Guessing: The First Step in Mathematics

by Jonathan Osters, The Blake School

“To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.” — George Polya

Often, some great understandings can be made through the most menial of scenarios. Early in our geometry course, we make sure to tell our students that one of the most important things they can do is form guesses. Hunches. Conjectures. If those conjectures work out in the end, that is wonderful, but if they don’t work out, we get the chance to go back and revise.

Yesterday we started with a seemingly trivial question: If we create a circle and place points along the edge, then connect all the points with line segments, how many regions can be formed in the circle?

We begin with one point on the circle. One point, one region.

C1
(Figure 1)

Add another point and the segment connecting them.

C2
(Figure 2)

Two points, two regions.

Conjecture time! At this early juncture, the conjectures can be varied. The most common are either (a) the number of regions is the same as the number of points, or (b) for each additional point, the number of regions doubles. How, I ask, can we tell if either of these conjectures is correct? Students, now invested in the problem by being forced to make a conjecture about how this will play out, insist that I add a third point.

C3

(Figure 3)

Three points. Four regions.

Students whose conjecture is consistent with the picture holler and playfully taunt those who had conjectured incorrectly. But the game has just begun. I write the following chart on the board:

Points 1 2 3 4
Regions 1 2 4 ?

I ask if we have any guesses as to what will happen when we add a fourth point. Those who previously had conjectured the “doubling” invariably hold fast to this conjecture, as it has played out so well so far, deciding the regions will again double to 8. Those who had conjectured incorrectly scramble to amend, often eventually claiming that the differences are increasing by 1 (an increase of 1 between one and two points, an increase of two between two and three points) and deciding there will be 7 regions.

C4

(Figure 4)

Four points, eight regions.

More confidence in the “doubling” conjecture, less and less reason to doubt.

What about five points? Now, nearly all students have been converted to the “doubling” conjecture, and the choice of 16 is nearly unanimous. By this time, students are so invested that they clamor to draw this scenario for themselves.

C5

(Figure 5)

And they are correct.

At this point, I ask what the point is in continuing. Don’t we know what will happen for six points? Many students insist we investigate just to confirm.

They work. They count. They pause, then recount. Some re-draw. They are miffed. Sensing this, I ask if they got 32 regions like we “know” we should. Some students say they got 30 regions, while others insist it’s 31, though very few say there are 32 regions*.

As it turns out, both the students who said 30 regions and 31 regions are correct. In certain arrangements of points, three lines become concurrent and cause a region to disappear.

C6ab(Figure 6a)                                                 (Figure 6b)

Several students indicate they obtained 30 regions by spacing the points equally around the circle. This scenario is interesting to discuss, as this is not the only scenario that results in concurrence, but the point of concurrence is always the intersection of the lines connecting opposite vertices (in Figure 6a, the region labeled “21” would become a single point).

The purpose of the activity is twofold: first, you can’t get so married to a conjecture that you are unwilling to amend it. While I model such a cavalier attitude during class, I close by making sure they are fully aware that they can’t claim a conjecture as absolute truth. This gets at the second goal: the role of proof in mathematics. To prove the things in math have the properties that they do is absolutely essential, and such proofs establish that we won’t run into unexpected places where our conjecture no longer holds. This lesson remains with the students, and when a student asks if we can assume some geometric fact and “get by” without a proof, the class reminds that student of this activity and how there are no guarantees without a proof.

 

*Those students who said there are 32 regions did something wrong, like count a region twice, which is why it is always best to number as we count.

5 Things Teachers Can Do to Establish a Cooperative Classroom Environment

by Chris Robinson, The Blake School

It is no secret that every class is different. The school community can have a large influence and a single student can sway things positively or negatively. I’m most successful at establishing a cooperative classroom when (1) I am myself, (2) my students can be themselves, (3) students see themselves as a learner of mathematics, (4) the physical layout of the classroom expects conversation and (5) the mathematics is approachable and scaffolded properly.

(1) I am sarcastic. I quote movies and eagerly wait to see which students crack a smile. When I want to encourage students to change perspectives, I will stand on a chair and look at the board upside down to focus on a different base and altitude in a triangle. I chide students, and use insults that make no sense, “Mr. Robinson, isn’t that an isosceles triangle?” “Your face is an isosceles triangle.” I treat the job like it is 10% bad standup and 90% math education. I want students to laugh, and take risks comically, in hopes that they will soon take risks mathematically. This is who I am. You are not me, although I hope you have fun with your students everyday. I keep doing it because it is working for me. Students do take risks and I hear many different voices each day. My students see me as human. I make calculation errors in front of them without embarrassment, and if a student asks a question that I don’t know the answer to I tell them so, and say “let’s look it up,” or “I will get back to you about that tomorrow.“ I expect my students to have a growth mindset and I try my best to model one myself regarding my teaching and my own knowledge of mathematics. (If you are not familiar with Carol Dweck and the growth mindset, view this video immediately before continuing to read.)

(2) Despite my meaningless insults I expect students to respect both me and others in the classroom, so that students can take risks and be themselves. I make it clear in the beginning of the year, and call out students (or myself occasionally) on going too far with a joke or impatience with another student’s appropriate questions, my classroom needs to be a space where all levels of questions are accepted and we can learn and relearn together. I expect that students will listen to each other, and will remind them “don’t be the first person to speak at your table this time.” When I assign new groups I ask students to reintroduce themselves to their classmates, to make sure they know the names of the people they are working with.

(3) I make mistakes in the classroom because I expect students to make mistakes in the classroom. We learn from mistakes. If all of my students are getting an investigation question correctly than I didn’t challenge them enough, and if no one is, than I didn’t provide the appropriate level of scaffolding. All students need to feel like they can accomplish the tasks before them (again see Carol Dweck). This may involve confidence-boosting questions like, “What is an answer you know is too small, too large?”

(4) The physical layout of your classroom should encourage talking. If you want students to collaborate, they should be seated close enough to “cheat off of each other.” Partners are good, but I would suggest trying groups of 3-4.

(5) Getting conversational buy-in from all students requires comfort and interesting problems with a low start threshold. A great problem to start the year is the handshake problem. I ask students to stand up and shake hands with everyone in the room and introduce themselves. Afterwards, I ask “How many handshakes occurred?” I force them to think-pair-share. Often some student admits or questions whether all intended handshakes occurred which gives us a great chance to talk about assumptions, if no one brings it up I ask them about assumptions they think we are making. If I had instead said, “Assuming all of you actually shook hands with each other, how many handshakes occurred?” I would have removed a layer of the problem.

This is a problem that allows for easy hypothesizing. Guesses are generally rooted in some calculation like how many handshakes a single student had. The wrong methods often need only a slight modification to achieve the correct answer, and there are multiple solutions and many, many visualizations students might create. Encourage students to use Polya’s methods of solving a smaller problem, look for patterns. If students think they have the answer, have them explain their method carefully and then raise the stakes, “what if there were 100 students in a college lecture?” (or some public schools, am I right? My largest class was 43…ugg) Later on I ask them leading questions about Gauss’s method for adding the first 100 numbers.

Start the year with many problems that have multiple known approaches, and as the year moves on students may surprise you with devising a new method you haven’t seen before.

 

A cooperative classroom is full of people who believe that they have something to contribute and is structured physically and mathematically to entice conversation. Find good problems, and get students talking multiple times every day.

About that “5th Grade” Logic Problem…

by Jonathan Osters, The Blake School

Every once in a while, there will be math or logic problems that go viral on social media. When they do, these are fantastic opportunities to get students to think about their own learning. One such problem is the “Cheryl’s birthday” problem, supposedly from a fifth-grade textbook in Singapore.

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This problem is a great opportunity to engage students at any level, from high-school, down to – yes, even fifth grade, if the problem is framed in a way that students can approach it.

How could we make this problem “Experience First?” Here’s one way:

  1. Have students pick an answer off the top of their head. Do this by secret ballot, clickers, or some other way that forces students to “put some skin in the game,” but is not too intimidating. You will likely see the same set of dates chosen multiple times, and several dates not chosen at all. This brings us to the student experience.
  2. Allow students to act it out. Have a students work in groups of three. Have one student be “Cheryl” and the other two be the two other boys. Act out the scenario, having the “Cheryl” student select a birthdate from the list and telling each of the other students the information they get in the problem. Then ask the two students if either of them know the birthdate. The person receiving the month should always say “no,” but the person receiving the date will say “yes” for the birthdates of May 19 and June 18. This allows students to narrow down possibilities down to only the ones where neither of them knows the birthdate.
  3. Ask students why they didn’t know. Select specific dates, asking why neither of them knew the birthdate. Answers like “there was more than one 17th in the list” or “there are three choices in July and I didn’t know which one was it” allow students to explain their reasoning.
  4. Allow them to act it out again. This time, the student playing Albert has an extra job. For each date, he has to say whether he knows, and whether he’s sure Bernard knows or not. This allows students to see which dates result in the conversation we see in the problem.

This whole process is an exercise in both direct and indirect reasoning, a skill with which many students (and even adults) struggle, and the fact that this problem has been on social media highlights that struggle.

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But problems that go viral have pitfalls. First, many people fear getting the problem wrong – your response is out there for the whole world to see – so they don’t even attempt to solve it. Second, people online can get particularly stubborn – people have a tendency to pick an answer and are unwilling to listen to other opinions. That’s why it’s our job as teachers to take problems like these offline and place them in the classroom where they belong. Approaching this problem in an experience-first way removes the fear of failure from the problem, since in an experience-first classroom, making mistakes is all part of the learning process. It also allows students to not have to feel like they have to defend a wrong answer – they can acknowledge their incorrectness and understand the process by which the answer is truly obtained.

Viral math problems often underscore the lack of numeracy or critical thinking skills that many people have. By finding opportunities to address them in class, we can help our students become better thinkers so they can better interact with the world around them.

What does Experience-First Mathematics look like in the classroom?

Experience-First Mathematics (EFM) is a term we adopted to describe all approaches of instruction that encourage students to take ownership of their learning. This includes problem-based learning, inquiry-based learning, and project-based learning. All of these instructional techniques require the teacher to carefully design an experience for students.

The physical setup of my room consists of students arranged around tables in groups of four to better accommodate conversation. There are multiple whiteboards around the room for students to work on together and present solutions. The room also has a LCD projector and interactive whiteboard.

The execution of EFM will vary in every classroom and certainly varies in my classroom from day to day. The members of The Blake School mathematics department each implement their own version of the approach, but what is evident in each case is a student-centered approach where students are doing mathematics in all its forms: conjecturing, testing, justifying and applying.

In my Honors Algebra II classroom the inquiry takes the form of students working on sets of problems that are designed to help them develop algebraic techniques or observe properties of functions. When discussing our approach with with Barb Everhart at Minneapolis Public Schools, I realized that a great deal of what I do in the classroom is really just the classic Think-Pair-Share. Students work on problems individually, then discuss in small groups and then we discuss major themes as a class.

To know what the classroom experience feels like for a student, it may be easiest to start at home. A typical night of homework involves two types of problems: practice problems on nearly mastered techniques and topics and motivational problems that are meant to guide students to new conclusions. Whenever possible, I try to give immediate feedback on practice problems, whether it is an online HW using our Hawkes book or just a matter of providing the key on our class website.

The Motivational Problems are the heart of our problem-based experience. Students work on these problems to the best of their ability at home, and come to school with them attempted or sometimes even fully-completed.

At the beginning of class we might answer a few questions about last night’s practice problems, but quickly transition to the motivational problem discussions. Most of the independent thinking occurred at home, and so we can jump straight into the pairing and sharing. Sometimes I will give students a fresh motivational problem, in that case I ask students to work individually for a few minutes before discussing with their table-mates. This brainstorming is important so that the discussions at the tables can have many starting points.

During these Pair discussions I circulate looking to correct groups that are coming to wrong conclusions by asking clarifying and redirecting questions. In addition I am gathering information about which students or groups have come up with efficient, novel, or standard algorithmic approaches. I then make sure that the full group discussions (Share) do not pass without hearing from those students.

It is paramount that students feel comfortable speaking to each other and sharing thoughts to the whole class. In my next post I will discuss how we establish a cooperative classroom environment.

Written by Chris Robinson

The name…

Hello World, as they say…

My name is Chris Robinson and I am a teacher at The Blake School in Minneapolis, Minnesota. After a long week and an encouraging day at the MAIS Conference (MN Assoc. of Independent Schools), I have decided to start blogging about my work with Experience-First Mathematics.

I would be remiss to not have my first post address the name of this blog.

Experience-First Mathematics (EFM) is a term I adopted to describe all approaches of instruction that encourage students to take ownership of their learning. This would include problem-based learning, inquiry-based learning, and project-based learning. Like all instructional techniques EFM can be executed well or poorly and it is my hope that through this blog, I can continue to reflect on and improve my practice and assist other teachers in improving their practice.

I named the blog Experience-First Mathematics in honor of Dr. Arnold Ross who started the “Ross Program,” a summer number theory experience for high school students at Notre Dame and then Ohio State. This program inspired Dr. Glenn Stevens and Dr. Al Cuoco to not only start a similar program for high school students at Boston University called PROMYS, but to start a parallel program for teachers, PROMYS for Teachers. I joined this community of educators in 2010 and despite the miles, I feel very connected to them and have returned to Boston four of the last five summers.

I will end this first post with a heartfelt thank you to my colleagues. My journey into EFM would not be possible without the partnerships forged in our daily work.