Tag Archives: middle school

The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.

Asking questions. Making choices.

I have had the opportunity to work this year with the wonderful staff at Laura Jeffrey Academy, a girl-focused charter school in Saint Paul, MN.

I’m still undecided on the larger questions surrounding charter schools and their relationships to public schools and public K-12 funding, so I am in no way interested in picking up that thread of discussion here.

Instead, I want to reflect on my experiences in this setting-an urban, girl-focused, open-admission middle school.

This week, I spent my first full day at the school with school in session. I had worked with the math teachers over the summer, but students are really an abstraction when we’re talking about teaching in the summertime.

As I observed a couple of classes in the morning, I was reminded of some age-old questions in mathematics teaching and learning. In particular…

Do girls like math?

This question (and its bastardized forms, Are girls allergic to algebra? Are girls too sexy for math? etc.) almost seems worth debating in the real mixed-gender world.

But spend a few hours in a Laura Jeffrey math class and it becomes obvious that these are ridiculous questions not worth wasting time with.

When you get rid of the boys, you still have just as much variation in attitudes, interests, predispositions, etc. with respect to math (and pretty much everything else) that you would in your standard mixed-gender classroom.

Of course, right?

But it is so easy to put students into categories and make blanket claims about all students in each category. Get rid of one of the two categories, though? Now we realize what a crappy way this was to categorize kids in the first place-especially if we’re trying to understand their interests, motivations and goals.

Thanks Laura Jeffrey staff and students for the reality check.

It was a pleasure and I look forward to our future work together.

Words to avoid in the middle school classroom (continued)

I have this to add to the collection so far:

Long and hard.

In trying to put the Common Core mathematical practices into kid-friendly language, a colleague transformed this:

1. Make sense of problems and persevere in solving them.

Into this:

1. Think long and hard to solve problems.


Words and images to avoid (addendum)

We had a little fun back in April with words and images to avoid in the middle school classroom.

Those who do not learn history are doomed to repeat it, no?

Consider the following from a recent draft of a project that will remain nameless, but which is intended for sixth grade:

Two boys who live near a golf course search for lost golf balls and package them for resale.

How many packs of 12 golf balls can be made
from a supply of 6,324 balls?


If a supply of 6,324 golf balls is packed in 12 boxes,
how many balls will be in each box?

QMST V: Creating an inviting classroom

Today I’m turning over the blog to a fabulous middle school math teacher who comes from an English teaching background. All those smart, smart questions from middle schoolers over the last 6 months? Her kids asked those. Give her a hand in thinking something through, would you please?


I’m Alex Otto and I teach 7th and 8th grade math at Kodiak Middle School in Kodiak, Alaska.

I was looking for some feedback from my peers.  So much of what math teachers talk about is how to teach abstract concepts, yet so much of what is important to middle school kids is what is going on in their immediate surroundings.

Along those lines, I’m thinking about how to best arrange my classroom in terms of seating, bulletin boards, groups, “stations”, etc.  Does anyone have any ideas and especially links to pictures of “inviting” classrooms that facilitate the kind of learning that we want to take place?  (I feel like a lot of the “inviting” classrooms in my school are language arts classrooms with reading nooks, writing stations, etc. )

QMS XI: Calculus

From the usual source:

Can you post a calculus problem and solution so we can see what it looks like?  Also, how and where would you use it in the world?

There are three major ideas of Calculus. I’ll tell you about the first two, since they’re the ones that are easy to wrap your mind around. Then I’ll finish with the third. The first two are:

  1. Derivative and
  2. Integral


A derivative is a slope. Pure and simple. Just as slope in seventh and eighth grade is the rate of change of one variable with respect to the other, derivative is the rate of change in calculus. I don’t mean this metaphorically; they are the same thing.

In algebra, slope has to do with straight lines. If Emile walks 75 meters in 30 seconds, you have no trouble figuring out how many meters per second he walks. That’s his walking rate; it’s the slope of his line. In calculus, we would say that this is the derivative of his walking function.

But what if his rate isn’t constant? What if he starts off slowly and speeds up as time goes by? What if his walking function is more complicated than y=2.5x? You get a taste of how to figure rates (slopes) of non-linear relationships when you work on first and second differences in studying quadratic or exponential relationships.

But first and second differences give us average rates of change. Calculus is interested in instantaneous rate of change-not just what is the average speed between 2 and 3 seconds? but how fast is this thing moving right now?

Consider the video below:

How does the speedometer work? Calculus answers that question using derivatives.

If we model the car’s distance traveled as a function of time:

Then we write:

We read this as The derivative of f with respect to t. When we specify f as a function, the derivative is a function. When we also specify a number t, then the derivative has a numerical value. That value is the speed at the time we selected.


Going back to Emile who walks 2.5 meters per second. If you know his walking rate and how long he walks, you can figure out how far he walked. And you do that frequently in algebra.

But what if his walking rate is changing? Now it’s a much harder problem, and it requires more advanced mathematical techniques. Go back to the video:

How far did the car travel in going from 0 to 60? Now, strictly speaking you don’t need Calculus to answer this. You could mark the beginning and end of the run and measure it. But what if it’s an airplane and you need to know how long to make the runway? What if it’s the Space Shuttle? These would be harder to measure directly. The Calculus idea of integral solves these kinds of problems.

If we model the car’s speed as a function of time from time 0 to time a (meaning whenever the car gets to 60 mph):

Then we would write:

And we read this as The integral from 0 to a of g of t. After specifying g as a function and a as a number, this integral has a numerical value-the distance the car traveled.


Now think about what you know in algebra. To find Emile’s speed, you divided his distance by his time. No matter what two points you choose on Emile’s graph, the difference in distance divided by the difference in time will be the same. But when his walking rate is changing, that’s not true. The average speed will depend on which points you choose. And as a general rule, the average speed between two times will not be the exact speed he is traveling at either time.

So mathematicians get a better approximation of the speed at a given time by taking averages over small intervals of time. And they imagine an answer to the question, What if we averaged his speed over 0 seconds? This would mean dividing by zero, which is impossible. The third major idea of Calculus-limit-gets around this technicality.

Derivatives are defined in terms of limits.

And so are integrals for a similar reason. Our attempts to precisely answer questions about how far the car travels in getting to 60 mph inevitably lead to adding up infinitely many numbers. We literally do not have time for that. Limits get us around this problem, too.

Words (and images) to avoid in the middle school classroom

When I taught middle school, I sometimes worried that they taught me more than I taught them. Middle schoolers are masters of the double entendre and they love nothing more than twisting their teachers’ innocent words and drawings in perverse ways.

I leave the following list to your imagination. How might a middle school teacher have meant to use the following words and images, and how might a student take deviant pleasure in hearing and seeing them used?

Final warning

The following is completely immature and inappropriate. Yet each example comes directly from my own middle school classroom. Middle school teachers, back me up on this, please!

And, of course, let’s see your additions to the list in the comments.





Pull out (especially if what we are pulling out is balls)

In a Connected Math probability unit, there is a problem involving blue and orange blocks. Students are trying to list out all the ways two orange blocks and two blue blocks can be put into two containers. Students typically abbreviate orange with O and blue with B. This all goes well until they put all four blocks in one container and list them out: blue, orange, orange, blue.

The formula for area, A, of a square as a function of its side length, s, is A=s*s