Category Archives: Lessons

The hierarchy of hexagons, continued

Max asks in the comments of the original hierarchy of hexagons post (and, if you are new to this, see also the follow up post)…

In defining Bobs, Stacys, and the like, did you run into situations where your definition admitted shapes the students didn’t actually want. For example, once you defined a Stacey as a hexagon with three congruent acute angles, did you draw some other Stacys and have students blurt out “wait, that’s not a Stacy — that’s not what I meant!” If not, how did you privilege the *definition* over some sense of “I know a Stacy when I see one.” Is it because their definitions were based on some property they liked about one example, rather than trying to say what was the defining quality of some *group* of hexagons?

Yes, yes, yes.

This instructional sequence is all about moving through the van Hiele levels.

An important learning goal for these lessons is for students to separate what it looks like from its mathematical properties. That means we need to talk about this very issue.

Early in the process, a central challenge is identifying precisely the property the student had in mind. This requires a use of language that can be unfamiliar and strange. Does “has a right angle” include rectangles, which have several right angles? Or did you mean “has exactly one right angle”? That sort of work comes first.

Then we look for other shapes that have this property. If those early-discovered shapes violate the spirit of the original intent, the student may object and will be invited to revise. If she wants to add another property, then I will usually suggest that this is a second class of shapes, and that the one she was really after is the combination of these two classes.

So your shape is special because it has all sides the same length AND it has at least one right angle. Let’s do this…let’s call hexagons with all sides the same length equilateral and let’s give a name to hexagons with at least one right angle…Who has a name for us?

A “sally”? Good. From now on, a sally is a hexagon with at least one right angle.

Now this shape is both equilateral and a sally. Let’s give this special category of shape a new name.

Et cetera.

That’s OK on day 1. Our goal on day 1 of the lesson sequence is to have a set of between five and eight named classes of shapes that have interesting interrelationships.

I need to be the judge of when we have enough, and whether they are of sufficiently interesting variety (I have screwed this up before). I reserve the right to add in some properties that I know will be interesting. For example, I make sure concave and equilateral make it into the mix somehow—either by student introduction or by my own.

After day 1, we need to move away from what things look like. We will be operating only on the properties as we have defined them. If we accidentally left ambiguities, we can plug those holes as clarifications. But these definitions cannot otherwise change. It is important to notice that shapes with very different appearances can share important properties.

It is important to notice that mathematical properties of shapes behave differently from the look of a shape.

For example, this semester we defined a class of hexagons this way.

windmill is a hexagon with three acute angles and three angles greater than 180°.

The iconic windmill is this one.

Screen shot 2014-03-14 at 3.04.24 PM

Much later, as we were in the process of trying to decide whether a windmill can be a utah (a utah is defined as a hexagon with two sets of three parallel sides), we happened upon this most un-windmillish object.

Screen shot 2014-03-14 at 3.05.01 PM

In trying to push the limits of windmill-ness, we started to consider shapes that had the right properties but that looked nothing like the original shape.

Addendum

For what it’s worth, we have produced a proof—which some of us can reproduce and others cannot—that a windmill is never a utah. This proof depends on the Dani principle, which states that when two sides of a polygon are parallel and separated by a single side, then the two angles formed sum to 180°. This happens twice in a utah, which accounts for 4 angles less than 180°, leaving (at most) two possible angles to be greater that 180°, and thus a utah is not a windmill.

It should further be noted that the Dani principle needs amending to admit the possibility of angles that sum to 360° (when one of them is greater than 180°) or 540° (when they are both greater than 180°). And it should be yet further noted that standards for proof are socially negotiated in all areas of mathematics, and that we had become quite confident that these cases would not come up with a utah, thus the weak version of the Dani principle was good enough for our work.

Finally, if you have made it this far, you may be interested in the speculative origins of the hierarchy of hexagons. It has been a long road.

NCTM

I will be presenting the hierarchy of hexagons at NCTM in New Orleans. Friday morning at 8:00 a.m. See you there?

A few quick words on a function context

Geoff Krall did us all the favor of preserving a brief Twitter conversation about a lovely applications of functions example found by Taylor Belcher.

Go have a look. Won’t take you long.

Sameness in College Algebra

Two years ago, I began using unit as an organizing theme in the math content course for future elementary teachers. That led to many adventures, including a TED-Ed video and new ways of talking to my colleagues about fractions, decimals and place value.

That work continues, but it has become part of my instructional practice; one of my habits of mind.

This year, I am thinking about sameness, and about helping my students to notice and pay attention to sameness. The formal name is equivalence, but I am not so worried about the vocabulary and formal definitions here.

I am concerned with helping students understand something about how mathematics views and uses sameness.

It is awkward at first, as any new teaching moves are. But it got us some good stuff recently.

We are studying functions. Our grounding metaphor for functions is vending machines. We discussed the following collection of vending machines the other day. vending.machine.collage

  1.  This is my favorite vending machine of all time. The banana vending machine. It dispenses only bananas. It is like the constant function. More on this below.
  2. There are two ways to get the Pocari Sweat in a can. Two inputs, same output. That’s OK. It’s not one-to-one, but it’s a function.
  3. You put in a quarter, you turn the knob. Sometimes you get a die. Sometimes you get a top. Sometimes you get a ball. This is not a vending machine, really. Same input gets you different outputs. That’s a problem in the vending machine world, and in the world of functions.
  4. The battery vending machine is one-to-one. Each battery type has its own button to push.
  5. Put a dollar into this one, get a dollar out. Put in five dollars, get out five dollars. The output has the same value as the input. This is the identity function.

We discussed these in class one day. Then we opened the next class session by having students brainstorm with their partners specific functions with the traits exemplified by the vending machines. We divided up responsibilities for recording these functions on the classroom whiteboards.

Here is what our boards looked like after the large group (45 students) discussion. (Click to make legible.)

In order:

  1. Lots of good stuff here. x=2 is not a function because, as a vending machine, it would take your money and not put anything out. All input, no output. The idea that we can write y=5 as y=5+0x was important. More importantly, this led a student to ask* about y=5, “Can it be a variable if it’s always the same value?”
  2. Our example the previous day had been absolute value. They weren’t ready to venture much beyond this. As a class, they struggled to identify two x-values that would generate the same y-value. We need to work on that. But I have mentioned that this is College Algebra, right? Students have placed here, or worked their way here through developmental math. Either way, the idea of producing example points to demonstrate properties of a function has not been schooled into them yet. I’m on it.
  3. Again, +/– square root was the prior day’s example. I love +/– x as an extension of the technique. Love that. And square root of x is not right. We’ll come back to that. Having a permanent record of the difference will be helpful.
  4. Wow. Just wow. That was our example from the previous day. Not even a y=x+3 in the bunch! Work to do here.
  5. Now we’re having fun. I love the y=\frac{x}{1}. Same function, different notation. I finished off our work by asking whether y=\frac{x^{2}}{x} is the same as y=x.

Which (finally!) brings us back to sameness.

My students are highly accustomed to writing \frac{x^{2}}{x}=x. But they are not accustomed to thinking about what this means. Because when x=0, that equation is not true. The question then becomes, In what sense are these the same?

And that points us to the very heart of the discipline.

In mathematics, we decompose things according to their attributes, and we focus on one (or two, or…) of these attributes at a time, disregarding all of the others. Formally, when we write \frac{x^{2}}{x}=x, we mean “These two expressions are the same for all but a finite number of values of x.” We don’t say that, of course, but that is the essence of the equal sign here.

We returned to the sameness question with this video.

Are the two outputs the same? How? Are they different in any way? How? Again, mathematical sameness requires us to specify the precise ways in which two objects are alike.

We will return to machine number 3 above in class shortly. If you just want to get “a cheap plastic toy” out of the machine, then you get that every time. It’s a function. If you want to get “a top” out of the machine, then you get something different every time. Is it a function? Depends on what you mean by “same”.

Much more work to do. I’ll keep you posted.

*I recently argued that learning is having new questions to ask. This student was learning about what variable means, and had a question to ask that she maybe could not have articulated before this.

College Algebra with Christopher

If you assume that Double Stuf Oreos are doubly stuffed (which may turn out to be false), then using the Nutrition Facts labels, you can write the following system of equations, where x represents the number of calories in one wafer and y represents the number of calories in a single layer of stuf.

\begin{cases} 6x+3y=160\\4x+4y=120\end{cases}

The first line represents the caloric content of a single serving (three cookies) of Regular Oreos; the second line represents the caloric content of a single serving (two cookies) of Double Stuf Oreos.

Getting to this system represents some effort on the part of me and my College Algebra students. They are wont to represent their work arithmetically; my job is to help them to transition this arithmetic problem solving to algebraic generality. It is work that I love, but it is hard work.

We had gotten ourselves there, and we had discussed the importance of being very clear about the meanings of our variables when I presented the following graph in class yesterday.

The basic questions in front of us were, What does the blue line represent? What does the red line represent? What is the meaning of this graph?

We had a number of false starts and hesitations. After a few minutes of this, a student pointed our attention to the slope of the red line.

Student: The slope is –2.

Me: Why is it negative?

Student: Because it goes down.

Student: Because if you count the squares over and the squares down, and write rise over run, it’s negative 2, which means the line goes down.

Me: Right. But what does that have to do with Oreos?

A few moments of contemplative silence from 44 college students.

Student: There are twice as many wafers as stufs in the regular Oreos, and the red line represents the regular Oreos.

Me: Right. But why negative?

A few more moments of contemplative silence from the group. This is not a routine they are familiar with, but they are working hard to acculturate themselves to these new expectations.

Me: OK. Let’s do this. Write your answer to this question in your notes.”What is the meaning of x on this graph?”

I allow a few moments for this to occur.

Me: Raise your hand if you wrote that “x represents wafers”.

About 80% of hands go up. I contemplate this. Then…

Student: Isn’t it “number of calories in one wafer”?

Now we have something to work with!

Some lightly edited thoughts about teaching probability

Michael Pershan boldly posted How I Teach Probability recently. I have struggled mightily with the teaching of probability over the years, so I took his video as an invitation to share and discuss.

 

The following is a copy/paste of an email I sent him (with, perhaps, some light editing).

I have been working on a longer, better edited version of the basic idea that I lay out here. But I’m working on tons of other stuff, too, so I cannot promise that the other one will ever see the light of day.

So here you are. Enjoy.

An important bit of honesty first: I am crap at teaching probability. Many reasons, mostly having to do with the ephemerality and abstraction of the topic, in contrast to the cold, hard demonstrable reality of (say) fractions.

With that said, I seem to have been making progress with an approach that has one important extended feature to the sort of work you show in your video.

Your video has (1) a scenario, (2) guesses, (3) small group data collection, and (4) large group data collection.

To that I add (2.5) explicit discussion of students’ probability models that inform their guesses, and then (5) some related follow-up activities.

After collecting guesses, I ask someone in class to describe WHY they said what they said. My job, then, is to press for details and to capture what they are thinking as carefully and accurately as possible on the board.

Once I have that, I ask for someone to describe a different way of thinking about it. Even if slightly different, we get it on the board. My writing isn’t abaout me dispensing  wisdom, it is about making a permanent record of a student’s model in enough detail that we will be able to test it later. For anything even moderately complicated (such as rolling two dice and considering their sum), I am disappointed if we don’t get at least four different models.

Now the data collection doesn’t just tell us who guessed closest; it can rule out at least some of our models.

As an example, in your scenario I would expect something like this:

Model 1: Zero multiples of three and and one multiple of three are equally likely, so I’ll bet on either one. There are only four possibilities: 0, 1, 2 and 3 multiples of three, so the probability of each of these outcomes is 1/4.

Model 2: There is more than one way to get 1 multiple of 3. Our model should account for that. There are three ways to get 1 multiple of 3 (on Die1, Die2 or Die3), three ways to get 2 multiples of 3, and only 1 way each to get 0 or 3 multiples of 3. That’s eight possibilities, so the probability of getting 1 multiple of 3 is 3/8, while 0 multiples is 1/8.

Model 3: Each die can come up either “Mo3″ or “NotMo3″. Getting all “NotMo3″ has probability 1/8 (1/2*1/2*1/2), while getting one “Mo3″ and 2 “NotMo3″ is also 1/8, but there are 3 ways to do it, so it’s 3/8 altogether.

Et cetera

As many different ways of thinking about it as my students have, I will dutifully record. I encourage argument, as each new argument suggests a new model, which we can test.

Now before we roll those dice, we set up a way to test these models. In my experience, I have to devise that test. My students are not sophisticated enough to think this way yet.

In your example, and with these models, I might identify an important difference to be that Model 1 predicts equal numbers of 0 multiples of 3 and 1 multiple of 3, while Models 2 and 3 predict three times as many 1s as 0s. Hopefully we also have a model that predicts something in between.

So now as we roll, we are not just looking for which happens more often, but to relative frequency. Because the model that better predicts the relative frequency that actually happens has got to be the better model.

Much more to say, examples to offer, etc.

Skills practice [#NCTMDenver]

I attended E. Paul Goldenberg’s session on Thursday of NCTM in Denver. It was not at all, as advertised, in keeping with the proof strand. But that does not matter.

What matters is this. Goldenberg shared the video below. The whole video is worth your time, but I have queued it up to the 2-minute mark, where a beautiful classroom sequence unfolds (give yourself about 5 minutes for it).

My eyes tear up watching this sequence. I am neither kidding nor exaggerating. It gives me hope for quality classroom instruction in elementary mathematics.

Be sure to notice the transition to a new task at the 4-minute mark, and how the teacher deals with the struggle that occurs at the 6-minute mark.

Also please look in the kids’ eyes. Watch their body language and their waving hands. Watch them think.

Kids are practicing facts in this classroom. The teacher is providing instruction. Contrast with this.

[NOTE: As of 5/2/2013, the video referred to seems to have been removed from YouTube. My apologies. Go search YouTube for "EDI math" and you'll find plenty of examples that are essentially equivalent to the one I refer to below.]

You can flip this latter instructional sequence because it involves telling and choral response.

You cannot flip the first instructional activity because it involves  adapting instruction in response to student ideas, and it involves students justifying their thinking to the teacher and to each other.

You can’t flip that.

[NOTE: I have edited some of the comments below in order to focus on the practices that were exemplified in the videos (one of which is now private), rather than on the teachers in them. See my post on norms a while back. My apologies to anyone who feels their words have been altered in ways that do not convey their original meaning.]

Experiments with datasets

Go out and collect a modest-sized, discrete dataset. Name lengths of all of the students in your classroom, say, or the number of people in each of their households.

This bar graph is only tangentially relevant, being more of a case-value plot of four different populations. But it breaks up a texty post. So deal with it.

This bar graph is only tangentially relevant, being more of a case-value plot of four different populations. But it breaks up a texty post. So deal with it.

Now play with that data.

If we add one or more new (hypothetical) cases, can we…

  1. Increase both the median and the mean?
  2. Decrease both the median and the mean?
  3. Increase the mean while decreasing the median?
  4. Vice versa?
  5. Increase the Mean Absolute Deviation (MAD) while decreasing the mean?
  6. Vice versa?
  7. Decrease both the MAD and the range?
  8. Decrease the MAD while increasing the range?
  9. Vice versa?

If we delete one or more actual cases, can we…

  1. [same list as before]

Thanks to Susan Friel, Connected Mathematics and tons of other creative folks for getting me started with this. Future elementary teachers to tackle this shortly. I’ll report back.