Posts Tagged ‘teaching’


Occasionally, students ask me questions about why math works the way it does. This doesn’t happen enough, in my opinion, but it does happen.  The latest is why, when we perform synthetic substitution, do we use the number as it is, but in synthetic division, there’s some sign trickery.  I have an answer for this.  It’s not super complicated either, but whenever I’ve given the explanation in the past it’s been frustrating and incomprehensible to students.  So, I don’t know if it is worth the time it takes to do that, and that kind of understanding is not necessary for skill performance.  But then, what am I doing? If we don’t care about understanding or make that kind of tradeoff of “well, they don’t really need to know this” that feels like the opposite of mathematics education.  I’m pretty good at explaining things like this, so I don’t think it’s me.  There is such a thing as developmental readiness and that’s what’s happening here.  I think.

What do you do when explanations are over the students’ heads?  Is that just a normal thing that resolves itself with further mathematics study and we shouldn’t worry about it?  I find “that’s just how it works” a completely unacceptable response.


Micro rewards have weird effects on decisions

Micro rewards have weird effects on decisions

How would you implement micro rewards for wrong steps in a classroom?  The idea would be to either have students get savvy enough that they didn’t take the reward automatically or else learn from the experience of receiving a reward that was short-term good and long-term bad.  The easy answer is candy.  Like, “I’ll give you a piece of candy if you eliminate 13x instead of 2x” when 2x is the right thing to do.  The extension and real lesson here is that sometimes people are trying to get you to do something that’s better for them than you by offering you a short term reward.  This article is really about game design, but it has instructional implications.

It’s easy to see the implications of rewarding correct steps.  This is shaping.  But even that is questionable according to the paper in the linked article.  The paper concludes that when someone is rewarded for correct steps, their learning is not transferable to novel problems, only to identical or very similar problems.

Super Teacher Tools

Super Teacher Tools

Maybe the neatest site I’ve seen for whole class math games.

linear combination

At a meeting earlier this week, some of the other math teachers were saying that they didn’t teach their kids about the LCM or how to choose a variable to eliminate when solving systems by linear combination.  I was stunned.  Instead of using the LCM, they said they told the kids to always swap the x coefficients and multiply, being sure to make one equation positive and the other negative.  Let’s leave aside the sloppiness of the language here for a minute.  I can’t really imagine using an approach geared to the lowest level students.  It doesn’t seem fair to everyone else to present material that ignores what ought to be prior knowledge, and after all “Algebra 2” means something. 

But they went on to say that students picked up on the variations on their own.

So, this is a dilemma: should I present a severely dumbed-down approach, hoping that students will make the extensions themselves, or be more comprehensive in my presentation?

If I’m serious about students deciding what and how and how much they will learn, I have to be willing to take the risk of them being lazy.  So I did that today.  The thing is that they’ve seen this before, in Algebra 1.  So, I don’t know how it would have worked with a class that had not seen the technique before.  Luckily (hah) I have an Algebra 1 class too, and can try it there later this year.

curriculum choices

I’m trying to find ways to get my students involved in the curriculum that I teach them.  To that end,  I would like to ask this question:

Is there anything in your daily life that you think has something to do with math, but you don’t know what it might be?  Something that you’re wondering about?

The old way of using a question like this is to try and find matches between their interests and the planned curriculum.  That’s fine as far as it goes.  Teachers can also ask students about this kind of thing as the class progresses, which would be called “making it relevant to their world.”  I don’t think that goes far enough either, but at least it’s not me finding applications.

The danger here (danger!) is that the things they identify are over their heads.  That’s entirely possible.  So what’s the right answer to that?  You don’t know enough math yet.  Investigate that on your own (hah.)

And the challenge, if it’s completely unrelated to the planned curriculum, is to find time to address whatever part of these answers can be brought into their zone of proximal development.  Even assuming that they all have the same proximal zone is a huge error.

What to do…


What difference does it make if I’m more creative as a teacher?  There’s no guarantee that creative lesson planning leads to growth in creativity for my students.  I could use super creative methods to bring students along rails to an objective that I’ve designed, but it’s still me making the hard decisions for them.  If we’re interested in promoting creativity in students, we need to think about curriculum development.

Contrariwise, I could use the same approach every day with open ended tasks that allowed free form solutions, but those wouldn’t always go where I wanted them to.  So that’s not a problem unless you’re determined to control the outcome.  But that’s not how creativity works, right?  You don’t go in looking for a certain and specific outcome.  You don’t need to be creative to get an outcome that you can see in advance.

Taxicab geometry

The warmup activity was designed thus: I placed a coordinate plane on the slide before the kids arrived, and marked several points.  I then removed the coordinate plane and put the question on the slide: Which two points are farthest apart?

It’s not a bad question but it’s also not inherently interesting or natural.  Students looking at the cloud of points would not automatically want to know which ones were farthest apart.  So, the activity could be better (any suggestions?  Cities?) but it didn’t appear to demand any math knowledge to answer.  This is very appealing to some of the kids, and in fact it seems like you have to do this kind of thing to reach those kids at all.

After about a minute, I tell the kids that I’m curious how they’re putting this warmup on their paper at all.  They’re eyeballing.  If one pair of points is obviously farther apart than any other pair, eyeballing would be fine.  I have cleverly marked the points so that you do need some accuracy.

I ask for guesses on which points.  (students in bold) Kids point vaguely at the points.  Can we label them? OK what should we do to label them?  Can we just use letters? That’s a good idea and that’s how they do it in Geometry anyway, so let’s do that.  We label the points A-F and we are able to put guesses on the board.  There are 5 guesses.

But how can we know for sure?  Unless it’s a tie, but we would need a way to find that out too, right?  Can we measure them do you have a ruler? I have two rulers; do you want inches or centimeters?  Most of the yelling says inches and the kid who actually goes to the board decides to use centimeters.  Some of the students are disgusted by this.  So why centimeters?  Well, I just thought it would be easier. More groaning ensues because inches are so easy, and I say what do you mean by easier?  Well, centimeters are smaller so if the answers are close, it’ll be easier to tell with centimeters than inches because you’d have to use part of an inch. This gets some respect from the students and of course from me.  He measures his guess and hands the ruler to someone else, as is our custom.

Three of the guesses turn out to be 59 centimeters apart, which was not intentional but will be from now on.  Are they all exactly 59 centimeters apart?  No, probably not… So then what happened?  It’s hard to measure up there, and the dots are pretty big so if you measure from the center or from one side of the dots you get a different answer.  This is true and really good.

Still, we need more precision than we can get from the meter stick and eyeballing.  We’re back to how to mark the points on paper.  Let’s put a graph on it. A graph?  What do you mean by graph?  A grid. Really?  Grid?  Really?  (this continues for about 30 seconds) A coordinate plane. Aha let’s do that.  I put a coordinate plane (20×20) haphazardly on the slide and ask if that will suffice.  No?  Ok someone come up here and move the coordinate axes to a more convenient location.  Do you guys want graph paper?

I could narrate the rest of the period, but basically we develop distance formula from the Pythagorean Theorem (converse.)  This is nothing new.  I want to relate one other interesting exchange in detail.

So, what do we do with the vertical and horizontal distances now that we have them?  Add them.  Ok, so that’s 15 + 3 = 18.  Are these two points 18 spaces apart?  Yeah.  Let’s check.  I turn to the next slide and bring out another copy of the coordinate plane and draw the desired distance (AE) and another segment that we can count and see is 18 units long.  These are different, so the points aren’t really 18 spaces apart.  Maybe not, but if we do this with all the points, we can compare the results and tell which points are farthest apart.

This is really interesting, and in fact the sum of the x and y distances is called the Manhattan distance.  It’s a valid way to express the distance between two points, it’s just not what we’re doing right now.  I feel like I should have had him pursue that further, but how?  Or is that even worth anything?