Almost a decade into my teaching career, I’ve learned a lot—about recurring decimals, British slang, the life cycle of fidget spinners. But one lesson I seem to relearn in new ways every day: Deep thinking is a very, very delicate flower.

It blooms only under rare and perfect conditions, when you’ve given the seedling absolutely everything it needs.

There’s no perfect recipe. What gets my 6^{th}– and 8^{th}-graders’ thoughts blooming might flop with my 7^{th}-graders. This work is wonderfully and maddeningly specific. Each seedling presents its own unique and irreducible case. The best you can do is kneel down in the soil and try to help it along.

Even so, I find a few recurring themes: three crude reasons why deep thinking fails to bloom, and the hardy but colorless perennial of “rote learning” surfaces instead.

**As students, we seek the cognitively easier path.**

Earlier this year, I had a typical conversation. A 7^{th}-grader wasn’t sure why 3√2 + 4√2 should equal 7√2.

“Well,” I said, “what’s 3x + 4x?”

“7x.”

“Why?”

“Because you add the 3 and the 4, and keep the x the same.”

“That’s an accurate description of what you’re doing,” I said, “but let’s try to figure out why it’s true. What do ‘3x’ and ‘4x’ mean?”

They mean, of course, “three groups of x” and “four groups of x.” That totals seven groups of x, no matter how large or small *x* happens to be. That’s elemental, but not elementary. It demands that you (1) think about specific cases; (2) look past their superficial differences to the underlying similarity; (3) articulate a general principle; and (4) translate your discovery into algebraic notation.

Or… you can ignore all that, and just learn a rule for moving symbols around.

“Oh, I get it!” he said before long. “You just add the 3 and the 4, and leave the √2 the same.”

**As a teacher, I seek the administratively easier path.**

Back to my 7√2 student: why had I, his teacher, put him in such a pickle to begin with? Why was I asking this student to extend a rule that he didn’t even understand?

Well… because that’s what came next in the syllabus.

Sure, I could have found a better personal activity for him. Or I could have found a richer task for the whole class, so that he could explore this idea while his classmates explored others. Or I could have stayed with him—collating numerical examples in a table, producing visual and verbal models, testing his symbol-based rule on cases where it would fail—until he developed an actual understanding of why 3x + 4x = 7x.

So why didn’t I?

Because lessons are short, teaching is hard, and I had a classroom full of 7^{th}-graders to manage.

Symbol-pushing isn’t just easier for students. It’s easier for me. It takes less planning before class, less improvisation during class, and less mop-up with struggling students afterwards.

To help a room of students think deeply—that’s no easy task. To help them learn superficial facts and mechanical rules? Well, that’s a heck of a lot easier.

**As assessors, we seek clear-cut standards by which to rank students.**

It was there in the room, as I spoke with my 7√2 student. I’m not sure I can pinpoint where—hovering by the ceiling? lurking under my desk?—but it was there:

The system’s need to assess.

Schools play a lot of roles in society, and one of them is the crude business of sorting. In June, the school’s students all take a year-group test. Top scorers win prizes and are called on-stage in front of the whole school. Low scorers feel the gut-punch of failure, no matter how meaningful or meaningless the test is.

Three years after that, the same students sit the IGCSE exams. Top scorers have a better shot at prestigious university degrees. Low scorers have a worse one.

This stuff matters.

The tests are written to be “objective” and “fair,” which means they ask for scripted performances of technical skills rather than for flexible improvisation. On such tests, deep thinking can be more an impediment than an aid.

So what is there to be done? How do you help healthy flowers grow in a climate that can feel so ill-suited to them?

Well, that’s called teaching, and I’m still learning how to do it.

But I’ve got some ideas.

To overcome the first obstacle—students’ preference for easier thinking—I’ve got to give them motive and opportunity. I’ve got to help them see the steeper path as an exciting adventure, not a pointless side quest.

To overcome the second obstacle—my own preference for easier administration—I’ve got to play it smart and conserve my energy. I’ve got to avoid the mire of self-created busywork, and lay the groundwork (with routines and class culture) to make open-ended tasks go smoothly.

And to overcome the third obstacle—the system’s preference for clear-cut assessment—I’ve got to fight a multi-front war. I’ve got to seek better, richer, more varied assessments. I’ve got to help students see their results not as irreversible judgments but as guiding feedback. And—hardest of all, in a system that puts all kinds of pressures on teachers and students alike—I’ve got to know when to hold ‘em, and know when to fold ‘em.

Now, the goal isn’t for students to think deep thoughts every minute of every day. That’s as unsustainable as a full night of top-intensity dancing. You need cool-down songs, slow dances, chances to catch your breath.

But I know this much: I want my students dancing as hard as they can.

You make me think back to my time with freshman year algebra.

One of the impediments I faced was (in retrospect) dealing with fractions.

3x + 4x seemed to make sense to me, but when it got into fractions it got tougher.

3/x + 4/x made me nervous, but I could grasp that multiplication of the first example was being directly replaced by division in the second example.

However,

3/x + 4/2x flummoxed me.

I “wanted” it to become: 3/x + 8/x so that the “sameness” of the division would be true.

Crossing between the fractions, is, I hope the actual correct method:

6/2x + 4/2x = 10/2x = 5/x

The 3 had to become 6…in the “other” fraction.

I still may have it wrong!! I’m about to submit my anguish into public view.

And, the initial answer of 10/2x was not ‘good enough” for my teacher, as I recall.

That’s fine. You could also think of 3/x as (3/1) * (1/x) and treat the (1/x) the same way as just x or sqrt(2). So for your example,

3/x + 4/2x

3(1/x) + (4/2)(1/x)

3(1/x) + 2(1/x)

5(1/x)

Isn’t there a way for each student to learn the very basic concepts.those who get it move on to more deep, challenging probs.then the teacher has a smaller group of students to work with for learning the basic concepts?

It’s a good principle! Not always easy to implement. In particular, I find that developing concepts, even basic ones, is some of the slowest, hardest work in the classroom. I suspect it’s more common to have 20% “getting it” and 80% struggling than the other way around, although of course it varies enormously from classroom to classroom and day to day.

Well said and good advice. It is about making connections.

I think most of my mathematics education was symbol pushing until I got to college.

Through the first 4 grades we learn the algorithms of additions, subtraction, multiplication, with borrows and carries. Division has its own odd ball algorithm that has little to do with the others.

Then we learn algorithms for fractions again with separate algorithms for addition, subtraction, multiplication and division. We learn that about the decimal point that can do the same things that fractions do, but have generally more familiar algorithms, (but more digits to keep track of).

Then we learn how to replace numbers with letters, and what parts of the old algorithms still hold?

3x +4x = 7x and 4√2 + 3√2 = 7√2 are two examples of the same distributive property of multiplication. But, the importance of the distributive property was never really hammered home! We probably should have been talking about the distributive property every year since the second grade. It is THE key attribute of multiplication. But, instead, it is something novel when we get to algebra. So, we get to (x+1)(x-2), and that same distributive property comes into play again, and why isn’t it x*x – 2*1? And teachers pump the FOIL mnemonic, and more symbol pushing.

The next 4 years aren’t much better. More algorithms for factoring. More algorithms for exponentiation, more algorithms for trig angle identities. 12 years of mathematics education is learning which formula / algorithm to apply in which scenario. No wonder everyone forgets how to do math within 5 years of graduation.

Oooh, that was a good rant. Thanks.

Thank you for bringing up my favorite property of Algebra, the Distributive Property! I always introduce it as Euclid did, by slicing up a rectangle and factoring out the width. Then I make a big deal of Mixed numbers, because that’s really a plus in there between the two and the one-third. When we get around to multiplying binomials, I start off by going back to the long way of multiplying mixed numbers to remind them that every term needs to be multiplied by every other term. I use FOIL too, but only after we’ve talked at length about why everybody needs to come to the party. One good rant deserves another!

Heh, you mean the “New Math” then Doug! It’s been tried, through-out the world.

It was stopped because, in reality, it doesn’t work. 2nd graders can’t cope with that level of abstraction. And if it is drilled into them, as it was with me, young brains can’t make the leap in applying it from arithmetic to algebra anyway.

The bit about being taught exclusively algorithms rings no bells with me at all. It’s not how I was taught, and it’s not how I teach. Let’s not confuse bad teaching with poor syllabus or theories of instruction.

“New Math” means to me set theory, modular arithmetic, “that base 6 crap”, Boolean logic, etc. Everything that might be taught to elementary school students that is not addition, subtraction, multiplication or division. And some of the “new math” survives.

The distributive property of multiplication is not “new math.”

Personally, I was poor in math until I started Algebra, then everything started clicking. A little abstraction was good for me.

You should have told my teachers that then Doug, because I was the first wave of “New Math” and we got weeks of Commutative, Associative and Distributive properties. And that was at Grade 5.

The set theory stuff was actually OK, because being visually based even small kids could follow it. It turns out that it doesn’t actually help later Maths much, but at least it was fine to do.

I’m not sure that you can say modular arithmetic is “crap” but that it is important to teach the distributive property early. Modular arithmetic is what really taught me as a small child how the base 10 and base 2 number systems worked. It has been far more useful to me than any of the distributive etc ideas.

Personally, I was poor in math until I started Algebra,It may well be that you are confusing the root cause here. Most sensible school systems start teaching algebra when the students are just maturing enough to be able to do it. Your starting to understand may well be related mostly to the fact that your brain was now able to deal with that level of abstraction.

My experience is that teaching young students algebra is of little use. They simply can’t cope with the abstraction. If you are against learning being rote and based on “symbol pushing”, then teaching algebra early is a bad thing, as most students can only deal with it in a very rule based way.

I see these things and resolve to balance these conflicts every day. But what about hard technical skills? Like the properties of exponents? I laid out in fine detail the underpinnings of one of these the other day only to have the math coach suggest that I limit that because it takes up so much time. Deep thinking makes for durable learning and a better ability to persevere as well as synthesize knowledge to solve new and unique problems. I’ll take my time with you Billy, because you deserve it.

Could you explain symbol pushing further? This is not a term I am familiar with.

I also wanted to clarify the paragraph you said

“until he developed an actual understanding of why 3x + 4x = 7x.

So why didn’t I?”

1. Why do you believe this student didn’t develop an actual understanding?

2. There have been moments where I later realise the significance of something I was taught and appreciate the underlying connection, so to speak. I say this because, is it necessary for the teacher to experience the student’s ah-ha moment (obviously it is satisfying)? That is, it is awesome to know that the student has a deeper understanding, however, the important part is that the student has the deeper understanding.

Good questions! “Symbol pushing” refers to manipulating algebraic expressions without necessarily knowing what they mean, or why these manipulations are valid, but others would not be. The classic example is a person expanding (x + 3)(x + 2) via a mnemonic like “FOIL,” but not knowing why this isn’t just x^2 + 6.

On question 1, I’m afraid drafted this post months ago and can’t remember the moment well enough; lessons blur. On question 2, I’ve had interesting conversations along these lines with colleagues. Lots of them identify with the experience of learning symbolic manipulations, and then later coming to understand the meaning behind them. I agree that it’s a genuine path to understanding.

But I think it’s easy overrate it.

Most math classes are taught in this way – symbolic manipulations today, understanding “tomorrow.” I find tomorrow rarely comes. A minority of strong students (like my colleagues, who by definition went on to become math teachers) will make the connections. Most won’t. I chalk this up to two basic causes:

First, if you teach a student to perform manipulations without emphasizing the underlying concept, it becomes very hard to assess understanding. They can perform the task but there may be something missing. I think of a chef who can execute a recipe but doesn’t know what each step accomplishes or what each ingredient contributes. They can pass the test – i.e., cook the dish – but can’t adapt or innovate or troubleshoot.

Second, chronic manipulation-based learning can undermine a student’s sense that mathematics *has* a deeper logic. I find that many try to “explain” or “justify” a rule by simply repeating or paraphrasing it. I suspect that an overemphasis on manipulation disguises the conceptual layer from students altogether.

#1 means teachers struggle to spot a gap in understanding. #2 means that students may not even know they have one. That’s why only a handful of students seem to “flesh out” their understanding later on. Most just proceed with a brittle, shallow understanding until the house of cards collapses.

More related thoughts in this post: https://mathwithbaddrawings.com/2015/04/08/the-math-ceiling-wheres-your-cognitive-breaking-point/

I think you left out one very important contributor, which is poor quality teachers. Bad teachers teach by pure rules because that’s where they are safe. They struggle to explain in a way that aids understanding, either because they aren’t good at teaching or because their own understanding is poor.

If you want to reduce the amount of rule based teaching of Maths, then the biggest area of improvement would be to have Maths-specialist teachers in junior school.

I don’t know any teachers who are good at Maths and like Maths that don’t teach conceptually. It isn’t how people who like Maths want to teach, after all.

Pingback: OTR Links 12/08/2017 – doug — off the record

What’s three apples plus four apples? What’s three hundred plus four hundred? What’s 3 eggs plus four eggs? What if we write it like this: 3 “X” plus 4 “X” ?

Eggs are sold at the grocery store in dozens. What’s three dozen plus 4 dozen? Since you know a “dozen” is 12, can you calculate for me the number between one and one hundred that means the same as “3 dozen plus 4 dozen”?

Now a tricky one: What’s three half-dozen plus four half-dozen? Can you calculate that number?

Okay, an even trickier one: We write “1/7” and say “one seventh” so what’s three “one-sevenths” plus four “one-sevenths”? What number does that calculate out to be?

Let’s rest with an easy one. What’s three pies plus four pies?

In math we have a symbol for the number ratio of the width or “diameter” of a circle and the distance around, or the “circumference”. The name of the ratio number is “Pi”. It’s a weird number that doesn’t come out even as a fraction. It’s 3.141592653 and more. So we just write ϖ What’s 3 ϖ plus 4 ϖ ? DON’T TRY TO CALCULATE IT RIGHT NOW! If you have to estimate, estimate that pi is a little more than 3 and give me an estimate for the sum of 3 “pi” s plus 4 “pi” s.

So, now look at the square root of two problem again. What do you think?

I realized, just before I retired, and too late to do anything about it, that there is generally a big misunderstanding between teachers and students. The teacher thinks “I am giving you an important idea along with a couple of examples to show you how to use it.” The student thinks, “Great, he’s giving us an algorithm to solve some of those pesky problems.” I figured this out too late to find out how to deal with it, but for a start, let me suggest that the answer to your question “Why was I asking this student to extend a rule that he didn’t even understand?” is that you hope that the student will realize that it’s an example to illuminate the rule and to show how generally it applies, and therefore to help understand the rule itself. But that you weren’t suggesting that it’s a rule to be copied. So, for example, choose something ridiculous for x, then ask, “If you have 3 on half of your lawn, and 4 on the other half of your lawn, how many are on your whole lawn? Once the student stops laughing, that should get you an answer. And you reply, “Right! So x can be 3, it can be 4, it can be the square root of 2, and it can be a .” (I’d be more specific here, except that I think that my nomination for might not go down too well with your students’ parents.)

I see that the automatic editing software eliminated everything between a less-than sign and a greater-than sign, so it eliminated all my mentions of [something ridiculous], and made nonsense of end of the post. E.g line 5 for the bottom should read “If you have 3 [something ridiculous] on half of your lawn, and 4 [something ridiculous] on the other half…

I read the title of this blog to my 16 year old daughter (who loves math) and she said “it’s simple the answer is ‘exams’ and ‘time’ and ‘deep thinking doesn’t give you marks'”

Reblogged this on Gr8fullsoul.