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

At TMC14 (Twitter Math Camp 2014) this year I did not attend many sessions, because I was the co-lead or lead in several blocks of time. It was great, and the comments I received were very complementary. I think the teachers telling me that were just being nice a little bit, but I hope they did receive some benefit from attending. With that said, the first thing I want to do in my TMC Recap posts is communicate some of what occurred in the sessions.

First up, Algebra 2. I co-led these sessions (there were 3 days of 2 hours each) with Jonathon Claydon (@rawrdimas) who blogs over at InfiniteSums. The 3 days were split into the following structure. Day 1 was about how to teach algebra 2 with some structure and form so that you can connect all the disparate topics of Alg 2. Day 2 was about a different way of cycling through the topics to allow for constant review and building of knowledge (pivot algebra), while day 3 as all about modeling.

Day 1 started off with the question, “How do you currently teach alg 2?” We had several answer. Graphing all the parent functions and creating a hook to hang the rest of the year that way (Family of Functions), or solving the equations and connecting the graphs later (equations first), going through the textbook units and color coding them, and then I introduced my (h, k) format. There was great interest in the (h, k) structure so we spent the rest of the time on that method.

What is that method, you ask? Well, on my board under the heading of Algebra 2, I have the following forms written down:


First off, what do you notice and wonder about all these forms? Yes, I do ask that and spend some class time on the noticings and wonderings about this list. I actually have a “You are Here” note that moves from one to the next to the next as we go through Alg 2 and I make a big deal about that move.

The really nice thing about organizing the class in this way is that clearly the learners are learning ONE set of math operations, not 12. The amazing similarity between all of these forms encourages the learners to actually look at the math and ask “what is the same, what is different” and STOP thinking “all of this is different each time.” It takes some work, but the learners figure out that my 3 rules (the ONLY 3 rules I allow them to use/ write/ or say in class) are how ALL of these functions are solved. [make sure you read the comments too]

Also, shown (but not handed out) during the session was how I consolidate all of the maths for all of the functions and what I expect for every single function listed. It looks like this:


All of the links for the handouts and materials are on the TwitterMathCamp Wiki site. If you want this handout or any other handouts from TMC, please feel free to download them.

My goal with this process is getting the learners to think of math as ONE body of knowledge and not a segmented series of things we memorize. We LEARN how to factor, how to graph, how to identify points on a graph, and we USE that same knowledge over and over again.

I have had some success with this last year and I am looking forward to doing it again and blogging about it as I go. Yes this means I am planning on blogging more. That is one goal I have for the year. It was created because of this article on the secret to writing. (hint, there isn’t one.)


Jan 172014

[I really need to return to blogging. My lack of focus on reflection has hampered me this semester, and I need to fix that. To that end, I am making a commitment to blog and to jog. Those are the foci this year of the ellipse that is my world.]

Yea, how often does that happen that a class gets excited about logs? It has not happened to me in several years, but this year I found a way. We started the second semester with graphing again. We have a standard list of things we look for, identify, and document on every single graph. The list is:



Asymptototes (vertical and / or slant):






End state behavior:

Every graph we do, we have to document all of these items. If we graph a line, most of the list is “none” but it creates the connection between all the graphs. Every graph has the same questions, it is just that some of the graphs / functions do not have those features.

So, I am doing this file on Desmos, and we are documenting. They have done all these as homework, so really we are checking answers and ensuring learning. Then weird things happen. They notice the symmetry of the inverses.


Then they ask to see the graph of the line of symmetry. Even nicer. THEN! OMG. We put the translation into the h-k form of the line, and we see the translation of the line of symmetry.  [Okay, seriously. If you are not using the h-k forms to make connections, why not. See This post, or This post or any other of the several posts I have on this topic.]

And then I graph the exponential. …. …. They know there must be an inverse, but nothing we have done in class looks like that. …. And then, because I have the list of all the h-k forms on the board, someone asks, “Is that what the log thingy is for?”

And now they have a reason to learn logs. They are intrigued by logs. They are asking questions about logs. Because EVERYTHING in math has a forwards and a backwards, addition has subtraction, squares have square roots, and exponentials have logarithms.

They are interested and inquisitive about a topic that normally is not approached this way. I have done something good I think. Only time will tell if I can continue that on this topic.

Aug 182013

So the other day I am giving my Algebra 2 STEM learners (this is our Honors designation” the big picture of Algebra 2. You know, so they know where they are at and where the class is going. This is what I had written on the right of the screen.

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:











Then I asked what do they wonder. This took some time, and I was REALLY glad I gave them lots and lots of processing time. Some of the later wonderings took some time to develop.

THEN I asked for a show of hands how many people wondered the same thing. It was amazing that most of the learners wondered the same thing about the important math. I was very pleased.


I think the list of “vertex functions” will be written on the board as we go repeatedly so they always have that reference of where they are at in the process.

What do you Notice, What do you Wonder? is a very powerful tool to get the learners involved. Max Ray at The Math Forum at Drexel has done a lot of work (even applied for a trademark for the Notice / Wonder phrasing) in this area.

The “What do you notice?” and “What do you wonder?” questions are great ways to pull even reluctant learners into the process of engaging with the material. I know I will be using it again and again.

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

I spent the last week at the Silver State AP Conference this week with Josh Tabor as the instructor. First, let me just get this out of the way. If you ever have the chance to spend some time with him, do it. Do not pass Go, do not collect $200, just go directly to the event. His knowledge of statistics and the pedagogy of teaching statistics is amazing. I will have several posts coming in the next day or so. Some of what I am posting will be pedagogy, some will be content, but all will be useful to the AP stats instructor (namely, me.)

I want to begin with some basic questions and formulations for asking question for the entire course. I think one of the best thing Josh did pedagogically was asking the same two question over and over again.

1. Is there evidence for <blank>?

2. Is there CONVINCING evidence for <blank>?

He started every course of discovery with these two questions. We would look at an AP question, or any question, and this is how it would start. Do we have evidence? Do we have convincing evidence? The next step, however, is evidence for what?


Let me lay out a problem for us to look at. I take a deck of cards and tilt it heavily or completely towards red cards. Make it look like it is a brand new deck, however, so the learners don’t initially think you are cheating. Then, tell them if they pull a black card out of the deck they will get a candy bar, or extra credit, or something.

The first person pulls, and of course, they get red. Aw shucks, no big deal though. Person 2, person 3, etc. At some point the class is going to accuse you of cheating. Of course they are, you ARE cheating, after all.


So what are our options that could be taking place?

1. The deck of cards is a fair deck and the learners are unlucky.

2. The deck of cards is an unfair deck and is Mr. Waddell is cheating.

These are the two options we have, and towards the end of the year we will recognize these are Null Hypothesis and Alternative Hypothesis statements, however at the beginning of the year (heck the first day of class!) these are easy and accessible statements to write down.

Next, I ask the class, do we have evidence for one of these statements? Yes, we clearly do have evidence that Mr. Waddell is cheating. 5 learners in a row got red cards.

Do we have CONVINCING evidence for one of these statements? Now, in the first week of class, we can have a discussion of what convincing means without getting into discussions of alpha or significant. We can think statistically without the math.

This line of questioning is repeated all year long on every question.

1. What are our two options?

2. Do we have convincing evidence for one of these two options?

And so begins the adventure and journey called AP Statistics. I will show more of this structure on questions to come.