The Structure of Parabolas

A year ago, I first envisioned and then taught the parabola as the sum of a parabola and a line.The standard form parabola, ax² + bx + c, is the result of a line with slope b and y-intercept c added to a parabola with a vertex at the origin, with vertical stretch a.   This insight came after my realization that a parabola is the product of two lines (although I wrote this up later than the first).

I didn’t teach algebra 2 last semester, so I’ve only now been able to try my new approach. I taught functions as described in the second link. So the students know the vertex form of the parabola. Normally, I would then move to the product of two lines, binomial multiplication, and then teach the standard form, moving back to factoring.

But I’ve been mulling this for a few months, and decided to try teaching standard form second. So first, as part of parent functions, cover vertex form. Then linear equations. As part of linear equations, I teach them how to add and subtract functions.  As an exercise, I show them that they can add and subtract parabolas and lines, too.

So after the linear equations unit, I gave them a handout:

I don’t do much introduction here, except to tell them that the lighter graphs are a simple parabola and a line. The darker graph is the sum of the parabola and the line. What they are to do is explore the impact of the line’s slope, the b, on the vertex of the parabola, both the x and y values. We’d do that by evaluating the rate of change (the “slope” between two points of a non-linear equation) and looking for relationships.

Now, I don’t hold much truck with kids making their own discoveries. I want them to discover a clear pattern. But this activity also gives the kids practice at finding slopes, equations of lines, and vertex forms of a parabola. That’s why I felt free to toss this activity together. Even if it didn’t work to introduce standard form, it’d be a good review.

But it did work.  Five or six students finished quickly,  found the patterns I wanted, and I sent them off to the next activity. But most finished  the seven parabolas in about 40 minutes or so and we answered the questions together.

Questions:

1. Using your data, what is the relationship between the slope of the line added (b) and the slope (rate of change) from the y-intercept to the vertex?
   Answer: the slope (b) is twice the slope from the y-intercept to the vertex.
   ^b⁄_{2= rate of change}
2. What is the relationship between the slope of the line added (b) and the x-value of the vertex?
   Answer: the x value of the vertex is the slope of the line divided by negative 2.
   ^b⁄-_{2= x value of the vertex}
3. What is the relationship between the y-intercept of the line and the y-intercept of the parabola?
   Answer: they are the same.

Note: I made it very clear that we were dealing only with a=1, no stretch.

The activity was very useful–even some strong kids screwed up slope calculations because they counted graph hash marks rather than looking at the numbers. Some of the graphs went by 2s.

So then, they got a second handout: 

Here, they will find the slope (rate of change!) from the y-intercept to the vertex and double it. That’s the slope of the line added to the parabola (b!). The y-intercept of the line is the same as the parabola.

The first example, on the left, has a -2 rate of change from vertex to y-intercept. Since a=1, that means b=-4. The y-intercept is 8. The equation in standard form is therefore
x² -4x + 8. In vertex form, it’s (x-2)² +4.

Tomorrow, we’ll finish up this handout and go onto the next step: no graph, just a standard form equation. So given y=x² -8x + 1, you know that the rate of change is -4, and the x-value of the vertex is 4. Draw a vertical line at x=4, then sketch a line with a slope of -4 beginning at c (0,1).

This may seem forced, but students really have no idea how b influences the position of the vertex. I’m hoping this will start them off understanding the format of the standard form. If not, well, there’s the whole value of practicing slope and vertex form I can fall back on. But so far, it’s working really well.

By late tomorrow or Monday, we’ll be formalizing these rules and determining how an increase or decrease in a changes these relationships. So I hope to have them easily graphing parabolas in standard form by Monday. Yes, I’ll show them they can just plug x to find y.

Then we’ll talk about factored form, and go to binomial multiplication.

I’ll try to report back.

The Product of Two Lines

I can’t remember when I realized that quadratics with real zeros were the product of two lines. It may have been this introductory assessment that started me thinking hey, that’s cool, the line goes through the zero. And hey, even cooler, the other one will, too.

And for the first time, I began to understand that “factor” is possible to explain visually as well as algebraically.

Take, for example, f(x)=(x+3) and g(x)=(x-5). Graph the lines and mark the x-and y-intercepts:

Can’t you see the outlines of the parabola? This is a great visual cue for many students.

By this time, I’ve introduced function addition. From there, I just point out that if we can add the outputs of linear functions, we can multiply them.

We can just multiply the y-intercepts together first. One’s positive and one’s negative, so the y-intercept will be [wait for the response. This activity is designed specifically to get low ability kids thinking about what they can see, right in front of their eyes. So make the strugglers see it. Wait until they see it.]

Then onto the x-intercepts, where the output of one of the lines is zero. And zero multiplied by anything is zero.

Again, I always stop around here and make them see it. All lines have an x-intercept. If you’re multiplying two lines together, each line has an x-intercept. So the product of two different lines will have two different x-intercepts–unless one line is a multiple of the other (eg. x+3 and 2x+6). Each of those x-intercepts will multiply with the other output and result in a zero.

So take a minute before we go on, I always say, and think about what that means. Two different lines will have two different x-intercepts, which mean that their product will always have two points at which the product is zero.

This doesn’t mean that all parabolas have two zeros, I usually say at this point, because some if not all the kids see where this lesson is going. But the product of two different lines will always have two different zeros.

Then we look at the two lines and think about general areas and multiplication properties. On the left, both the lines are in negative territory, and a negative times a negative is a positive. Then, the line x+3 “hits” the x-axis and zero at -3, and from that zer on, the output values are positive. So from x=-3 to the zero for x-5, one of the lines has a positive output and one has a negative. I usually move an image from Desmos to my smartboard to mark all this up:

The purpose, again, is to get kids to understand that a quadratic shape isn’t just some random thing. Thinking of it as  a product of two lines allows them to realize the action is predictable, following rules of math they already know.

Then we go back to Desmos and plot points that are products of the two lines.

Bam! There’s the turnaround point, I say. What’s that called, in a parabola? and wait for “vertex”.

When I first introduced this idea, we’d do one or two product examples on the board and then they’d complete this worksheet:

The kids  plot the lines, mark the zeros and y-intercept based on the linear values, then find the outputs of the two individual lines and plot points, looking for the “turnaround”.

After a day or so of that, I’d talk about a parabola, which is sometimes, but not always, the product of two lines. Introduce the key points, etc. I think this would be perfect for algebra one. You could then move on to the parabolas that are the product of one line (a square) or the parabolas that don’t cross the x-intercept at all. Hey, how’s that work ?What kinds of lines are those? and so on.

That’s the basic approach as I developed it two or three years ago. Today, I would use it as just as describe above, but in algebra one, not algebra two. As written,I can’t use it anymore for my algebra two class, and therein lies a tale that validates what I first wrote three years ago, that by “dumbing things down”, I can slowly increase the breadth and depth of the curriculum while still keeping it accessible for all students.

These days, my class starts with a functions unit, covering function definition, notation, transformations, and basic parent functions (line, parabola, radical, reciprocal, absolute value).

So now, the “product of two lines” is no longer a new shape, but a familiar one. At this point, all the kids are at least somewhat familiar with f(x)=a(x-h)²+k, so even if they’ve forgotten the factored form of the quadratic, they recognize the parabola. And even better, they know how to describe it!

So when the shape emerges, the students can describe the parabola in vertex form. Up to now, a parabola has been the parent function f(x)=x² transformed by vertical and horizontal shifts and stretches. They know, then, that the product of f(x)=x+3 and g(x)=x-5 can also be described as h(x)=(x-1)²-16.

Since they already know that a parabola’s points are mirrored around a line of symmetry, most of them quickly connect this knowledge and realize that the line of symmetry will always be smack dab in between the two lines, and that they just need to find the line visually, plug it into the two lines, and that’s the vertex. (something like this).

For most of the kids, therefore, the explanatory worksheet above isn’t necessary. They’re ready to start graphing parabolas in factored form. Some students struggle with the connection, though, and I have this as a backup.

This opens up the whole topic into a series of questions so natural that even the most determined don’t give a damn student will be willing to temporarily engage in mulling them over.

For example, it’s an easy thing to transform a parabola to have no x-intercepts. But clearly, such a parabola can’t be the product of two lines. Hmm. Hold that thought.

Or I return to the idea of a factor or factoring, the process of converting from a sum to a product. If two lines are multiplied together, then each line is a factor of the quadratic. Does that mean that a quadratic with no zeros has no factors? Or is there some other way of looking at it? This will all be useful memories and connections when we move onto factoring, quadratic formula, and complex numbers.

Later, I can ask interested students to sketch (not graph) y=x(x-7)(x+4) and now they see it as a case of multiplying three lines together, where it’s going to be negative, positive, what the y-intercept will be, and so on.

At some point, I mention that we’re working exclusively with lines that have a slope of positive one, and that changing the slope will complicate (but not alter) the math. Although I’m not a big fan of horizontal stretch outside trigonometry, so I always tell the kids to factor out x’s coefficient.

But recently, I’ve realized that the applications go far beyond polynomials, which is why I’m modifying my functions unit yet again. Consider these equations:

and realize that they can all be conceived as as “committing a function on a line”. In each case, graphing the line and then performing the function on each output value will result in the correct graph–and, more importantly, provide a link to key values of the resulting graph simply by considering the line.

Then there’s the real reason I developed this concept: it really helps kids get the zeros right. Any math teacher has been driven bonkers by the flipping zeros problem.

That is, a kid looks at y=(x+3)(x-5) and says the zeros are at 3 and -5. I understand this perfectly. In one sense, it’s entirely logical. But logical or not, it’s wrong. I have gone through approximately the EIGHT HUNDRED BILLION ways of explaining factors vs. zeros, and a depressing chunk of kids still screw it up.

But understanding the factors as lines gives the students a visual check. They will, naturally, forget to use it. But when I come across them getting it backwards, I can say “graph the lines” instead of “OH FOR GOD’S SAKE HOW MANY TIMES DO I HAVE TO TELL YOU!” which makes me feel better but understandably fills them with apprehension.

Teaching: My Retrospective

Okay, I’m rolling along on my task of drawing clear lines of demarcation between my particular brand of squish and traditional progressive education (heh–traditional progressive. Get it?). First up was my new no homework policy.

I then decided to take on sitting my kids in groups (as opposed to group work), which led me to look back at some old post, which forced me to look back at my practice over the years, and that’s been a trip. So much of a trip that I decided to do the retrospective first.

The introspection kicked off when I reread one of the first posts I ever wrote on this site, over 3 years ago, halfway through my third year of teaching. Some key observations:

1. I focused almost entirely on classwork, even then. The essay doesn’t even mention homework which, at that time, I assigned in much the way I describe in my last essay.
2. At that time, the school I worked at used a traditional schedule of 60 minute classes, so the 3 day span per lesson is about two days at my current school. Additional evidence I was focused primarily on what kids learned in class, although as I said, my original homework policy goes back even further than this post.
3. Here’s a real change. Me on low ability students three years ago:
   I’m so cheered to realize how much I’ve improved. I had good student engagement back then, but in rereading this I can remember how many students I had to nudge endlessly, how I had to constantly pick up pencils and hand them to kids to get them to work. Recall I was teaching algebra and geometry, and had just begun what is now my bread and butter class of Algebra 2. So my experience at the time of writing those words was with a lower level of math class, which will always mean lower engagement. Nonetheless, that simple paragraphs reminds me of the struggles I had to get total engagement. I’ve come a long way. Yay, me.

4. Interesting to see my off-hand mention of EDI. No one seeing my teaching would think of me as using the direct instruction mode, but in fact I always, at some point, give kids specific, explicit instructions on the concept at hand.
5. While I talked about differentiation and my need to challenge top students, I have actually moved away from different assessments for different students. At that time, I was just three months of teaching out from year two, all-algebra I-all-the-time, and I basically taught 4 different classes. I’d tentatively planned on continuing this approach, but learned that year (and confirmed in later years) that this wouldn’t work for any class but algebra I.

I wrote this post on January 8, 2012, at almost exactly the same time I began an experiment that utterly transformed my teaching. I speak, of course, of Modeling Linear Equations, which I’m amazed to realize I wrote just one week after the “How I Teach” post. So shortly after I began this blog and described my teaching method, I started on a path that took my existing teaching approach–which was pretty good, I think–and gave it a form and shape that has allowed me to grow and progress even further.

I haven’t really read this post in over two years—I tend to link in Modeling Linear Equations, Part 3, written a year later (two years ago today!), when I’d realized how much my teaching had changed. So reading the original is instructive. I talk about the Christmas Mull, something that stands very large in my memory but don’t remember quite as described here:

The part that’s consistent with my memory: Christmas 2011, I was depressed by the dismal finals in my three algebra II classes. In the first semester, I had gone through all of linear and quadratic equations, including complex numbers, at a rate considerably slower than two colleagues also teaching the course. Yet the kids remembered next to nothing. Every single person failed the multiple choice test–the top students had around half right. I had experienced knowledge fall-offs in algebra and geometry, but nothing that had so sublimely illustrated how much time I’d wasted in three months. So I came out of the Christmas break determined to reteach linear and quadratic equations, because to continue on teaching more advanced topics with these numbers was purely insane. And I wasn’t just going to reteach, but come up with an entirely different, less structured approach that allowed my students to use their own understanding of real-life situations.

What I hadn’t remembered until reading this closely was my rationale for ignoring the regular curriculum requrements. At the time, Algebra 2 was considered a “terminal” class; students weren’t expected to take another course in the college-prep sequence. This has changed, of course–these days, algebra 2/trig is, if anything, experiencing a fall-off in favor of a full year of each course. But at the time, I justified my decision to go off-curriculum based on the student needs. These students’ primary concern, whether they knew it or not, was what happened to them in college. How much remediation were they going to need? Could the best of them escape any remedial work and go straight onto credit bearing courses? This, of course, still remains my priority–I’d just forgotten how linked it was to my initial decision to try something new.

Also interesting that I described this approach by the specific method I used for linear equations–using “inherent math ability”. That’s not how I describe my approach these days, but I can see the germination of the idea. At the time I wrote this, I had no idea I would go beyond linear equations and use this approach consistently throughout my instruction.

I think the best description I’ve come up with for my approach is modified instructivist, which comes in one of two forms: “highly structured instructivist discovery, and classroom discussions with lots of student involvement”.

As for the latter: I don’t lecture, with or without powerpoints. When I do explanations, they are classroom discussions, and you can see this demonstrated in all my pedagogy posts. However, I am constantly migrating my classroom discussions to structured discovery.

What’s structured discovery? Imagine a teacher and students on a cliff, with a beach below. There’s a path, but it’s not visible.

In a traditional lecture or classroom discussion, the teacher shows them the path and leads them down to the beach.

In a discovery class, the teacher doesn’t even tell them there’s a path or even a beach. In fact, to the discovery/reform teacher, it doesn’t matter whether there’s a path or not—the kids will all find their own way down. Or maybe they’ll just find some really cool flowers and stop to examine their biology. Or maybe they’ll just kick back and have a picnic. It’s all good, in reform math. (sez the skeptic)

In what I call structured discovery, the kids are given a series of tasks that use their existing knowledge base and find the path themselves. They may not yet know there’s a beach. They may not know what the path means. But they will find the path and recognize it as a consistent finding that makes them go “hmm”. In some cases, an interesting finding. In other cases, just something they can see and understand.

Sometimes the path they’ve found is the concept–for example, modeling linear equations or exponential functions, or finding gravity in projectile motion problems.

In other cases, the model just introduces an inevitable observation that leads to the new concept. For example, I teach my kids about function operations when we do linear equations–adding and subtracting are good models for simple profit and loss applications.

So I kick off quadratics by asking my students to multiply linear functions, which they can see clearly as an extension of adding and subtracting them. This is an activity they can start off cold, with no intro (I haven’t written it up yet). I designed this because parabolas just don’t have a natural “real life” model other than area, which gets kind of boring. Plus, I need to cover function operations anyway, so hey, synergy. In any event, the kids are seeing an extension of a concept they already know (function operations) and seeing a new graph form consistently emerge. Then we can talk about factors (the zeros) and realize that we are looking at products of two lines. Could a parabola exist without being a product of two lines? Well, this is algebra 2 so they are fully aware that parabolas don’t have to have zeros. But what does that mean in terms of multiplying lines being factors of parabolas? Well, they must not have factors. So are all parabolas the product of two lines? And we go from there.

Understand that my classes still have lots of practice time where kids just factor equations and graph parabolas, learn about the different forms, and so on. But rather than just saying “now we’ll do this new thing called a parabola”, I give them a task that builds on their existing work and leads them into the new equation type. I don’t define the path. But nor do I let them go off on their own. I give them something to do that looks kind of random, but is in fact a path.

And all of this came from the results of the Great Christmas Mull. The previous Christmas had been productive, too–it’s when I came up with differentiated instruction for my algebra class.

So what can I say about my teaching, 5.5 years in? What’s consistent, what’s changed?

1. I never lectured. I always explained, with increasing emphasis on classroom discussion.
2. I have always been focused on student work during class, emphasizing demonstrated test ability above everything, and minimizing (or now eliminating) homework.
3. I have always tried to move the student needle at all ability levels, from the no-hopers to the strugglers to the average achievers to the top-tier thinkers. I’m not always successful, but that’s consistently my stated priority.
   
4. I have always designed my own curriculum and assessments.

5. My teaching was transformed Christmas of 2011, when I realized I could introduce and teach topics using existing knowledge, forcing students to engage immediately with the material and start “doing” right away, increasing engagement and understanding. I have evolved from a teacher who mostly explains first to a teacher who only occasionally explains first. And that is a huge change that takes a lot of work.
6. The observer might think that this change makes my classes student-centered, but I disagree. My classes are definitely teacher-centered, and let’s be clear, I’m the star of my teaching movie.
7. Thanks also to the Great Christmas Mull, I’ve become far less concerned about curriculum coverage than I was in my first two years of teaching.
8. I have always been a teacher who values explanation. It’s the heart of my teaching. I’ll explain through discussion or demonstration, but I’m not a reformer letting kids “construct” the meaning of math. I’m there to tell them what it all means.

I have plenty of development areas ahead. I’m working on tossing in the occasional open-ended instruction, just to see if I can come up with ideas that don’t waste hours and have some interesting learning objectives. I still have many concepts waiting to be converted to a “path to the beach”. And I’m now teaching something other than math, which gives me new challenges and more opportunities to see how to construct those paths without running off the cliff.

The Negative 16 Problems and Educational Romanticism

I came up with a good activity that allowed me to wrap up quadratics with a negative 16s application. (Note: I’m pretty sure that deriving the algorithm involves calculus and anyway, was way beyond the scope of what I wanted to do, which was reinforce their understanding of quadratics with an interesting application.) As you read, keep in mind: many worksheets with lots of practice on binomial multiplication, factoring, simpler models, function operations, converting quadratics from one form to another, completing the square (argghh) preceded this activity. We drilled, baby.

I told the kids to get out their primary quadratics handout:

Then I showed two model rocket launches with onboard camera (chosen at random from youtube).

After the video, I toss a whiteboard marker straight up and caught it. Then I raised my hand and drop the marker.

“So the same basic equation affects the paths of this marker and those rockets–and it’s quadratic. What properties might affect—or be affected by—a projectile being launched into the air?”

The kids generated a list quickly; I restated a couple of them.

Alexandra: “What about distance?”

I pretended to throw the marker directly at Josh, who ducked. Then I aimed it again, but this time angling towards the ceiling. “Why didn’t Josh duck the second time?”

“You wouldn’t have hit him.”

“How do you know?”

“Um. Your arm changed…angles?”

“Excellent. Distance calculations require horizontal angles, which involves trigonometry, which happens next year. So distance isn’t part of this model, which assumes the projectile is launched straight….”

“UP.”

“What about wind and weather?” from Mark.

“We’re ignoring them for now.”

“So they’re not important?”

“Not at all. Any of you watch The Challenger Disaster on the Science Channel?”

Brad snickered. “Yeah, I’m a big fan of the Science Channel.”

“Well, about 27 years ago, the space shuttle Challenger exploded 70 some seconds after launch, killing everyone on board when it crashed back to earth.” Silence.

“The one that killed the teacher?”

“Yes. The movie—which is very good—shows how one man, Richard Feynman, made sure the cause was made public. A piece of plastic tubing was supposed to squeeze open and closed—except, it turns out, the tubing didn’t operate well when it was really cold. The launch took place in Florida. Not a place for cold. Except it was January, and very cold that day. The tubing, called O-ring, compressed—but didn’t reopen. It stayed closed. That, coupled with really intense winds, led to the explosion.”

“A tube caused the crash?”

“Pretty much, yes. Now, that story tells us to sweat the small stuff in rocket launches, but we’re not going to sweat the small stuff with this equation for rocket launches! We don’t have to worry about wind factors or weather.”

“Then how can it be a good model?” from Mark, again.

“Think of it like a stick figure modeling a human being but leaving out a lot. It’sstill a useful model, particularly if you’re me and can’t draw anything but stick figures.”

So then we went through parameters vs. variables: Parameters like (h,k) that are specific to each equation, constant for that model. Variables–the x and y–change within the equation.

“So Initial Height is a parameter,” Mark is way ahead.

Nikhil: “But rocket height will change all the time, so it’s a variable.”

Alissa: “Velocity would change throughout, wouldn’t it?”

“But velocity changes because of gravity. So how do you calculate that?” said Brad.

“I’m not an expert on this; I just play one for math class. What we calculate with is the initial velocity, as it begins the journey. So it’s a parameter, not a variable.”

“But how do you find the initial velocity? Can you use a radar gun?”

“Great question, and I have no idea. So let’s look at a situation where you’ll have to find the velocity without a radar gun. Here’s an actual—well, a pretend actual—situation.”

“Use the information here to create the quadratic equation that models the rocket’s height. In your notes, you have all the different equation formats we’ve worked with. But you don’t have all the information for any one form. Identify what information you’ve been given, and start building three equations by adding in your known parameters. Then see what you can add based on your knowledge of the parabola. There are a number of different ways to solve this problem, but I’m going to give you one hint: you might want to start with a. Off you go.”

And by golly, off they went.

As releases go, this day was epic. The kids worked around the room, in groups of four, on whiteboards. And they just attacked the problem. With determination and resolve. With varying levels of skill.

In an hour of awesomeness here is the best part, from the weakest group, about 10 minutes after I let them go. Look. No, really LOOK!

See negative 2.5 over 2? They are trying to find the vertex. They’ve taken the time to the ground (5 seconds) and taken half of it and then stopped. They were going to use the equation to find a, but got stuck. They also identified a zero, which they’ve got backwards (0,5), and are clearly wondering if (0,4) is a zero, too.

But Ed, you’re saying, they’ve got it all wrong. They’ve taken half of the wrong number, and plugged that—what they think is the vertex—into the wrong parameter in the vertex algorithm.. That’s totally wrong. And not only do they have a zero backwards, but what the hell is (0,4) doing in there?

And I say you are missing the point. I never once mentioned the vertex algorithm (negative b over 2a). I never once mentioned zeros. I didn’t even describe the task as creating an equation from points. Yet my weakest group has figured out that c is the initial height, that they can find the vertex and maybe the zeroes. They are applying their knowledge of parabolas in an entirely different form, trying to make sense of physical data with their existing knowledge. Never mind the second half—they have knowledge of parabolas! They are applying that knowledge! And they are on the right track!

Even better was the conversation when I came by:

“Hey, great start. Where’d the -2.5 come from?”

“It’s part of the vertex. But we have to find a, and we don’t know the other value.”

“But where’d you get 2.5 from?”

“It’s halfway from 5.”

Suddenly Janice got it.

“Omigod–this IS the vertex! 144 is y! 2.5 is x! We can use the vertex form and (h,k)!!”

The football player: “Does it matter if it doesn’t start from the ground?”

Me: “Good question. You might want to think about any other point I gave you.”

I went away and let them chew on that; a few minutes later the football player came running up to me: “It’s 2!” and damned if they hadn’t solved for a the next time I came by.

Here’s one of the two top groups, at about the same time. (Blurry because they were in the deep background of another picture). They’d figured out the vertex and were discussing the best way to find b.

Mark was staring at the board. “How come, if we’re ignoring all the small stuff, the rocket won’t come straight back down? Why are you sure it’s not coming back to the roof?”

“Oh, it could, I suppose. Let me see if I can find you a better answer.” He moved away, when I was struck by a thought. “Hey….doesn’t the earth move? I mean yes, the earth moves. Wouldn’t that put the rocket down in a different place?”

“Is that it?”
“Aren’t you taking physics? Go ask your teacher. Great questions.”

I suggested taking a look at the factored form to find b but they did me one better by using “negative b over 2a” again and solving for b (which I hadn’t thought of), leading to Mark’s insight “Wait–the velocity is always 32 times the seconds to max height!”

The other kids had all figured out the significance of the vertex form, and were all debating whether it was 2.5 or 2 seconds, generally calling me over to referee.

One group of four boys, two Hispanics, one black, one Asian (Indian), all excellent students, took forever to get started, arguing ferociously over the vertex question for 10 minutes before I checked on them to see why they were calling each other “racist” (they were kidding, mostly). I had to chastise the winners for unseemly gloating. Hysterical, really, to see alpha males in action over a math problem. Their nearly-blank board, which I photographed as a rebuke:

The weaker group made even more progress (see the corrections) and the group to their left, middling ability, in red, was using standard equation with a and c to find b:

My other top group used the same method, and had the best writeup:

Best artwork had the model wrong, but the math mostly right:

• All but one group had figured out they wanted to use vertex form for the starting point.
• All but one group had kids in it that realized the significance of the 80 foot mark (the mirror point of the initial height)
• All the groups figured out the significance of five seconds.
• All the groups were able to solve for both a and b of the standard form equation.
• The top three groups worked backwards to find the “fake” zero.
• Two groups used the vertex algorithm to find b.
• All the groups figured out that b had to be the velocity.

So then, after they figured it all out, I gave them the algorithm:

h(t)=-16t² + v₀t + s₀.

Then I gave them Felix Baumgartner, the ultimate in a negative 16 problem.

And….AND!!!! The next day they remembered it all, jumping into this problem without complaint:

Charles Murray retweeted my why not that essay, saying that I was the opposite of an educational romantic, and I don’t disagree. But he’s also tweeted that I’m a masochist for sticking it out—implying, I think, that working with kids who can’t genuinely understand the material must be a sad and hopeless task. (and if he’s not making that point, others have.) I noticed a similar line of thought in this nature/nurture essay by Tom Bennett says teachers would not write off a child with low grades as destined to stack shelves –implication that stacking shelves is a destiny unworthy of education.

The flip side of that reasoning looks like this: Why should only some students have access to a rich, demanding curriculum and this twitter conversation predicated on the assumption that low income kids get boring curricula with no rigor and low expectations.

Both mindsets have the same premise: education’s purpose is to improve kids’ academic ability, that education without improvement is soulless drudgery, whether cause or effect. One group says if you know kids can’t improve, what a dreary life teaching is. The other group says dreary teaching with low expectations is what causes the low scores—engage kids, better achievement. Both mindsets rely on the assumption that education is improvement.

Is it?

Suppose that in six months my weakest kids’ test scores are identical to the kids who doodled or slept through a boring lecture on the same material. Assume this lesson does nothing to increase their intrinsic motivation to learn math. Assume that some of the kids end up working the night shift at 7-11. Understand that I do make these assumptions.

Are the kids in my class better off for the experience? Was there value in the lesson itself, in the culmination of all those worksheets that gave them the basis to take on the challenge, in the success of their math in that moment? Is it worth educating kids if they don’t increase their abilities?

I believe the answer is yes.

Mine is not in any way a dreary task but an intellectual challenge: convince unmotivated students to take on advanced math—ideally, to internalize the knowledge for later recall. If not, I want them to have a memory of success, of achievement—not a false belief, not one that says “I’m great at math” but one that says “It’s worth a try”. Not miracles. Just better.

I would prefer an educational policy that set more realistic goals, gave kids more hope of actual mastery. But this will do in the meantime.

I have no evidence that my approach is superior, that lowering expectations but increasing engagement and effort is a better approach. I rely on faith. And so, I’m not entirely sure that I’m not an educational romantic.

Besides. It’s fun.