The Monty Hall Problem: Intuition in the Math Classroom

When I began my observation days last year I was sitting in on a grade eight math classroom that was covering geometry. Students were working on calculating the area of a circle, and one task was the pizza problem. The pizza problem's reputation proceeded it, so I was surprised when it was simply a question out of the textbook that boiled down to whether you should buy two large pizzas or three medium pizzas. Students were provided with the diameters and prices for each size. I don't recall the specifics, but it worked out to there being about 180 cm² more pizza for the option that cost $5 more. More pizza, more money, and no surprises yet. But the question asked which you should buy---what is the better value?

Students were stumped by this, and some even made some very convincing arguments about how they wouldn't need that much pizza for their family, or how three medium pizzas would have more crust and fewer toppings. On it went, but much to my surprise no one figured out to use rates to solve the problem. Maybe it was because my mom took me grocery shopping a lot when I was little and I got to help her compare the unit prices of different sizes and brands, but it seemed so obvious to me. How much pizza do you get for a dollar?



180 cm² didn't seem like very much pizza for $5, so that couldn't have been the solution---it just wasn't worth it. And I agreed---it didn't sound like much! But no one ever tried to show me how much 180 cm² actually was. What does 180 cm² look like? I imagine it to be about one slice of a reasonably-sized pizza, and so I definitely wouldn't spend $5 on it. And this was where the activity delved into students' feelings about the solution.

The students intuitively knew that it wasn't a good deal to pay $5 for so little pizza, and it is this type of reasoning that is very important, yet underappreciated, in math. Math might deal almost exclusively in numbers, but those numbers have a context in the real world, and for students to say that they wouldn't pay that much for the extra pizza was a strong connection to self. But intuition in math goes well beyond pizza sizes because it is responsible for how we decide to approach problems with the information we have at hand. What do we do with our information? What results are we looking for?

Number talks are a great way to promote mathematical flexibility with mental math, but flexibility also applies to the procedures and formulas we use in all areas of mathematics. For example, we could use the sine law to solve for an angle in a triangle, but we might also use the knowledge that all three angles have a sum of 180°. And while this might seem overly simplistic, it is exactly these types of understandings that are lost by students when they focus on following formulas blindly. There are often better, faster ways of doing things, and when it comes to applied mathematics in areas such as chemistry or physics, there are many situations in which those beloved formulas don't hold. What then?



The Monty Hall Problem is one of those paradoxes that truly pries at human and mathematical intuition. Picking a door, we see ourselves as having a 1/3 probability of selecting correctly, and mathematically this is true, but the trick is that when a second door is opened to show a goat, the entire scenario is changed on us without our knowledge. People erroneously assume that there are two doors, one of which contains a goat, and therefore the probability of having a car is 1/2 for each. But what the latter example in the video shows is that the probability of all other doors condenses onto the remaining door after the goat is shown. There has been a lot of debate about this problem and it stems from the fact that mathematical and human intuition are divergent when that goat is revealed.

The one issue with the Monty Hall Problem as proposed in the video is that Monty Hall never allowed his guests to change their door. Contestants were allowed to take cash instead of sticking with their door, but were never allowed to actually switch. And while the chances of winning a car are higher if you could switch, what would be a good strategy if you weren't allowed to? Would it be worth taking the guaranteed cash rather than sticking with your 1/3 chance of winning a car? It's suddenly a much more human problem.

Intuition drives mathematics. It dictates how we understand and approach problems, and sometimes, as with the Pizza Problem, it assesses and breaks down the human components of math when put into context. Mathematicians may sometimes call it conjecture, but is it possible to just know that something is true without being able to properly articulate it? Why not? I am certain that Fermat knew his Last Theorem would hold despite not being able to offer any proof of it. And if others didn't feel the same way, then why would we have spent over 350 years trying to prove it?

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
---Pierre de Fermat, 1637

Technological Devolution

I was sitting in grade nine math class, ten weeks ahead of the other students, when Mr. M came and sat beside me. Mr. M was the resource teacher for the gifted students at my high school and was therefore in charge of planning all IPRC meetings. Mr. M never had much to say to me, and this day was no exception, so as he watched me progress through the learning guides he sighed and said, "Back in my day you had to use a slide rule for calculations like that."

I was unimpressed. A slide rule? How old are you?

My parents had shared any number of stories like this over the years, and it just seemed like such a grown-up thing to say. I did the polite thing---smile and nod---and went on with my work, never stopping to actually inquire as to what a slide rule actually was.

Flash forward fifteen years and here I sit in my attic, surrounded by boxes brought home after cleaning out my grandparents' house, and what do I find but a slide rule. The funny thing is: no matter how long I look at this device, I can make no sense of it whatsoever.



Like this slide rule, technology has a time and a place in education, and it is of no use to students if they are not trained on how to use it properly. Or if as in the case of Mathies, the technology runs on Adobe Flash, which is famously incompatible with iPads, the most common form of classroom technology today.

It seems to me that sometimes we use technology just for the sake of using technology. We feel pressured by the grand ideas of 21st Century Learning and try to compensate for our shortcomings by throwing things like math video games into the mix thinking that they will in some way benefit students, when really we have failed to vet the material and blindly assumed that students will find it engaging. I know that when I sit down and play Demolition Division as a grown man that knows his times tables and has nothing at stake in the game, I still find it extremely stressful. How should students feel?



I learned most of my times tables through Math Castle, which was still considered "new" when I was in elementary school, and I have no fond memories of the game whatsoever. As far as being in the gifted class went, I was the worst at mental math, and so I had to sit and suffer through the constant embarrassment of my castle being loudly obliterated over the computer's integrated sound card for everyone to hear. It was as if those little UFO lasers were not only eroding my castle walls, but my self-esteem as well. Where was the evidence of learning there?

That is not to say that all technology is bad. It was incredibly useful to have a Smart Board during my first practicum. To be able to pull up pre-made materials in Google Slides, then work on the board next to the screen gave me great flexibility in my lessons, and the Smart Board made collaborative data management tasks incredibly effective. And as for my students, Kahoot! was never frowned upon. I used Kahoot! quizzes as check-ins every Friday afternoon, and created them myself so that I knew they were directly linked to our week's work. The response to this was so enthusiastic I have since used Kahoot! for a variety of topics, even reproductive anatomy.



The key to implementing technology in the classroom is planning. There are so many models and tools and acronyms out there that it can sometimes be overwhelming, but I have found that the SAMR for technology is very practical. The actions of substitution, augmentation, modification, and redefinition really put an emphasis on what role technology is playing in the classroom. For example, when students play Demolition Division, technology simply is being used as a substitution for rote learning. The game still looks and feels much like rote mathematics, but with the added unpleasantness of time limits and anxiety. And while there might be a time and place for learning by rote, we often leave it at that---technology simply acts as a substitute for traditional instruction and the learning is therefore still very traditional in style. Games like Ratio Blaster never go any deeper than basic knowledge and understanding, and maybe this is the key to implementing technology in the classroom. How does the use of technology align with the achievement chart categories?

When we think in terms of technology and the achievement chart, it is very easy to use the aforementioned games to address knowledge and understanding. But as we all know, setting up tech in a classroom can be very time-consuming, so why go through all of that trouble to substitute learning by rote with learning by rote? It is only once we begin the augmentation, modification, and redefinition of tasks through the use of technology that we get into the 21st century skills that we want students to develop as well as rich tasks for assessing communication and application. The purpose of technology in the classroom is for enhancing learning, and we must be careful not to treat it as a shortcut for teachers to minimize planning. Technology does not make rich tasks, teachers do.

At Odds with Number Sense

One of the most humbling experiences in my time as an undergraduate student was when I took a first year mathematics course in my fifth year. It had been listed in the course calendar as a statistics course, and I enrolled hoping to get a better understanding of number crunching for the vast amounts of data I was compiling in my electrochemical experiments. As it would turn out however, the professor for the course dealt strictly with number theory in his research, and we soon found ourselves not doing Q-tests, but developing proofs for an infinite set of primes.

Up until that point I had conquered five senior courses in atomic theory and quantum mechanics, and I thought a first year course on number theory was going to be laughable. I was not prepared however to have everything I had ever learned in math put into question. Mathematical reasoning, as it turned out, was a language of its own, and did not look like anything I had ever done before. That is not to say that I did not understand it, but that I struggled with the strict rules and conventions that at times seemed almost arbitrary to me.


Euclidean division algorithm
My experience in this course made me reflect on my entire understanding of mathematics. Yes, I knew what even, odd, and prime numbers were, but when it came to the rules around them I was not so certain. I felt like I was starting from scratch---that there was some hidden truth underlying mathematics that had been hidden away from me all along. And this, I suppose, is how most students feel in the math classroom. Lost, confused, and as if the inner-workings of math are always obfuscated behind cryptic rules and language.

Steven says that when you add two odd numbers with an even number, the answer is always even. Is Steven correct?

If like me you learned your even and odd rules by rote, you're probably thinking that yes, it's always even---because the sum of two odd numbers is even, then the sum of two even numbers is even. But while we quickly come to Steven's defense, many of us probably don't know why this is true. And this is where the way number sense has traditionally been taught is failing us.

Consider some arbitrary set of numbers, S = { 0, 1, 2, 3, 4, 5, 6, ... n }

We all probably know that numbers are either even or odd, and that even numbers are divisible by 2. This is what most of us learned by rote! But what about zero? Is it even or odd? Students will probably wonder this, and I must admit to having never thought much about it until now. Can we divide zero by two? Yes, we can, therefore it is even.

Our even and odd numbers therefore alternate (i.e. even, odd, even, odd, etc.) if we list out all of our integers, and each odd number is always greater than and less than an even number by 1. If we represent an even number using A and an odd number using B, we could say that B = A + 1, for all odd numbers. But what does this have to do with Steven? We haven't forgotten about him, have we?

What Steven is saying is that the sum of two odd numbers and an even number is always even, so let's take a look at the general expression:

A = B + B + A
A = (A + 1) + (A + 1) + A
A = A + A + A + 2

What we end up with is that we can rewrite our two odd numbers so that they are equal to even numbers plus 1. We now have three even numbers, which have an even sum, and two extra 1s. That extra 2 is also even! Would this work with three odd numbers? You would have three 1s, or 3, and....no, it's not even! And thus is the beauty of number theory, because these rules apply to all of the integers.

Elizabeth says that if you add a negative to a positive integer, the sum is zero or less. Is Elizabeth's statement true for all integers?

Elizabeth's predicament is not as clear as Steven's. It might seem obvious that it can't always be zero or less, because it's easy to find two numbers with a positive sum, but can we prove it? Coming at the problem from a number sense perspective, what she is really saying is that for two integers, one positive and one negative:

a + b ≤ 0

Where a is positive and b is negative. However, adding a negative integer is the same as subtracting a positive integer, so let's use some math rules to rewrite the problem a little more clearly:

a + b ≤ 0
a + (-1)|b| ≤ 0
a - |b| ≤ 0
a - b ≤ 0
For all positive integers, a and b.

What we now have is an expression that states that for any two positive integers, the difference is always less than or equal to zero. That can't be! And this line of reasoning is how an understanding of number sense can enable students to succeed in the math classroom, rather than just passing off rules they had learned previously.

EQAO: Meeting the Needs of an "Average" Student

In my previous post I expressed concern over the nature of EQAO testing and its use of closed-ended questions. At the time I felt that no opportunity for differentiation would be afforded by this design, however after attending a professional development workshop concerning EQAO assessment, I now have a better understanding of the role the EQAO serves in our schools.

I should begin by mentioning that we had a guest speaker from the Dairy Farmers of Canada during a recent health and physical education class, for which the presentation concerned the teaching of healthy eating habits for grades 7 and 8 through the use of Canada's Food Guide. While the DFC have worked closely with the Ministry of Education in developing such resources, there was voiced a concern over potential biases and the pushing of an agenda. As someone that does not eat meat of dairy products I can see how one might take issue with the presentation, however it occurred to me while I listened that the presentation was not for me. The presentation as well as Canada's Food Guide are intended to provide information regarding healthy eating to the average Canadian. That is, the average meat- and dairy-consuming Canadian that lives in close proximity to a big chain grocery store. The Guide was developed based on the eating habits of Canadians, and these habits were in turn used to outline ways of meeting our physiological needs from the foods we already eat.

(Note: there are available web- and mobile-based applications of the Food Guide that can be tailored to your eating habits as well as versions for FNMI communities)



I found many similarities between the concerns voiced over the DFC presentation and my feelings towards EQAO testing. How can we create a one-size-fits-all model that suits all of our eating habits or learners? The idea flies in the face of diversity and differentiation. But while I was willing to look past my differences in the case of the Food Guide, I found myself more resistant with respect to EQAO. Perhaps it is my background in mathematics or my experience having an IEP that have made me skeptical of standardized testing, but I never really saw a place for EQAO in education.

The role of EQAO however is much like the Food Guide. Like the Food Guide, EQAO testing is based on a set of underlying standards (nutrition and curriculum, respectively) and serves as a way for us to examine and direct instruction for the average learner. Students with accommodations or modifications outlined in an IEP will find these needs met for their sittings of EQAO, and resources such as manipulatives are also made available for all students. Some degree of differentiation therefore exists in the process component of writing EQAO, however the product remains the same as a result of the aforementioned closed-ended questions answered via pencil and paper. Fortunately, work is being done that will hopefully allow the use of technology to deliver EQAO to students, thereby affording more opportunities to incorporate new ways of differentiating not only process but product as well.

While I remain somewhat skeptical of testing such as EQAO, I have come to see what its intended role is in our education system. I do however take issue with the practice of "teaching to the test." As EQAO is based on the Ontario curriculum, teaching with the test in mind will direct students towards meeting the required overall expectations. The problem however is that in preparing for the test it is not uncommon to have students solve EQAO-like questions (or questions from previous EQAO tests), and this only serves to propagate a very traditional approach to mathematics in which closed-ended questions reign supreme. For us to use EQAO data reliably the test will need to be updated in order to better reflect the teaching and learning practices of the 21st century. Gone are the days of pencil and paper and learning by rote, so why are we still relying on data that is produced this way?



An interesting component of the EQAO workshop was an opportunity to review the assessment tools and attempt assessing a handful of questions (including that pictured above). Open-response questions are scored on a scale from 10 to 40 that seems to roughly correspond with the typical levels of 1 to 4 we would see on a report card. In addition, assessment includes the letters B and I, representing blank and illegible, respectively. I was happy to see these latter components included in the assessment guidelines, as I believe the distinction between a level 1 and having made no attempt at a solution is important.

The assessment conversation became heated when discussing the solution pictured above. Workshop participants were split between a score of 30 and 40, with most favouring a 30. The argument for a 30 was that the student did not show a "complete" solution process, as they had not written out 24-20=4 despite that the question required them to show their work. As with any open-response assessment, some degree of uncertainty will arise, however the solution was assessed as a 40, and I am inclined to agree. Had the question dealt with decimal numbers or fractions, one could make the argument that the last step of subtraction should be shown. However, as the student clearly used the strategy of counting---as evidenced by the marks on the diagrams---the difference is easily noted without this arithmetic. The solution is complete.

So now the real question---and I have been wondering about this ever since watching a video about how the tests are created. EQAO problems, or "items," go through rigorous screening and those that---as it was put during the workshop---"perform nicely" are eventually placed on the test. But when only 50% of students are meeting standards, what does it mean for an item to perform nicely? Are we pulling questions that students are struggling with and replacing them with ones they perform better on? If items are screened so well, why do some fare better than others? Are we introducing bias?

Related: How the "average" person does not exist.

Herding Students in the Wrong Direction

It may come as no surprise that 75% of the students featured in the video were unable to reason their way through the shepherd problem. EQAO scores have hit a new low under the latest Ontario math curriculum, and the Ministry is pushing for renewed strategies that include mandated daily math instruction and the designation of a math lead teacher at each school to help fix this.

While some will quickly identify inquiry-based learning and the new curriculum as the root cause of the decline in test scores, EQAO assessment remains rooted in the closed-ended questions of years gone by and is at odds with how math has been taught recently. It should therefore come as no surprise that students are not succeeding when blindsided by problem sets unlike anything they have seen before.

It is clear that there exists a disconnect between knowing math and understanding math, and it takes no more than a quick show of hands for one to find that people think it is more important to understand math than to know it. This is representative of the shift in mindset that has been occurring in recent years with respect to mathematics education, however this push for renewed mindsets is contradicted by demands for standardized testing and rote mathematical skills. How will students achieve if parents do not value the new methods of mathematics instruction?


Last year a simple math problem marked as incorrect sent the internet into a furor. People quickly came to the defense of the student, pointing out that five times three and three times five had the same solution that we all know as 15. The article however, parsed the definition of multiplication, defending the teacher and their professional judgement.

5 x 3 = 3 + 3 + 3 + 3 + 3
As much as I was raised to respect and think in terms of the strict rules of mathematics, this aspect of multiplication was never harped upon. Sure, if you write it as you would teach it for primary students you would have five groups of three, which is indeed what the definition puts forth. This however ignores the fact that students have traditionally learned multiplication with the help of tables (hence times tables), and that one can quickly notice certain trends with the help of such tools.


The commutative property of multiplication is certainly apparent through the use of such tools, as it would be if students were simply expected to memorize the values of each product. While five groups of three and three groups of five are certainly distinct from one another, I vehemently disagree with the teacher's actions. Context is important in mathematics, and while we do not know if the student had learned previously the commutative property of multiplication, there is no context guiding the question in such a way that the distinction is important.

a x b = b x a
Consider the array the student drew in the next question. I see four groups of six and six groups of four, and we cannot know for certain that the student did not count the groups vertically instead of horizontally as the teacher insisted. At this point I feel the teacher is leading the student astray and sticking to a very traditional mindset, telling the student that there is only one way to solve a math problem. A 4x6 and a 6x4 array are identical without some sort of guiding parameters---for example, a fence that is four feet wide and six feet tall is distinct from a fence that is six feet wide and four feet tall---and rather than penalizing the student, it would be an excellent opportunity to ask deeper why or how questions to probe for understanding.


Approaching multiplication from an inquiry- or exploration-based approach, we must tread carefully around such scenarios. Students will quickly discover the commutative property on their own, and may use this to simplify problems and make their lives easier. I certainly would not go about multiplying 5 x 3 by adding five threes when I could much more easily add three fives. It's a shortcut, a mental math strategy of dissecting a question in a way that makes it simpler, and we should be happy to see students drawing such connections and discovering these relationships.

Are these triangles congruent if one is upside-down?

Est-ce que c'est un cadeau? Quelqu'un m'a dit.

Quand j'étais jeune, je croyais qu'il y avait certains qu'étaient meilleurs en mathématiques. J'étais bon en maths, et je pensais que c'était un don que tout le monde n'avait pas. Aussi, je pensais que tout le monde ne pouvait pas faire bien en maths, et semblablement, je pensais que je ne pouvais jamais faire le français.

Cet été mon épouse m'a fait prendre un cours d'immersion français aux Québec. Je n'étudiais jamais le français sérieusement, donc j'ai pensé qu'il serrait trop difficile pour moi, et je n'y voulais pas aller. J'avais une mentalité figée.


On dit souvent que les mathématiques sont comme une langue, et si cette idée est vrai, on peut comparer les deux. J'ai passé seulement deux mois aux Québec, mais chaque jour je pratiquais la langue verbalement et en langage écrit---je n'avais jamais le temps pour lire beaucoup. Je pratiquais tout le temps et j'améliorais.

J'améliorais beaucoup, mais au début j'ai pensé qu'il ne marchait pas pour moi. Après deux semaines, j'étais découragé et j'ai réfléchi à ma situation. Sans mon épouse, ni ma maison, ni mon chien, ni mes amis. J'étais sans espoir, et il restais trois semaines de l'école.

Évidemment, j'apprenais un peu de français, et peut-être plus d'un peu. Je suis certain de faisais beaucoup d'erreurs---probablement, dans cette phrase---et ce post me prend une éternité d'écrire. Mais j'essaie.

Si les mathématiques sont comme une langue, on a besoin de pratiquer souvent. On a besoin de croire qu'on peut faire les mathématiques. On a besoin d'une mentalité de croissance, parce que c'est une mentalité de croissance qui permet à quelqu'un d'apprendre vraiment les mathématiques. Car si tout le monde étudie les mathématiques chaque jour à l'école, pourquoi est-ce que c'est difficile pour beaucoup de gens? Pourquoi est-ce que le plupart de gens deteste les mathématiques? C'est les mentalités figées qu'arrêtent la connaissance. Comme moi avec le français, des élèves croient elles ne peuvent pas faire les mathématiques.

Nous avons besoin de changer nos mentalités et les mentalités de nos élèves, parce que si je peux apprendre le français en deux mois, des élèves peuvent apprendre les mathématiques à l'école.