I was at the Anja S. Greer Conference on Mathematics and Technology at Phillips Exeter Academy this summer, and I loved it.  Not only did I get to think about math and math teaching all day, but then Happy Hour meant refreshment and more great problems.   Like when Frank Griffin posed this problem:
 
How many regions are formed inside a circle when you draw chords from a given number of points on the circumference? 
 
With one point on the circumference, there are no chords and one region inside the circle. 

With two points on the circumference and there is one chord and two regions. 

Add a third point and draw all possible chords.  How many regions?  There are four regions inside the circle.

How about 4 points?  And so on. At some point there are multiple answers, so we are always looking for the maximum number of regions you can create for a given number of points.  Can you predict how many regions for 10 points? 45? n points?
 
 
That’s the problem.  As I began to work on it, I found myself on the same kind of path I often follow when working on a rich and challenging problem (in fact, it is the reason that we often refer to such problems as “rich”, like a rich vein of ore): not a direct path to the solution, but a branching course, with a lot more trial and error than my students probably imagine math problems are supposed to involve.  I decided to document my path through the landscape of this problem, both for myself and as an example for my students to follow.  So what follows will involve a lot of spoilers.  Stop here if you want to work on the problem on your own.
 
For my students, I want to describe the process of mathematical investigation just a little further.  It is like trying to cross an unfamiliar landscape, where some paths may take you from your camp to a cave on the ridge, but many more will not.  You have to find the right path.  So you look up at the ridge you want to reach, and look at what is just below it.  Then you make a guess:  what looks like a likely path to the top?  Now you have a new task – try to get to the bottom of that path.  In order to get to the bottom of the path you have to cross the river right in front of you.  So now the problem becomes how to get across the river.  You can see some rocks to step on close by, but you can’t see what is beyond that.  So you walk out as far as you can…
 
And that is how it goes.  You keep changing your focus, from far away to the step right in front of you.  You make guesses about what might be useful, and you make discoveries as you journey further along that may or may not change your path.  Your one question – How do I get from here to there? – sprouts numerous smaller tasks and problems, and you try to solve those.  In the meantime, depending on how curious you are and how much time you spend exploring, you get to know the lay of the land, more than just the path you are searching for.  This is how all the various subjects of mathematics (in fact, we often refer to these as “fields of mathematics”) have been discovered and developed, by people following the breadcrumb trail of questions that lead to further questions.  It literally never ends.
 
 [THE SPOILERS START HERE]
So, the first thing that I discovered was that the pattern that appears is not the pattern it appears to be.  1, 2, 4, 8, 16, 31… Wait a second. How does that work?  In looking at the differences between the terms, and looking at those differences, I realized this was not a geometric function, but it was a polynomial of degree 4.  Okay.  That’s weird.  But it also was my first discovery and new idea… or rather my first new question.  This suggests that just because I have 7 terms of 2^n, or even 17, that doesn’t mean I have the function 2^n.  And here come the questions:  is that generally true?  Can I make the change at any term?  Can I change that term to any number?  Can I do it to any power sequence?  This seems to suggest that polynomials can approximate a geometric sequence to arbitrary accuracy… is that useful to anyone?  I don't know, but it is some mathematics that I understand in a different way because I walked here by myself.  I haven't yet walked much farther down that path, but I may come back to that someday.

When we teach math, what are we teaching?  I think of it in layers.
 
On the surface, students learn to solve problems of increasing complexity and sophistication, using mathematical tools of increasing sophistication.  They learn how to multiply, add fractions, solve equations, factor quadratics, etc.  But the goal of math education has to be more than that.  In fact, if that is all we have achieved, we have literally failed.
 
So what is below the surface?  Mathematical thinking.
 
There is a reason that we have fields of study that have endured for thousands of years.  Literature, music, science, philosophy, mathematics all contribute a useful and unique way to view and process the world.  Mathematical thinking is a way to deal with quantities and patterns that can be valuable in many contexts.
 
Broadly, I think there are three areas of mathematical thinking we deal with in primary and secondary education.
 

• First of all we teach numbers.  What is 3, 4/5, -7, or sin(2π)?  Starting with whole numbers, we help students understand what these quantities represent, and what their existence implies.  There are relationships within and between number families that students need to explore and investigate on an ongoing basis.
  

• Secondly we teach operations.  We have to remember that the symbols on the page are code for something else.  Think about what is written here: 4/5.  There is a 4, a line, and a 5.  The symbols don’t tell you anything about what 4/5 means.  When we teach operations like addition or multiplication, we can teach those operations in a purely symbolic way: what to do with the various numbers in different positions on the page.  What does it mean to divide 4 by 5?  What does it look like?  When we multiply two fractions, what does it mean to take 2/3 of 4/5?
  

 
I want to be clear: I am not saying we need to teach students why the various algorithms work (perhaps with the purpose of having them remember them better).  I am saying we want students to understand what the operations we use in mathematics represent.  When we divide 4 by 5, what are we actually doing to the 4, and what role does the 5 play in the operation?  Mathematics describes and uses certain operations on numbers (and algebraic expressions) that our students should explore and understand.

• Finally, we teach generalization.  Sometimes people refer to this as abstraction.  The process of noticing patterns and making up “rules” or routines that apply in general occurs on so many levels – all students have an opportunity to do it on a daily basis in math class.  The process of observing something, finding patterns, and making generalizations, is a process called inductive reasoning.  The concept is powerful but not perfect - the further we get from pure numbers the more carefully it should be applied.  But this is the crown jewel that mathematics and we math teachers have to offer the world.  I would say that there is no point to teaching math if we do not teach this.  And we have important choices to make about our teaching which will determine whether it is present or absent from our classrooms.  If we train students in manipulating symbols on a page, they have no opportunity to experience the process of generalizing, let alone learn to do it themselves.  But if we talk to students about the process and ask them on a daily basis what they see, what they notice, and help them grow in the ability to make conclusions and develop algorithms, we can grow their skills in generalization.
  

 
What is most important?  I think we have to have a zen-like duality in our minds, keeping a close eye on both computational competence and mathematical thinking.  Both measure our progress, but either one alone is not enough.  And they are certainly not mutually exclusive.  Does it hurt students to teach them an algorithm?  Not if you also build a classroom culture in which students understand that doing mathematics means exploring and reasoning and building generalizations for themselves. 

On Science Friday last July they did a segment about lollipops, which included discussion of a formula for how long it would take to reach the center of a tootsie pop (an old ad campaign slogan).  I know there are probably a number of teachers who use lollipops as a context for some data collection and analysis, and it made me think I would try that.  If anyone has some good sources for materials, I would love to hear in the comments.
 
It also brought to mind a possible new species of Math Talk/3 Act.  One could present a scenario/video/picture, and ask students about the relationships between the quantities.  I have seen questions like this in other materials, like the qualitative graphing in the Shell Centre materials.  But I realized you could revisit this kind of qualitative analysis regularly using a 3 Act type of presentation. Some possible roads of inquiry:
 
-What quantities do you see?  Which are important?
-What quantities do you see that are changing?
-Which quantities are related, and how are they related?
-Take a pair of related quantities and sketch a graph of their relationship.
-If you were to write a formula for this, what variables would you use?  What operations do you think might be involved?
 
Once you have introduced and worked with some formulas, you could start to expose the students to formulas from various fields and they would have a better appreciation for what they represent and how they came to be.  The students can explore and get a feel for the difference between direct and indirect variation, and they could begin to see why some formulas included constant coefficients.  And, perhaps most important, these kinds of discussions would help build their sense of what a variable is all about, why you sometimes have more than one, and why we bother with variables, functions, and equations in the first place.

A few days ago, we were coming to the culmination of our discussion of division for the year.  We have taken each of these topics in turn and explored them:
sharing a whole number of objects with a whole number of people (5÷7)
the equivalence of a division problem and a fraction  (7/8 = 7÷8)
finding a fractional group of a whole number (2/3 of 12)
finding a whole number group of a fraction ( 5 groups of 3/7)
how many groups of a fraction go into a whole number ( 12 ÷ 1/4)
how do you divide a fraction into smaller units (1/4 ÷ 5)

I put this up on the board (without the colors or the words, those came out of the discussion):

I asked the students to notice any patterns in the problems as a group.  What was the same and what was different?

S:  All the numbers are even.  Oops, no they're not.
S:  They all have fractions in them...except for 4 · 2
S:  A lot of the fractions have 1's.
S:  Wait a second... aren't 1/3 ÷ 4 and 1/3 · 1/4 equal to the same thing? OHHHH, I get it!

At this point, I realized, once again, that their minds just don't work like my mind.  They don't see what I think they should (didn't we just spend 20 minutes figuring out that these problems were equivalent?), and more importantly, they don't think the way I want them to... yet.  The pattern we are looking for here is not adding squares to a figure, or a decreasing sequence.  We are not comparing obtuse triangles and acute triangles. We are looking for a pattern much more abstract, involving operations and numerical relationships.  And learning to think about this kind of pattern is one of the biggest reasons we teach mathematics (or should be!).  I was pretty stoked.

So we continued to discuss, and we found that the labels, especially the word "divisor", came in handy here, and one student suggested that you could get the answer to a division problem by "multiplying by the reciprocal of the divisor."  I was thrilled.  We had arrived at the ageless trope, bane of anyone who dreads top-down, teacher driven math instruction, through the work of the students.  I wanted them to get there; it seems a useful tool, especially in later algebra classes.  But I wanted to reap the fruit of having them think it out as well.  And I will do it the same way next year.

HOWEVER..
The next day I gave them some problems to practice the skill. 
S:  Can you remind me how to do this again?
S: Yeah, do you flip the front one or the back one? 
S: Don't you flip both?

REALLY?! And of course, if your classroom is like mine, and we are completely honest, this happens time and again year after year.  I want the students to understand math and always grow their mathematical thinking, their map of the math landscape.  And I want them to continually add to their mathematical toolbox, have computational competence.  I want both.  And I have always believed as many of us do, that introducing the computational procedures at the start ends the thinking.  So we explore first, and the computation evolves from what we discover.  THEREFORE, the exploration and understanding must enhance the student's competence with computation.  If they understand it, they will remember it better and use it with more facility. 

I am not so sure anymore.  Why does it keep happening that we derive the formula for the area of triangle over 3 days of work, and half the class needs a reminder the next day?

WHAT'S GOING ON
1) The connection between Understanding and Recall is not straightforward.  We assume that understanding why will reinforce remembering how, but I am not sure it is that direct a relationship. I suspect those two parts of the brain communicate only loosely with each other.  And I suspect that we make important, but perhaps erroneous and misconstrued assumptions about how these work, like assuming that building understanding is enough to give students lasting skills. 

2)  The symbols are different than the math.  That means the students often don't see the same relationships in the symbols that they do in the math or the models.   Consequently, the students often aren't thinking of the symbols the way we think they are. We want students to grow their mathematical thinking and understanding and experience with mathematical ideas.  We also want them to gain operational knowledge for using mathematical symbols, and again I think the connection is much less direct than we imagine.  I know that if I ask my students to show me 3/5 of 60, I get this:

And if I write 3/5 x 60 on a piece of paper, I get this:

And in the second case, half of the students will want a reminder or want to be reassured that they should put the 60 over 1.  Or if I ask them to reduce 45/7, they will easily give 6 and 3/7 as an answer, but if I ask them to find 45÷7, they will become dismayed as the decimal answer drags on, or if I ask for a fraction answer, will round off the decimal or simply say they can't do it.  The symbols are different than the math for all of us, but even more so for students encountering both for the first time.

So I think we need to give more nuanced thought to the connection between mathematical thinking and remembered procedures, the purpose and role of each and the nature of how they interact with and reinforce each other... or don't.

Misconception #2:  “Multiplication and division are opposite operations.” 
This is true in a sense, but the implicit message is that multiplication and division go in opposite directions.  This becomes a problem when we could use the equivalence of m/3 and 1/3 m.  Both are algebraic representations of this relationship:

At a recent conference, a presenter was talking about pretesting students so that she could provide differentiated instruction.  She was looking for, among other things, students who could recognize unreasonable answers with the knowledge that division makes numbers smaller.  But of course, when you divide by a number less than one, you get an answer larger than the dividend.  As we introduce the concepts of multiplication and division to students, there must be a way to do it without embedding ideas that are dead ends, or incorrect. The equivalence of ÷ 3 and x 1/3 or ÷ 1/4 and x 4 is an important concept that we should be aiming at from the start.

Underlying both multiplication and division is the process of grouping quantities.  We can make whole number groups, and we can also make fractions of groups; parts of groups.  Are there are two different ways to represent the same event with symbols.  If I have 9 objects and I separate them into three groups and focus on one of them, I can represent that action symbolically as 9 ÷ 3 or 9 x 1/3.  Would it be possible to use both representations from the beginning?  In fact, fractions themselves could be interpreted from the beginning as division problems with just a little tweaking I think.  Instead of only focusing on 1/5 and 2/5 as one out of five and two out of five, could we highlight the concept of division when we introduce fractions?  When you take a whole, or a group, divide it into 4 equal shares and focus on one of them, that is 1/4.  When you take 3 wholes or groups, divide each into 4 equal shares and focus on one of them, that is 3/4.  We certainly don't expect 3rd graders to master this concept, but they could explore simple examples.

The difference it would make in my class is that I would not have to unteach the clunky operations for 4/5 x 95 that many of my students cling to.  And more importantly, I think it could help shrink the divide between the students who feel confident and may easily adopt mental math strategies and routines on their own, and the students who might understand strategies like dividing first with 4/5 x 95, but still feel more comfortable putting a 1 under the 95 and "multiplying straight across."

I have become suspicious of conscious thought.

Here are my case studies:
Learning to hear better whether a note is sharp or flat - I am learning to play double bass, and without frets, it is important to be able to hear whether a note is sharp or flat relative to the other notes you are playing.  It used to be impossible for me to tell.  I had to move the note around and hear if it got worse or better. I have been working for months to improve my ear, doing exercises for a few minutes a day, and I am noticeably (though unfortunately not dramatically) better.  I am not "figuring" this out.  I am learning this in a totally unconscious way, and when I "know" the note is flat, I just know, I can't tell you how I know.

Similarly, when learning to ride a bike or hit a fast ball, any of the physical skills, there may be some conscious thought going on, but the skill is largely non-verbal and out of our conscious control.

But physical skills are not the only phenomenon where unconscious thought prevails.  When solving problems using bar models, suddenly the sequence of steps makes sense and the relationships literally appear in front of you, relationships that were perhaps obscured by the language or mathematical symbols used to pose the problem.  When performing fraction operations using visual models, or studying sequences using visual patterns, you can uncover connections and have flashes of insight that seem to come from nowhere.  How were those constructed?  And how permanent is that new knowledge?

So much of mathematical thinking feels like something you consciously figure out and then you should know it for a lifetime because you understand it.  But this other kind of learning ("processing"? - it doesn't seem right to call it thinking) that is more gradual and less predictable, may have a bigger role in learning mathematics than we think.  How would that change our teaching?

By the time the students get to my pre-algebra class, they’ve  learned "facts" about multiplication and division that lock them in, and actually make it harder to learn the next bits.  The misconceptions or limitations in their thinking show up when we begin to solve equations that involve fraction answers or operations.  I wonder if we shouldn’t start differently at the beginning.  From simplest to most subtle, these are three of the misconceptions that cause the most trouble:

1. There are some numbers that don’t go into other numbers.
2. Multiplication and division are opposites.
3. Multiplication is the same forwards and backwards.

I will tackle each one in its own post.
1) There are some numbers that don't go into other numbers.  This shows up when we get to 5x = 13, and the students tell me “5 doesn’t go into 13.”  We use bar models before we use algebraic notation, so the problem looks like this:

And with the problem visually represented this way, the students can conclude for themselves that dividing by 5 will get them the value of one of the unknown units.  They will happily answer when the division comes out evenly, but when it doesn’t, some will report “I don’t know what to do.” (Have you noticed this? Sometimes, rather than say “I know I have divide by 5 but I don’t know how” the students’ brains seem to back out of what appears to be a dead-end, and deny that it ever existed in the first place.  … Makes me wonder about my own invisible dead-ends.)
 
I have learned to frontload work with division problems with fraction answers at the very start of the year so this doesn’t happen.  We use things like granola bars and sandwiches and we also use bar models to explore cutting things up.  The students need to see what it means to divide 13 into 5 equal shares, or, for 5x = 3/7, how to divide 3/7 into 5 equal shares.  

In a later post, I will talk about the problem with seeing 5 • 3 as the same as 3 • 5, but it comes up here.  There is another way to interpret 13 ÷ 5... or more accurately, 13 ÷ 5 can also represent this question: how many times does 5 go into 13?  Again, this is an easy question that becomes more challenging once fractions are involved.  The students need time to explore what it means for 5 to go into 13 two and three fifths times. 

And here is the rub:  one might ask why bother with all of the exploration and confusion?  Why not just teach the students how to multiply and divide fractions and be done with it?  Because that's not mathematics (see NIxtheTricks).  The symbols represent mathematical situations and mathematical thinking.  If students can only apply the rules to a problem you wrote on the page, and they can't think about it for themselves, or use the rules appropriately in a situation or word problem, then they haven't really learned anything useful. Division and multiplication describe what we do to stuff, and the symbols need to represent those operations, not replace them with math facts.  We will see just what those operations are in the next post.

I am always looking for better ways for the students to expand their understanding of multiplication and division, and be able to transfer what they know about operating with whole numbers to fractions.  Along the lines of Tina Cardone's book Nix the Tricks, I have always hated the phrase "of means multiplication" because it doesn't always, as in "3 pounds of strawberries and 2 pounds of bananas."  But it suddenly occurred to me that "groups of " does often mean multiplication, and that would be worth sharing with my students.  Why does this matter?

- We have been looking at linear equations and the different roles numbers play in multiplication.  3 x 4 looks the same as 4 x 3, but 3 groups of 4 is different than 4 groups of 3.  Going 65 miles per hour for 4 hours is different than going 4 miles per hour for 65 hours.  "Groups of" could be part of our conversation to help distinguish the different roles (or jobs) the numbers play in the problems. 

- 2/3 of 5 is where "of" comes up most often, and is usually followed by someone declaring (with a great show of relief) "Wait, doesn't 'of' mean times?" and the algorithm spreads across the room and the thinking stops.  And 2/3 x 5 seems so far away from 2 x 5, when they are so close.  Partly because no one says "2 of 5,"  and neither 2/3 of 5 or 2 of 5 has much meaning.  But 2 groups of 5 and 2/3 "groups" (we can discuss "of a group") of 5 both give us something we can picture or act out and consider, and allow us to build on our understanding of 'groups of."

- We can carry that on to area (6 rows of 15) ... and I am sure other things will pop up if I commit to using "groups of" or "rows of" etc. 

So we can go beyond a trick with words to language that provides an image, a relationship that we can evolve our understanding around.  It took me 20 years to figure that one out?

Michael Pershan has been writing a thought-provoking series on teaching students about equations this summer.  I find bar models to be a very powerful way to nurture students' understanding of algebraic symbols and the process of solving equations.

How I Use Bar Models
Visual representations make it easier for the students to reason for themselves what steps to take and the order of the steps when solving for unknown quantities.  Algebraic symbols obscure the important relationships for the unitiated, and before I used bar models, I struggled to make those relationships tangible for the students.  And it seemed I was always out in front, coaxing them on.  I couldn't find a way to have them be able to take the lead and really discover things for themselves.  But if I put a problem like this up on the board:

and ask the students what the value the x's represent, almost immediately a very high percentage of the class can jump in and figure it out on their own.  They can tell me why we subtract 47 from 175, and why we have to do it before we divide by 4... and why those are the operations we do.  Once the students get the hang of it, they can solve  some pretty complex word problems, including problems that could be represented as simultaneous equations, with insight and creativity.  Some students prefer bar models to algebra as we start to make the transition, because this makes so much more sense, and contains more information. 
Here is another example:

Representing subtraction can be done in different ways.  This is the way I do it.  The students pick up on that pretty quickly.  Question 2 is part of the way I build bridges between algebraic notation and bar models.  And notice that there are a number of problems that this diagram facilitates.  Suppose I had told you that Andrea had 51 more mangoes than Bill.  Do you see where that would go in the diagram, and how you would find the number of mangoes that Andrea has?  Or the number of mangoes Andrea has, or the number Bill has.  Any of those could be the starting point.  And notice that we aren't actually solving for x, we are asking a slightly more involved question, that might take a lot of groundwork for the students to untangle using algebra, but is almost trivial in the visual context.  I could instead ask how many Bill has, or how many more Andrea has, or what the total is, etc. depending on the information we start with.  This is a pretty deep understanding of how all of these quantities are related to each other, and how to get one from another.

How We Get There
We begin by translating words to bar models, learning to represent "5 times as many" or "6 more" or "shared among 6 vases" and their inverses.  We also represent them algebraically, so immediately the students see a connection.  I also give them algebraic expressions and talk about what bar models represent those relationships accurately.  Then when we are solving equations later, I often reference a quick sketch when a student is stuck.  It seldom has to be a full model of the problem, just an illustration of the part they are hung up on.

I don't use algebra tiles, so that the students don't need manipulatives when there may be none available.  They can just make their own drawing (or I often encourage them to start imagining without drawing as we get further into the year).  I also don't get bogged down in rigorously modeling negative quantities or general approaches to fractions (there are some situations, like 2/3 x = 28, which work very well with bar models... 3x = 5/7 not so much).  For instance, I would handle something like Michael was discussing, a problem of the form b - ax = c, this way: 
    
First, we would treat problems like -7x = 56 as a simple extension of 7x = 56.  If I have 7 x's, and I want to know how much 1x is, so I divide by 7.  So I ask the students what about -7x = 56? Someone suggests dividing by -7, and we check the answer, and it works.  We don't worry about how to represent negatives with bar models.

We talk a lot about the equivalence of 4 - 6 and 4 + -6... that the two terms are being combined, and that can in practice involve either addition or subtraction.  So the above situation is 4 combined with -6x to get 7.  So the reasoning is the same as in 4 + 6x = 7.  Subtract 4 to be able to see the value for -6x by itself.  And if -6x = 3, then we have to divide by -6 to find out what 1x is. 
Likewise with a problem like 3x = 5/7, the bar model would simply be:

and would simply illustrate that we need to divide 5/7 by 3.
I use the bar models when they make it easier, and work by analogy at other times.

The Benefits
The students' work with bar models gives meaning to all of those old maxims:

• Do the opposite operation
• What you do to one side, you do to the other
• Simplify the equation
• Get x by itself

In fact, it allows the students to really discover the principles by themselves.  They see why they need to subtract when there is addition in the equation.  They recognize that we are trying to get a simpler picture, and as they simplify the bar, the equivalent value changes.  They see that the goal, or a sub-goal, is often to find the value for x, and why dividing or multiplying often needs to be done after we have a value for just a certain number of x's by themselves.  (It would also be easy to introduce distributive division with bar models).  With the principles in place and reasoning behind them, they are easier to apply in a purely symbolic algebraic setting.

One extra benefit is that they see the problems more broadly and flexibly, as in the mango problem above.  We could change the given values or ask for something besides just x.  The relationships in the expression are the essential piece.  They could use the information to write their own problems.

What Comes Next?
Algebra should serve exactly the same role - be a compact picture of a set of relationships that can be manipulated to make new relationships - so I build connections between the two representations by going back and forth.  I ask the students to draw bar models from equations, and to write equations from bar models, and write both from word problems.  I draw parallels between the way I want the students to show their algebraic work and the work we do with bar models.  And as we do more and more work with algebraic notation, I will go back to a bar model whenever a student seems to be fishing for what to do next without really understanding why.  My hope is that by the end of the year the students understand that algebraic equations don't just materialize in the homework pages of a math textbook.  They represent numerical relationships that the students understand and can manipulate knowingly and effectively.