My elective math class has all grade levels represented within it. While not ideal, this is a feature of our scheduling system. So I have freshmen in Algebra I all the way up to seniors that have already completed Calculus AB. This poses a huge classroom differentiation challenge each Monday, Wednesday and Friday we hold class. Here is an instructional strategy I use that gets students writing about the mathematics they do in class.

Students worked on one of four things in the computer lab last Friday.

1. Construct 2013 Probe I Problem 7 diagram (2D)


2. Construct 2013 Probe I Problem 23 diagram (2D)


3. Construct 2013 Probe I Problem 11 diagram (3D)



4. Work on Alcumus problems independently

After we spent approximately 55 minutes in the lab, we returned to my classroom for a writing activity.

Here is the writing prompt I put on the board:
(Writing exercise on a separate sheet of paper to turn in to me)
Think about what you learned about the diagram or diagram(s) you built in Geogebra. Write a letter to the you of October 12. How did building diagrams in Geogebra help you understand the problem better?

I put eight minutes on the clock and informed students they would need to continuously write for the eight minute timeframe. Here are some samples of student writing from the activity. For convenience, I have inserted another copy of the problem. Immediately below each problem appears student writing associated with the problem.















The Students revealed their thinking about these math problems throughout their writing. While some students chose to concentrate on the construction process in Geogebra, others also revealed some of the mathematical structure they encountered while making the diagrams.

Writing samples from students that worked on Alcumus:













These writing samples revealed to me the depth of student thinking going on in the classroom. If I could have a superpower, I would be a mind reader. Then I wouldn't have to guess at what my students are thinking. Having the students write for an extended period of time gives me insight into how they are seeing the mathematics and gives me ideas on how I can help further their understanding and guide them as they struggle.

I collected these writings immediately after students completed them. I ran the pages through my ScanSnap scanner and converted them to a PDF for me to review later. I told students we would get these writings back out in a month's time, emphasizing the need for specificity on what they were working on and what they learned that day.

Going forward, these writings help me be more efficient with respect to differentiating classroom instruction. We don't need to be working on the exact same thing at the exact same time at the exact same pace for the students to engage in meaningful problem solving.

My wife is a pharmacist at a hospital. She and her co-workers often provide me with some really interesting math problems. Clinical pharmacists often need mathematics to effectively work with complex patient cases. Modeling drug interactions becomes tricky as the number of prescriptions for a particular patient increases. As the number of prescriptions increases, the amount of time a pharmacist must spend doing drug interaction research also increases. Time is money, so herein lies the problem.

One of my wife’s co-workers asked her to ask me, “What is the number of possible permutations of seven days?” After looking at a handwritten note and talking to the pharmacist directly, here’s a more thorough description of the problem.

Coumadin is a drug used to treat issues associated with blood clots.

Risks come with any anti-clotting drug. If the patient has a car accident, for example, bleeding risk increases dramatically and can have dire consequences. The benefits associated with Coumadin usage must be weighed with the bleeding risk. Computing the correct treatment scheme – a schedule and selection of doses – is important to the patient’s safety. Doctors and pharmacists determine dosage based on a target value: INR target value of 2.5, target range of 2.0 – 3.0 (a confidence interval!!), and if the INR goes above 4.0, there is no greater therapeutic benefit to the dose and patient bleeding risk increases beyond any benefit.

Dosages for this particular drug vary dependent on many factors. However, for the sake of this problem, the pharmacist in question wishes to investigate how much time it will take to write an Excel spreadsheet to determine the different possible treatment schemes. We will assume the simple case: either a patient takes a dose (pill) on a particular day, or a patient does not.

Back to the original question: why does the pharmacist ask for possible “permutations” of the days of the week? Because these would correspond to patient dosage schedules. For example, if the patient takes a dose three days a week, they might take that dose on Monday – Tuesday – Wednesday and not take a pill the rest of the days of the week. Or they might take a dose Tuesday – Thursday – Saturday. All the possibilities would not correspond to permutations, despite the wording of the original question. What we really need to consider are combinations. Permutation means order matters, so we would treat Tuesday – Thursday – Saturday and Thursday – Tuesday – Saturday as different events, when in reality they would be the same treatment schedule in a given week.

If the patient takes a dosage of Coumadin three days a week, then the possible number of treatment schedules would be 7 choose 3.




There are 35 different possible ways to choose 3 days out of the 7 days in a week.

How would we count the number of possible treatment schedules assuming a patient either takes a dose or does not take a dose each day? We need to consider all the ways to select different groupings of 7 days. Enter Pascal’s Triangle & binomial coefficients.








The bottom row above corresponds to each case. 7 choose 0 would correspond to a patient not taking any Coumadin. There’s only one way for that to happen: the patient takes no doses. This would be the trivial case. We aren’t concerned with the patients not taking any doses. The blue number, 128, is the sum of the values in the row with 1, 7, 21, 35, 35, 21, 7, and 1.

Written a little more formally, we have





This value, 127, is 128 – 1, which is , where the 2 corresponds to the number of daily outcomes, either dose or no dose. The 7 corresponds to the number of days in week, and the subtracted 1 corresponds to the trivial case where of the 7 days, the patient takes doses on 0 days.

(Tangential side note: This value, 127, is the difference of a power of 2 and 1. I am immediately reminded of the original Legend of Zelda game on the Nintendo Entertainment System. The maximum number of rupees – currency – a player could obtain is 255, which is 256 – 1, and seems to be related to binary storage limitations of the game.)








This was the answer the pharmacist needed to communicate the number of different outcomes his Excel spreadsheet would need to consider. Some of the treatment schemes might be impractical, so instead of considering 127 different possibilities, he will argue they should boil the cases down to 10 or 12 common dosage schemes.

But for me, the math doesn’t stop there.

What if a person is taking a drug with a more elaborate dosing scheme? I wanted to put a structure on the next case up in complexity, the case where a patient might take one of two different doses on any given day. Here is the structure I used when reasoning through this case initially.

Case 2: Two different dosages on any given day
A = 5 mg dose of a drug (an arbitrary concentration)
B = 2.5 mg dose of a drug

1 = Monday
2 = Tuesday
3 = Wednesday
4 = Thursday
5 = Friday
6 = Saturday
7 = Sunday

A patient might miss a dose. Let C = no dose. Or, they might be instructed to take nothing on a particular day. Either way, the patient takes no dose. The table below describes all the cases for a week.


To count all the possible outcomes efficiently, we have 3 independent choices each day. At least, we will treat each treatment choice as independent although this may or may not be practically true. The total number of possible treatment schedules, then, would be




But we would also throw out the trivial case (no treatments on any day) by subtracting 1. So our total number of possible treatment schedules would be 2,186.

I then wondered if I could write a function to count the number of treatment schedules for any possible number of different dosages.




d = the number of different dosages a patient may take
t = number of days in the timeframe of reference
7 above for days in a week; we could change this to 30 for days in a given month

The number of cases grows quickly as the number of different dosages increases.







This problem is a great practical example of mathematics used in an authentic example. I can extend this problem to exponential decay and dosing with antibiotics. Kids need to know there's a reason a person needs to take an entire course of antibiotics, even if they are feeling better midway through the treatment course.

I will pose this question to my students when we start our unit on counting theory this coming school year. I need to spend more time this summer finding authentic applications locally.

When we teach children to add, subtract, and multiply, we commonly see a vertical format used in school.

+ 6

Then, the student learns division. Students usually see the division algorithm represented horizontally.

1 HorizontalDivide

A student may not, beyond a worksheet with dozens of problems, be comfortable with a vertical format for division.

It almost feels strange to me to the type the problem vertically. Then comes fractions. We did the following exercise in a workshop I led today. I wanted participants to consider the question, “Does the size of a fraction bar matter?”

Below is an image of the problems we worked on in groups. Calculators were allowed. Spirited discussion and abundant disagreement followed.

2 WorkshopFractionProblems


As the back of the textbook often says, “Answers will vary.” The participants’ responses, split up by problem, appear below.

3 WorkshopAnswersGiven

How can we resolve this conflict? Let’s type it into a reliable computation source, like Wolfram Alpha. It depends on how we type the input.

4 Wolfram Alpha First Entry


Above is one way to type the input. Below is another way to type the input.

5 Wolfram Alpha Second Entry


Why the difference? Because multiplication is commutative, but division is not. 3 x 4 is the same as 4 x 3. But 3 divided by 4 does not yield the same result as 4 divided by 3.

In the original input, where I typed 3/4/1/2, the computer interpreted this as the product of 3 and the multiplicative inverses of 4, 1, and 2, like this:



In the second example, we see the computer associates the fractions ¾ and ½, treating the middle fraction bar as a grouping symbol.




The confusion for me stems from converting between vertical format and horizontal format. We teach students to follow order of operations, to divide in a horizontal statement as they encounter it from left to right. In school mathematics, we commonly write the fractions vertically when we mean to take the ratio between two ratios, specifically ¾ and ½ in this case.

Here is a state assessment practice item, taken from the Nebraska Department of Education website, that demonstrates this understanding of the vertical format when dividing fractions.

8 NDE Site Image 19 NDE Site Image 2













10 NESA Item



Let’s look at this Word document, specifically this problem. The author of the item intends for students to invert 7/3 and multiply. I worry a student may see this problem in a horizontal format as 11_NeSA_Item_Mult_Invthe work shown to the left, which gives a result which is obviously not an available choice, but if we are examining item reliability, I wonder if we should have the conversation about whether this expression should be typed in the following way:




Again, I’m just posing the question about the test item. I want to be careful in how I represent problems with my students. I want my own students to understand the context matters. I want my students to be procedurally fluent with fractions when they solve problems in the world. The conversation in this morning's workshop was great. Participants reflected on how what they say to their students while they teach the students can have unintended consequences on how students view procedures with fractions.

Here’s a link for a great blog entry on this topic:



One thing I hope my students learn in school is how a person can unlock their creative potential and pursue their passions into adulthood. To this end, our Math Theory & Problem Solving (MTPS) class took a field trip to do some data collection. I would like to give special thanks to Daryl Payne for allowing our MTPS class to enjoy data collection (racing cars). Daryl's creativity and passion for racing inspired the students to re-imagine what is possible in the world outside school. The video below shows the electric car race track where students raced.

We spent our lunch period having a pizza party prior to racing cars as a reward for the work students have been doing in class. My original motivation in this trip was collecting data and trying to determine how a person could use statistics to potentially detect cheating through exceptional lap times. However, there are also many other mathematical and statistical ideas we can explore with what we learned on this trip.

Here are some things that students wondered about and could lead to mathematical investigations:

  • How much longer is the outside lane than the inside lane?
  • How does the electronic timing system work?
  • How much voltage/current is being supplied to each car?
  • What would a person have to know about electronics and circuitry in order to build such a track?
  • What is the difference between cars with magnets and cars without? (Cars with magnets can maintain higher speeds around the turns, for example)
  • What amount of voltage causes a car to fly off the track?
  • How should we determine the best racer? The fastest lap time? The best median lap time?
  • Is there a difference between the performance of the blue guest car and the silver guest car? If so, how could we detect this difference numerically?
  • How does the "KILL POWER" switch work?


Below, Mr. Payne gives the students some guidelines to follow while they practice racing on the race track.

This short video shows the beginning of a head-to-head race between students.


I am interested to see the types of mathematical investigations that spring up from our field trip. We will take our race data and use it to determine the best racers.


Disclaimer: The following is an idea I have been thinking about this week. I have absolutely no idea whether it would work. I don't know if anyone anywhere is approaching principal work in this way. Some of this has been daydreaming or thinking during a long drive to a conference. These ideas may seem disconnected, but I will try my best to explain the relationships I see between these ideas.

We had some professional development days to start this week. I enjoyed two presentations Monday by David Webb from the University of Colorado Boulder and the Freudenthal Institute. His morning and afternoon sessions focused on formative assessment in mathematics. When he saw many morning participants planned to stay for the afternoon session, he quickly talked about something he and his colleagues use to teach early computer programming concepts to middle school students.

Dr. Webb posed this question to the audience: how do we design intelligent ghosts that will actually chase Pac-Man? The mathematical process, known as collaborative diffusion, describes a possible method for programming ghosts to effectively chase Pac-Man. Here's a link to an academic paper on collaborative diffusion by Alexander Repenning. A screenshot from the paper appears below.


Think of the spaces around Pac-Man as the yellow mountain above. The ghosts want to climb the mountain - and effectively destroy Pac-Man - by climbing to the top of the mountain as quickly as possible. I was thinking about this idea of how the ghosts are chasing down Pac-Man. Then, I thought about how we often in school try to chase down behavior. For example, when a teacher is in the hallway greeting students, sometimes amorous couples try to hide from the teacher's line of sight. If the teacher has to help a student in the class, and cannot man the hallway post, then the threat of punishment is gone. Speeding tickets then came to mind. I thought about how punishment rarely works well as a behavioral deterrent. Drivers may choose not to speed when a police officer is nearby, but once the police officer leaves, look out.

To tie this stream of consciousness back to teaching, think about how often teachers must identify, on the fly and while making mental decisions regarding content delivery, students misbehaving in the classroom. Proximity works well as a deterrent - walking near the student, pointing at the open book on the student's desk as the teacher walks by - but this technique also has its limitations. As soon as the teacher walks away, the student may misbehave again.

Then I thought about how tough it can be to be a principal. Here's a great post on how to navigate the frequent interruptions a principal faces. The principal position can sometimes be very similar to the function of a police officer - a deterrent. But, as a principal leaves, so does the threat of getting into trouble, and the idea is the same as the teacher that walks away from the student's desk. How do we address this behavioral piece while teaching? How do we keep students on task?


The possibility of being called on randomly. 

While thinking about police officers, I thought back to another article I read in a grad class about The Santa Cruz Experiment. The article, which appeared in Popular Science, described predictive police work. <think Minority Report> A mathematician designs an algorithm based on data for allocating patrols. Though random phenomena may be wildly unpredictable in the short term, long terms trends and patterns emerge.












Tying this back to the principal idea... if the ghosts chase Pac-Man... doesn't the principal chase the behavior? Suppose we try to incorporate a random mechanism into the principal's behavior in an effort to make chasing this behavior - just like the patrols in Santa Cruz - more efficient. Let's set up an imaginary simulation. We will declare the following events as things the principal could do.

0 = observe 1st floor hallways
1 = observe 2nd floor hallways
2 =  observe 3rd floor hallways
3 = observe 1st floor classrooms
4 = observe 2nd floor classrooms
5 = observe 3rd floor classrooms
6 = observe school entrance / parking lot exit
7 = monitor stairwells
8 = monitor cafeteria
9 = monitor library

Then, we could use some sort of random process to generate a random behavior for the principal.


Looks like today's focus will be first floor classrooms. Because all outcomes are equally likely, we now have a mechanism like the Popsicle sticks in the classroom. This will be a more efficient approach to deterring negative behaviors among students as well as teachers. This would also give the impression to students that the principal could be anywhere. Thinking back to collaborative diffusion, and Pac-Man emitting a scent that can be chased down by ghosts... the metaphor places data in the role of the scent. We have plenty of sources of data on student misbehavior. Also consider certain events more likely given certain days of the week and months of the year. Isn't a student more likely to get a discipline referral close to a vacation, after a long block of no days off from school, because teachers' behavioral tolerance is lower? Isn't a staff member more likely to violate dress code on a Friday? Aren't students more likely to be off-task close to passing periods? We could use data (and a different random digits assignment scheme) to make an attempt at 'predictive principalship' much like the predictive policing in the Santa Cruz Experiment article.

It would be interesting to see whether this is a viable strategy for administrators to use.

P. S. If you've made it this far in the article, please be sure to read the disclaimer at the top of the article a second time.

P.S.S. I know the title doesn't quite jive with what was discussed here... since the metaphorical principal is the ghost and the metaphorical Pac-Man is the behavior... but the title is way more catchy this way.