 8 − 5 would look like (8 = old position, 5 = movement):

In this case, subtracting means to travelling backwards along the movement arrow to reach a new position.

The arrow is drawn with its head at the old position while its tail indicates the new position.

This is almost tragically different from the previous illustration for 8 − 5. Apart from my own kids, I've never taught arithmetic to children, but I cannot help but think that the difference will confuse a student. Moreover, with the position-minus-movement interpretation, the overwhelming temptation is to reverse the direction of the movement arrow, thereby conflating 8 − 5 and 8 + (−5), which may lead to further confusion.

So what?

Most people work with numbers in an abstract way: 8 − 5  has no meaning unless the 8 and 5 refer to something "real," yet we have no hesitation in saying that 8 − 5 is 3. We do this without reference to any "real" setting whatsoever, and we do not revert to using a model to do the computation. We have completely abstracted the operation of subtraction from real life.

However, it seems that we need to give meanings to numbers in order to learn even the most basic arithmetic facts. But when we do assign meanings to numbers, we encounter the strangeness described in this post, namely:

• Two numbers can measure exactly the same thing but it may be nonsensical to add them. 
• Two numbers can describe quite different concepts, and yet it may make sense to add or subtract them.  
• Two numbers that don’t make sense when added can make perfect sense when subtracted.

I can't recall being taught about this strangeness when I was a student, and I don't know if it is explicitly dealt with by today's teachers, but it is conceivable that it could be a source of confusion.

To some extent, the number line model deals with this strangeness, but I haven't seen it used to this end.

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¹ Some number line models use only arrows. Positions are replaced by arrows whose tails are at zero.