Feb 072017

Math majors who are interested in teaching are the toughest group of learners. They really are. They are in a mixed science / math class, so they band together. They reinforce each other’s beliefs that the way they have been taught math is a great way, because they have been successful in learning math that way. They then fight against any notion that math can be taught any other way than the way they have been taught.

The struggle is real.

Except last week, in an introductory class, I had a break through. One of the learners asked if a better way would be to have a learner go to the board and do a problem.

I had an aha moment. I asked THEM what they did when a teacher had a learner at the board. They unanimously agreed they tuned out.


Then, I asked how many of them tuned back when the teacher took over.

They agreed that maybe 30% of them did tune back in. The rest (these are all science / math majors who were successful, mind you) said they just relaxed and let the teacher work.

Next, I asked, “If you are the successful learners, how many of the rest of the class tuned back in?” The agreement was unanimous, no one.

My last question sealed the deal for them.

“If only 30% of the successful students tune in, and none of the unsuccessful students tune in, why do you think the way you have been taught was successful?”

The silence was deafening.

That small exchange finally made them think about what success and failure is in teaching.

Success is not the teacher working and the learners listening.

Aug 042014

2014-07-25 12.48.04

My Favorites are some of the best part of the TwitterMathCamp experience, and this year was no different. I know one favorite I had was walking into this building and seeing that even a public high school could afford to build a dedicated Science & Math Center!

But inside the building, we were offering our own My Favorites. I had one my favorite that I offered, which is a cheap (free) way to record your class so you can observe yourself.

Take an old smartphone and remove all the apps. It is best to use a phone that has a SD card, but if you can clean enough space off of an internal memory phone that works too. Once you have at least 4 gig free, then you have enough space to record 45 minutes of video at 720p or 30 minutes at 1080p. Ideally you would want at least 8 gigs so there is extra space.

Once you have this phone ready, you can use whatever you have on hand to construct your own stand. Lego’s work great, a coffee cup, or even a paperclip. Learners will freak out at a tripod and video camera set up in the back of the room (I know, take it from personal experience) while they will not even think of the phone sitting on a shelf recording them.

Two other My Favorites that were offered by others that I really liked are Plickers and a very interesting and annoying problem that has incredible extensions.

Plickers are “Paper Clickers” and it is genius. Using a cell phone or tablet with camera and the paper funky symbols you can poll the class on a question and have the responses immediately tracked and recorded. You can show the class results in bar graphs, and later can use the results for data tracking and demonstrating what you are doing for your admin. Great discussion and engagement in class and  data tracking for later. It is a win-win.

Finally is this problem. IT is tricky, fun, amazing, and all around a well designed problem.


What proportion of the triangles is shaded?

That is it, just find the shaded area. The solution has extensions all over the place and is a great problem to try and work through.

I hope you Enjoy!

Aug 032014














Jan 172014





Asymptototes (vertical and / or slant):






End state behavior:








Aug 182013


2013-08-15 11.41.37 2


After writing this down, I did the following exercise:


What do you notice about this list?

Here is a list of everything I wrote on the board:

















Jun 292013

A great question to understand why we check conditions in stats, “What happens we have a sample size greater than 10% of the population?” One of the themes of Tabor’s institute was what happens when we violate the conditions, and on day one we asked this question. Another way to phrase this question is, “Does the size of the population we are sampling from matter?”

To explore this question we started off with the Federalist paper exercise (a first week exercise in his class). This is very similar to the Gettysburg Address exercise focused on the Central Limit Theorem, where we are sampling from a population of words. The key to checking the condition is that the population we are sampling from is of a limited size.

In this case, we have a population size of 130 words, and we are sampling different sizes of samples.


Sample size (n)


SE of xbar of simulation samples


image= 1.296



image= 0.648



image= 0.290



image= 0.255



image= 0.254


At a sample size of 5, image is a good approximation of the simulation standard error we calculated. But notice, as we increase the sample size, the standard formula for the Central Limit Theorem breaks down. The difference between the two values grows wider and wider as the sample size increases. Clearly the CLT breaks down at some point and is no longer a good estimator of the standard deviation.

What we need to do is adjust the formula for the fact the sample size (n) is approaching the population size, or N. This adjustment is called the Finite Population Correction Factor and is  image.

Sample size (n)


SE of xbar of simulation samples



image= 1.296




image= 0.648




image= 0.290




image= 0.255




image= 0.254




Wow, notice now that the simulation approaches the corrected value! That is wonderful. But why the maximum sample size condition of n < 10% of the population? Let’s graph image on the domain from 0 to 1 (since n/N will approach 1 as n approaches N) to see if that gains us any insights.


So why do we say that n < 10% of the population? Because between 0 and 10% there is only a 5% drop in the adjusted standard deviation, but between 0 and 20% there is an 11% drop (approximately). The curve drops off at a faster rate from there.

Why n < 10%? So we don’t have to worry about adjusting more than dividing by the root(n) and don’t have to worry about the Finite Population Correction Factor! I was blown away by this explanation. It solidifies so much about the reason why we check conditions for me.

But, and this is a very important but, there is more to come on the checking of conditions. Is this condition an important one? Not so much it turns out, and the reasons why are so informative to teaching and understanding stats.

The Fathom File used to generate the simulation of samples:

Jun 292013



1. Is there evidence for <blank>?

2. Is there CONVINCING evidence for <blank>?






So what are our options that could be taking place?







1. What are our two options?

