Sometimes we are so deep in a particular worldview that we can’t possibly imagine there is any different way to look at things.  Or even if we believe there is another way, what we are imagining is really just derivative of what we already see.  You have probably heard about the differences between a U.S. mathematics classroom and a Japanese mathematics classroom – in particular, the amount of time students in Japan might be left to struggle with a problem on their own.   And while the following might seem obvious, this morning I really got them (anyone remember “grok” from Stranger in a Strange Land?) – I really grokked it.
 
I was doing tai chi. And I was thinking that there probably things that my tai chi teacher wouldn’t want to teach us directly, because when he saw us doing them, he would know that we had figured something out, that we were experiencing the positions correctly.  And it occurred to me that I do the same thing in my classroom.  There are things about math problems that I am always pleased to see students figure out on their own.  When they do, it tells me something about what they understand.  And there is a huge difference between knowledge that students gain on their own and knowledge that I, or other students, give them.  The former directly indicates growth in the student’s capacity to think and problem-solve.  The latter may also contribute to that, but the connection is less direct.  And, I think, only the former contributes to the confidence and growth mindset that we know is an important factor for success in anything.
 
So in math class, there are times when I am deliberately not explaining something to students that I know would be useful to them, in order to allow them to figure it out on their own.
 
Duh.  So what’s the point?  I think lots of our students' parents nod their heads when we say at Back to School Night that we aren’t going to explain everything to their children, that the students need to figure things out on their own.  But in practice, I think many of those same parents would be confused or even stunned to hear a teacher say “Steve is still really challenged by finding the areas of triangles.  I am working on some good problems that I think will stretch his thinking.”  We know in the U.S. that figuring things out for yourself is a good idea, but we still have the expectation that teachers explain things, and the best teachers explain things well, meaning quickly and painlessly.  This is a deeply rooted priority, and what appears to be a superficial gap between teacher as explainer and teacher as challenge-giver and coach is really more of a gaping chasm. 
 
I write this to warn young teachers about two possible results of this chasm, in the hope that making them explicit may make them more manageable.  First, you may forget the difference.  In an effort to maintain relationships with parents, administrators, and students who have the “teacher as explainer” model in their heads, you may lose sight of the value of letting students struggle.  That does not mean you are deserting the cause, or that you are losing ground.  The challenge of managing how much struggle and how much explaining you are going to do in your classroom is just part of the job.  You will have to make choices about that every day, they won't be perfect, and you will get better at it.
 
Second, just as you are doing the very thing that you think is the most valuable for students, when you see yourself as teaching at your best, they may be resentful and disappointed, and see you as an inept, ineffectual teacher.  That's the students themselves, their parents, even other teachers or administrators.  The gap between the two approaches is both vast and nearly invisible.  Everyone seems to be nodding their heads at the same time, but the pictures in their minds of what is good teaching are vastly different.  You may want to think about how to talk to both parents and students about the value of struggle, and about why you are going to challenge the students with difficult problems without providing a clear explanation or walking them through the steps.  Highlighting your instructional decision and outlining the carefully thought out reasons behind it may go a long way in bridging the gap.  Find allies who are eager to support you at your school, conferences, or online (check out some of the My Favorite blogs or follow them on Twitter).  Take time to reconnect with the big picture, however that works for you.
 
I speak from experience.  I am always little embarrassed when this suddenly becomes clear to me ... again.  It always feels like something I just figured out and something I just remembered at the same time. But that's how it works.  We are all captains of our own ship.  The course may not always be known, but we keep sailing and doing our best. 

***Note:  I am aware that the terminology for Common Core states that "sequences" are a high school topic, while "patterns" can be studied starting in elementary school.  I use the term "sequence" in an informal way, interchangeably with "pattern".

Starting with the 7th grade, we ask students to develop an understand of the relationships behind linear expressions.  We ask them to write linear expressions and solve linear equations from real-life situations. Whenever I did this work with my students, I found that their understanding was often very shallow and disconnected.  For instance, with the table on the left, my students could easily write the relationship as 3x.  The table on the right was more difficult. 

I realized that the students were not seeing the numbers in the right hand column as a sequence, and that the kinds of questions I wanted to ask them required that they be able to imagine that sequence and reason about it.

I don't use a textbook for my prealgebra class.  I tossed the textbook out because it finally dawned on me that the most important idea in the year was just a one-class two-page spread (back in the days of that kind of textbook).  That was the lesson when students were asked to write a function for a linear table.  Linear functions seemed to be the concept behind 70%+ of the book: evaluating and solving linear expressions, graphing equations, and ratios and proportions; and this seemed to me to be the essence: understanding linear functions as a pattern of steady growth or decrease. As I struggled to find a way to have students make the connection between the patterns in the numbers and the functions, I tried The Pattern and Function Connection by Fulton and Lombard, the Creative Publications' Algebraic Thinking: First Experiences, and other places where students were looking at visual patterns, making tables and writing functions.  (This was long before Fawn Nguyen had created Visual Patterns).

The students could see the patterns in the visual sequences, and that made it easier to write functions from the shapes, but writing a function from a table was another matter.  As I wrote above, they saw how the numbers in the right column (the range) were increasing or decreasing by a constant amount, but they couldn't understand why that appeared in the function as multiplication, not addition or subtraction.  They didn't see how the function could both describe the pattern and predict other values when there were gaps in pattern, if x = 9, 32, or 105.  And there was another trouble spot.  When we got to graphing linear functions, the students could find the rise and the run, and they could even use slope and the y-intercept to write a function, or find those values from the table, but it seemed those connections were just correlations, and they weren't really seeing the pattern of growth behind them.  They were just numbers. 

That's when I figured out that the students did not see the numbers in the range as a sequence at all.  They were focused on the (x,y) number pairs.  Or if they did see the sequence, they had a difficult time extending it mentally, or filling in gaps.  The students needed time to look at linear sequences and reason about them, find missing terms, and find connections between the generating function and the sequence.  Using Dan Meyer's metaphor, if linear functions are the aspirin, then missing terms in a sequence should be the headache.   I also realized that I should start with just sequences, lists not contained in a table, then introduce the concept of term numbers, then combine the term numbers and the sequence to make a horizontal table, so that the students could develop this way of seeing and understanding linear patterns and graphs as a pattern of growth in constant steps.

Proceeding in this linear fashion :), my students develop a much deeper understanding of linear relationships.  I do a lot of work with situations that are linear - writing linear functions, exploring sequences, filling in tables, etc. - before we start writing linear equations and solving.  You might say that gives the students a context for solving equations, but I think more importantly it gives the students a mathematical model and a tool that allows them to do independent thinking about all of the above. 

I hope at a later date to post some of the materials I use with my students for exploring linear growth situations and functions.  You can take a look at some of the sequence activities I use in the powerpoint I mentioned in my previous post.  The powerpoint outlines a progression of increasing challenge that takes place over a couple of months for me.  You can use it at any pace that works for your students.  I use sequences the same way one might use Visual Patterns: present a couple of examples at the start of class once or twice a week, have a good discussion about what people did and noticed, then extend the ideas to a slightly more challenging problem the next time. 

I had a great time at CMCS 2017.  Thanks to everyone who attended my session.  I will follow up with more about the presentation, but for now, here is the powerpoint.

And here is the link to my Desmos activity.

I have been thinking a lot about discipline recently, and I have also been watching a lot of baseball.  I occurred to me that the outcome of a class is a lot like the outcome of a baseball game, or even the microcosm of a single at-bat.  Success depends on the performance quality of the batter, the pitcher, and a good dose of luck.  And in the classroom it’s the same – the outcome depends on whether I have “got my stuff”, how my students are doing, and some luck.  It isn’t entirely in my control. 
 
When I set my expectations and hold myself accountable as if it is entirely in my control, I lose my sense of humor, tend to take things personally, and can fall into making power struggles out of student confusion or misbehavior.  When I remember that it’s only 1/3 me, I can ease up on myself and gain some distance.  I am still accountable, but that shift in perspective makes it easier to stay calm and respond with my best stuff.  And when things don’t go the way I want them to, it is sometimes comforting to know I get another chance tomorrow. 

Two years ago Elizabeth Statmore (cheesemonkeysf) wrote a blog post reflecting on the book How People Learn.  She shared the blog with us in her session at TMC17, and I have been thinking a lot about the post and some of the comments she got.
 
Blaw0013 wrote: “I cringe when the term "discovery" is used. It suggests to me that the teacher has a certain way of knowing or thinking in mind for the student to use. This seems to me to be a coercive way to interact with people. C. Kamii uses "re-invent" which seems better to me (don't know why). I most prefer invent--maybe because it is my way to remind myself to allow students to make the meaning that they do, it does not need to be mine.”
 
I have struggled with this question of how free the thinking should be in an ideal math class, and I think the first thing I realized is that I don’t teach in an ideal math class.  There are time constraints, and there are people constraints.  And I have a curriculum, ideas about what concepts I want the students to work with, a broad sequence that I think helps to build understanding in a productive way, and some goals for insights I want them to have. 
 
I think of that as “mining student thinking.”  I agree with Elizabeth: “I don't believe in playing ‘guess what I'm thinking’ because I find it psychologically and emotionally insulting.”  I would add exhausting and usually discouraging, both for me and the students.  I pose questions that highlight the ideas I am interested in, and then I accept whatever work they do in response, again highlighting the work I see as most pertinent.  So it feels like a good balance of my leadership and their initiative.  If something comes up that looks like an interesting detour, and we have time (that’s just the reality) then we will pursue it.
 
I also give the students opportunities to explore and investigate.  We don’t spend nearly as much time in this open environment, partly because of the time, and partly because some students (I would say most) really struggle with the lack of structure.  Clearly that is because they don’t get much practice with it, but there are also students who just aren’t that interested.  Maybe that is a problem with schools, but as I said, I don’t teach in an ideal school.  But students investigating is just one format for getting students to think about and discuss mathematics.  Engaging the students in mathematical reasoning pretty much constantly is my broader goal, and that can occur in all sorts of ways. 
 
I would caution beginning teachers from getting too caught up in trying to create an environment where the students follow a “pure” path of discovery.  Students don’t have to discover all the math that they learn.  But the process of discovering math should be a regular part of both what and how they learn.

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.