Moving the Focus

One of the things that I've been reflecting on this summer has to do with student-driven focus on getting the answer.

Recently, I watched Phil Daro's Against "Answer-Getting".  I like the phrase answer-getting because it sums up exactly what my students are trying to do in math class.  I am aware of this and I have used various strategies over the years to move my students farther from just answer-getting, and towards other things mathematical.

Here in Quebec, students are evaluated on their ability to reason mathematically and on their ability to solve problems.  Reasoning is evaluated throughout the year and students sit a final exam.  Lots of answer-getting here.

Students are also evaluated on their ability to solve problems, without a final exam.  Here was my chance to expose my students to forms of evaluation which didn't focus on getting the answer.

This year, I gave my students some evaluation tasks based on Dan Meyer three-act tasks.  I've used these tasks in the past for instruction but never for evaluation.

In the first instance I used Dan Meyer's three-act task called Finals Week.  I showed the students only the first act.  Multiple times.  As many times as they wanted, actually.  Then I gave out the sheet below.  I wanted them to do a number of things: describe the problem, list what they would need to know, list what they would not need to know, describe/list the problem solving process they would follow, and estimate an answer.

The brief rubric at the bottom of the page is from what Quebec students are expected to be able to do under the problem solving competency.

The second  task I used was Dan's Leaky Faucet.

The third task I used this year was Dan's Meatballs.

I do not have student samples (note to self: conserve student samples!), but I did make a number of observations about the student work.

Some students had success because the numbers weren't in the way.  Sometimes students just make up math to arrive at the "answer-getting".  This forced them to think about the process not the product.

Some students were able to show their problem solving process not by way of a series of steps, but in one compact formula.  They created a new formula which incorporated all of the elements required (here I'm thinking about Meatballs).

This is definitely something I will keep doing, and do more of in the future.  I'm proud that I moved my students a little farther away from the focus on answer-getting.

Taking the Time

My students completed the Des-man project through Desmos.

The original, original Des-man project was created by Fawn.  She is so inspirational.

The post by katenerdypoo at in pursuit of nerdiness, was my starting point.  Kate also teaches at an IBMYP school, and so the language and her assignment set-up made sense to me.

I used her assignment to develop my own.  I also evaluated Communication within the IBMYP framework.  I teach secondary 3, grade 9.

I let the students have one 75-minute period to work on their project.  At the end of the period, I realized that they had just barely had enough time to get used to Desmos, figure out what the project was, and begin to work.  At the end of the period I promised them another, even though on the site they suggest that it is a 45-minute lesson.

I have spent the weekend grading these assignments.  As I evaluated them, I looked through what they had produced during the first period.

One pair of students produced the following:

The first effort.

The final product

Two 75-minute periods were devoted to this assignment.  To some this may seem like a lot.  Looking back it was time very well spent.  A number of things struck me as I was grading their assignments. 

The first period allowed them to get used to the graphing calculator and do some trial and error.  The Teacher Dashboard really helped here because I was able to put it up on the SMART board, and the students were able to identify who was doing what.  I encouraged them to visit other groups and ask for equations and ways of making certain lines and parts of the faces. 

The first period also let the students figure out what I was asking them to do.  I asked them to use at least two linear functions.  I asked them to restrict the domain for at least one of the linear functions.  The written part asked them to identify why what they had chosen was a function.  I had not done a lot of direct teaching in these areas, and so during the time that they took, they were actually learning, from their notes, from each other, from Google.

At the end of the first period, once we were back in the classroom, I asked how many knew if they had used a linear equation in their Des-men.  Even though I had seen many in their Desmos faces, not many people raised hands.  I knew I needed to reinforce this.

A second pair of students did the following:

The first effort.

The final product.

During the second period let the students start again.  I let them see their previous attempt and use the equations.  I put up the Teacher Dashboard again.  I let them visit again.  They finished during the second 75-minute period and were able to begin on the written report.

A third pair of students produced this:

The first effort.

The final product.

I was struck by the students' creativity.  I had let them play, and they really did.  In the side-by-side Des-people above, I can see the transformation to the wink and crooked smile, the addition of the hat, and feminine aquiline nose.  

I am happy that I devoted two periods to this learning, and I am glad I took this time.

Learning from Students' Whiteboards

This was the second-ever time that the students used the whiteboards.  It was magic.

My plan for the whiteboard session was based on an introduction to functions.  I had decided that I wanted the students to represent a situation using a table, graph, function rule and description. Previously they had practiced the individual parts, and now I wanted them to put it all together.

I randomly arranged them into groups of three (thanks Fawn!), and they got organized into the groups with their boards in the vertical position (thanks Alex ).

I gave them a situation.  I told them that the first situation had to do with the grocery store.  They were to call out input values (independent variable values) and I would give them the output (dependent variable values).

The results were something like the following:
them: 3, me: 3.75
them: 16, me: 20
them: 50, me: 62.5
them: zero!, me: zero!
them: 1, me: 1.25

After this it suddenly became quiet as the students realized that they had enough information in front of them to begin to figure out the function.  The also asked: "Can we begin working on the graph?"  Of course!

What was interesting to see at the end of this first session was the choice of written descriptions.  One group chose the cost per Granny Smith apple, another chose the cost per piece of beef jerky.  Another group said it was a 25% tax per item purchased.  None of these examples were in the textbook...

I took the opportunity at this point to introduce them to new notation for a function rule f(x)=1.25x instead of what they had been using which was y=1.25x

For the second round, after I had the students erase the boards and re-randomized the groups of three, the situation had to do with public transportation.

The results for the input/output went like this:
them: 0, me: 47.25
them: 1, me: 47.25 (them: wait a minute.....)
them: ok, how about 3 miss? me: 47.25
them: 2, me: 47.25
them: 10, me: 47.25

Again they got straight to work creating the table of values, the graph, writing the rule and writing the description.

And then I was blown away.

As I wandered looking at the work I noticed that many of their representations of functions looked a little non-traditional to the part of my brain which is the math-is-traditionally-represented-in-this-way-always part.

I was seeing stuff like:
f(x) = 47.25 +0x

f(x) = 47.25 + (x-x)

and, f(x) = 47.25x/x

I came to my senses and realized that the students were merely representing the function in a way that made sense to them.

When questioned, each group who had used one of these types of strategies explained that they were trying to find a way to "get rid of" the variable x.

Miraculous!

Board Practice

I have a huge double blackboard in my classroom which I will never give up.

When my SMART Board was installed in my classroom, I wanted it to be installed on another wall because I didn't want it to interfere with my beautiful blackboard.

One of the ways that I use my beloved blackboard is for Board Practice (I didn't really have a name for this, I just made it up for the purposes of this blog, and to get it in print, because this is my first homework assignment for the MTBoS).  For my students, I usually refer to it as the-thing-that-we-do-with-the-board-split-into-sections.  Or I split the board into sections and they get all excited: "Miss, are we doing the thing with the groups and the practice, and the coaches?"  Yes, we are.

My students work in groups, and have been since the beginning of time.  Since before it was a thing.  I've always asked my students to talk, share and coach.  I think it may come from my teacher beginning as a science teacher, and my love of labs.  I've carried this forward to my math classes.

So the Board Practice works the following way in my room.  I've taught/reviewed the skill, I split the board into sections, they get excited, I throw a problem on the board, they work it alone (without calculators, "...but Miss..."), then they turn around and talk, share, coach in their groups of four.  When my gut tells me they're ready, I pull a number from 1 to 4, and that person, one from each group, goes to the board and shows work for the problem.  When it's particularly difficult, or when we're at the very beginning of the skill, they get to bring a coach, if not they go alone.

The photos above show the board split into sections, labelled A through G for their groups, and the problem (simplifying powers).

These show the worked problems for another expression.

Board Practice is one of my students' favourites because they get to practice in a "fun" way.

For me, it is so much more than fun.  Students are practicing, talking about math and being coaches to their peers.  I believe I have created a safe environment for this practice, in which students are allowed to make mistakes and correct with the help of their peers.  I like the way the same problem is to be simplified, but often small differences occur, or even mistakes.  At the end, when each section is full, and the students are re-seated, I tell them: "Ok, there is a mistake in one of these, who can find it...".

Once de-briefed from the current expression, I erase a section, write the next expression, and say, "You're up, copy, get to work."

And they're off again.

Penny Pyramid

After reading about Dan Meyer's three act Penny Pyramid, I was keen and ready to give it a shot.  I had read through the acts and watched the videos.  I had watched one of the workshop presentations he had given on this topic.  I had also read Fawn Nguyen's posting on her site.  In addition, I thought my students were ready for the estimation part because we had begun some work with the Estimation 180 site, and my students were familiar with making reasonable estimates and making reasonable high and low estimates.

And it was fantastic.

I began by showing the video immediately following the bell.  The video is pretty impressive, and so it got their attention right away, and I immediately heard lots of comments, including "cool" and "why would someone do that".  I showed it a second time, and asked them to write down some of their questions.  Then I collected them on the board.

Unlike Dan, who said he wanted the first question, I wanted all of them, and I spent some time wringing them out.  We then went through the list to determine which were mathematically related, and checked them off.  I told them that we would get to some of the "who" and "why" questions later (and I did), and that I'd like them to focus on how many pennies there were in the pyramid.

I then put the photo of the pyramid back up and asked the students for their estimates, including their reasonable estimates for the high and low bounds, and I got this:

Next was to ask them what else they needed from me to be able to figure out how many pennies there were in the pyramid.  They wanted the height of the stack, and the dimensions of the bottom layer.

I split them into random groups of three (thanks Fawn!), and they were off.

Many of the groups went straight into number crunching with the calculator, and when I visited their groups, complained that it was taking a long time.  Some tried to find some sort of pattern to make things easier:

Some groups calculated what they were calling the area of each triangle, and when I asked them if there were pennies in the middle of the pyramid, there was recognition that this needed to be calculation of the inside of the thing, not the outside.

We wrapped up and I told them that next time we'd be going to the computer lab to have a look at what could have made this calculation much more efficient and faster.

Extra Time

One of the yearly struggles I have is with students who are done early.  Whether they are working individually and finished the practice, or whether it is a group of students who are finished after working on a group problem, I have never come up with a constructive way for them to spend their time when they are done early.  I have experimented with different options over the years, but I have never been satisfied with any I have tried.

One of the options I have tried is to let the students who were finished early get up and move around to "help others".  This, of course, turned into giving others the answers, showing them how to do it, or fooling around, none of which is satisfactory to me.

I recently read that in some countries, the classroom culture is that of having students help others when they are done.  It is expected, and part of what goes on in a classroom.  This makes sense to me.

Then I thought about how I help students.  I ask questions.

With these ideas in mind, I thought that if I came up with a list of questions that I ask, and posted these, then the student helpers could ask these questions while they are helping others.

1. What is the math concept/topic for this problem?
2. What information is given?
3. What is the meaning of the word ___________?
4. What do you think you need to do?
5. Have we already done a problem like this one?
6. What have we learned that may be helpful here?
7. What drawing/diagram could you make to show this problem?
8. What other strategy (table, equation, etc.) could you use here?
9. How would you describe the problem in your own words?
10. What would be your best guess/estimate for the answer?

I will model these questions.  I will post them.  When I explain how I will expect students to help others when they are done, I will refer to the posted list of questions and encourage them to use only these. 

Maybe this will work.

To Blog

I came across Drawing on Math's blog post introducing 15 new bloggers who have begun to blog, due to this challenge.  I am scared of the permanence of adding my own words to the universe even though I have read in so many places that it is one of the best professional things I could be doing.

Let me see if this time I can make it a habit.