In Pursuit of Great Mathematics Teaching

Famous Mathematicians Choice Board (Differentiation Idea)

A few years ago I made a project that I used with my year 7 (eleven year old) students. I introduced it by talking for about two minutes about Blaise Pascal, a mathematician I find personally interesting.

Then I told my students they were going to choose a mathematician to learn about and it would be their choice how to show their learning. There are nine options for them to produce, and they needed to do three of them which form a vertical, horizontal, or diagonal line from the choice board. (Here is the choice board file.)

I arranged the nine items so that the lines require some diversity of products. The two times I have used this project my students have responded very positively to the choices. The second time I did it, after they had made their three items, I made larger groups of those who had chosen the same mathematicians to put together a display board of their work. It’s a bonus for me that this project makes for a lot of good display work if you need/want to change your classroom displays.

I think if I did this project again, I might allow more choice on the person as well, perhaps by adding the option, “Choose another mathematician of your choice and have it approved by Mrs A”.

I am posting about this today because I was reading Mr Bigger’s post about differentiation using choice boards. It also makes me think that I should try to use a choice board for a more “meaty” mathematical topic. Has anyone done this before and have an example to share?

Lesson: Ferris Wheel Exam-Style Question

Sometimes I just want to remind myself and others that not every lesson has to be “special” or involve a game or video. Lessons that are successful are those where students learn to think better and experience mathematics at work. Here is one of those; it’s not flashy, just solidly successful.

It is time to review our work on trigonometric functions in my grade 11 class (IB Mathematics SL year 1). I made this document with an exam-style question closely modelled on recent exams. The question is about a Ferris wheel rotating at a constant rate.

The first page is the question, which I copied onto half sheets and gave out. These instructions were on the board.

I told my students to try to work in exam-style conditions for the first step, promising that under the black box on the slide there were other steps we were going to go through with this question. Then when it seemed like everyone had time to attack all the parts of the question, I moved the black box on the slide to reveal the next set of instructions. Students worked in pairs to create their best answers.

And finally, I gave out the mark scheme (the second page in the document) and we moved the answer papers around so every pair marked answers from someone else.

We then returned each student’s paper and they got to glue their corrected answer and the mark scheme into their notes.

This lesson mirrors the discussion idea called Think-Pair-Share and provides good exam practice. Students appreciate getting to see the mark scheme and how it applies to a their own and another’s answers.

My lessons are like this a lot of the time. I would say I teach a whizz-bang-special-game-or-video lesson once a week, or less when I’m tired. But I always try to get students talking, working together, and going deep into mathematics.

How much of the time do you teach whizz-bang-special lessons?

Using Exit Slips: an #eduread post

My grade 11 class (Mathematics SL year 1) are getting ready for an exam in a week and a half. I was reading this week’s #eduread article about exit slips while they were doing a quiz. I got to the end of the article a few minutes before they finished and I was pondering the last two sentences of the article:

Exit slips are easy to use and take little time away from instruction. Many teachers use them routinely—even daily—and attest to their positive influence on student achievement.

It’s been a while since I used exit slips so I thought, well, there is no time like the present! And I wrote these three questions on the board to use immediately with my grade 11s.

I passed out some small pieces of scrap paper, and voila!, exit slips.

The article mentions four main uses for exit slips. First, to get formative assessment data. My first two questions are of this type. Students give feedback on what they have learned. I now know that my students feel somewhat prepared; the median and modal level was 3. I need to plan more review about trigonometric functions and applications of differentiation.

Secondly, exit slips can be used to have students reflect on their learning strategies or effort. An example question would be “How hard did you work today?” I am planning to use this question soon–it could be illuminating.

Thirdly, the slips can be used to get feedback about my teaching. In the past I have often asked how my pace was during that lesson. My third question today is also of this type. Some students asked for more exam-style questions, several others want me to do tricky stuff on the board.

Last, exit slips can be a place for open communication with the teacher. In the past, I have frequently asked, “What is your foremost question or concern?” This prompt allows students to say whatever it is they want to about mathematics, our class, or anything else. The responses have ranged from useful to hilarious.

This post is for a group of mathematics teachers who read an article and chat about it each week using the hashtag #eduread. You are welcome to join in; our chat about exit slips is on Wednesday night at 8pm in North America/Thursday morning at 9am in Singapore (and the time where you are).

What questions would you ask on exit slips?

Giving More Useful Feedback to Students: an #eduread post

How to give better feedback is always a goal of mine. I sometimes (maybe frequently) find it hard to keep up with the pace of teaching, assessing, giving feedback, and reflecting on it. What about you?

Today I was reading a short article titled “How Am I Doing?” with some pointers for effective feedback. Five very useful ideas were presented; have a look at the article to see them all.

The item that struck me the most was that students need a clear view of where they are heading with their learning and what their learning target is. Phrased in this way, it is a very simple idea and one I have been familiar with for years. However, I have found that at times students think their job in my classroom is to complete the activities I give them. They think that completion of activities is the success criteria. Instead I need to get across to them that their job in my class is to reach desired learning outcomes.

This relates to feedback in that there is no use giving feedback when students think they have successfully completed the task of “learning”. Feedback may seem like giving them more work to do, rather than helping them learn. But if students know their job is to achieve a learning goal. Feedback about progress towards learning goals helps students know how their efforts are leading to success.

In my classroom I am not the best (yet! growth mindset!) at giving indicators of where learning is heading and how we will get there. I was marking student work today and saw this comment (pictured above): “This is confusing because you have not explained what you are trying to do.” As it turns out, that is exactly what I need to hear myself! I appreciated this article as a reminder to expose the learning goals more frequently. And to provide personalised, specific feedback to students about how they are going towards meeting those goals.

#eduread is a group of mathematics teachers that read an article each week and discuss it on Twitter. Here is the blog that organises it. The chat is on Wednesday evenings in the US, which is Thursday morning for me. I can’t always participate in the Twitter chats but I can usually follow along later thanks to the hashtag. Would you like to join us?

Do you find it easy to give useful feedback to students?

The Most Popular Spot in My Classroom: Who Tall Are You?

At the back of the room, near the back door, is this brilliant height chart, titled “Who Tall are You?” It’s the most popular place for students in my room. Sometimes when I go out for tea at break times, I’ll return to see some young people have slipped in and are crowding around it. Here’s a link to a good quality image of it. “I’m as tall as Beyonce!” one person squeals.

Every once in a while students will be as tall as someone they don’t know: some of the celebrities now seem a little dated (singer Charles Barkley, for instance) and others are just people my students haven’t heard of yet (such as Alexander Pope).

Since I bought it about five years ago I’m having trouble finding somewhere to buy another copy. But I was thinking today that students could make one of these for display. They could have one featuring their classmates (and teachers?) and heroes of their choice. I reckon my current classes would include more sports people than the original chart.

I’m as tall as George Clooney. If I was making a new chart I would be sure to include Jensen Button, since I am also the same height as him. If you know your height in centimetres, please leave a comment telling us “who-tall” you are!

Building Collaboration Using Changing Partners Activities

A few classes needed to revise at the end of teaching units and I wanted them to collaborate while they did so. I printed off a class set of these small checklists with the first names of everyone in the class.

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I also printed some review questions onto colourful card. I separated my tables and spread the questions out over the tables, with two or three on each table. I put some tables facing the wall or the windows so that pairs of students would hopefully focus better on just their partner and the question at hand.

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I gave each student a name checklist to glue into their notes and displayed these instructions.

From time to time there were not two students looking for new partners at the same time. As a result, sometimes they would have to work in a group of three. Other times there was one person who had to wait a minute to start their next problem. However, this didn’t happen too much and overall I would say that the students did a lot of work. Sometimes in a revision lesson such as this one, students get bored and lose momentum. Not so during this lesson. They stayed on task until the end (80 minutes later) and completed loads of questions. In the last few minutes of class we checked their work as I displayed the answers on the board.

I have tried this with a few classes now and it has worked well every time. It goes most smoothly when the class size is 20 or more. Though I did try it one day when lots of students were at a field trip and I only had seven learners. That still worked but it was a little harder for students to begin and end questions at the same time as others. it devolved into mixed group work instead.

One time I tried this in my class that has a wheelchair student. This also worked smoothy. I positioned her in a place where she wouldn’t be bumped by others walking by. From time to time I prompted students to go over to her since she can’t come to them.

Some tips for myself and others to make this work well:

A class size of 20 or more works best.
It works best if you have enough tables to have at least two empty tables at the start for the first students who finish and find new partners to move to.
Having three questions on each table means students can sit down at the same table a second or third time with different partners and solve different questions.
Make sure the questions are numbered (or lettered) and students write these down as they solve. Then sharing the answers becomes easy.
Mixed revision from several units of study can be done this way. Just mix up the questions around the room.

Do you have ideas about helping students collaborate in math(s) class?

Peer Assessment: The “Production Line”

My grade 9 students did an individual investigation last week and I wanted to involve them in the process of marking it. I heard about a peer assessment structure called the production line from a colleague last year. In brief, the students mark each other’s work in groups, and each group concentrates on one aspect of feedback. The assignments travel around the group, gaining lots of detailed feedback.

Preparation

The investigation task already had an assessment rubric. It is based on the MYP year 4 criteria. The rubric focused on Investigating Patterns (MYP criterion B) and Communication in Mathematics (MYP criterion C). I divided the task specific criteria into themes – five in total:

B1: Finding and stating patterns
B2: Problem-solving techniques
B3: General rules
C1: Language and representation
C2: Reasoning

The Lesson

I rearranged the tables in my room and assigned groups of three students to sit together. I gave each group the details of one of the criteria. An example is shown below (and you can get them all in the investigation file).

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I had some instructions on the board (shown below) and we talked about what we were going to do. I had students help me staple an extra sheet of paper to the back of each task so that we could use it give feedback.

I gave each group three tasks and asked them to read them, as a group, one at a time. Then they were to discuss the criterion and give some feedback. When they were finished with a task, I passed it on to another group.

Reflection on the Lesson

Since the rubric was already set up for this task, the preparation was quite easy for me. Making a task-specific rubric is a job that can take some time! Thankfully it only needs to be done once.

The number of criteria I wanted to mark didn’t match the number of groups I made in the class. I opted to have three criteria marked by two groups each and two criteria marked by one group each. This turned out to not work very well. The repeated criteria groups finished all the class tasks much more quickly (obviously!) and the other groups had much more to do. When I realised this, I asked two groups to take on a different criterion so take up slack from the inundated groups. Next time, I need to think more carefully about this. I didn’t want to make bigger groups because I felt like they would not work together well. In that case, I should have more criteria so that each criterion is only assessed by a single group.

I just managed (in 80 minutes) to get all the tasks marked by all the criteria groups. Next time I will not need to explain the production line in as much detail and I expect it will be more comfortable in terms of time.

The feedback that was given was extensive, though, I think I need to talk with the students about what the criteria is looking for. I can see the value of a class discussion about what constitutes good reasoning or communication or pattern stating.

The students gained a much more clear view of what criteria they are marked against. (The same criteria are in use throughout MYP maths.)

How do you use peer assessment in your classroom?

Algebraic Fractions Quiz-Quiz-Trade Activity

One of my classes of students sat an exam lately and I realized they need more practice with adding and subtracting algebraic fractions. There were a collection of misconceptions on the exam, including not taking a common denominator, trying to cross multiply, and cancelling incorrectly. I made a set of cards (split over two files) to use to help practice this tricky manipulation.

In this quiz-quiz-trade activity, students start with one question each and make sure they can simplify it. I give them a few minutes to check with their partner or with me. My students like to use their mini whiteboards because it lets them change any errors easily.

Then they get out of their seats and meet someone from another table. They take their mini whiteboards (or their notebooks) with them. Meeting someone else, they quiz each other. After they are satisfied that they both got their questions correct, they trade cards, thus leaving with a different card then the one with which they arrived.

While the students are quizzing each other, I am able to circulate and address misconceptions. The students are quite good at helping each other. After this activity I think they will feel a lot more confident with adding and subtracting algebraic fractions.

When students sit back down, I ask them to do a few of the algebraic calculations in their notebooks so they have a record of what they learned.

I made these cards using Tarsia software, designed for making mathematics activities. Here are two files (pdf) of the cards.

One Easy Way to Never Teach a Boring Lesson Again

I have discovered that I can’t teach a boring lesson anymore. I really wanted to today. I was tired and it was Friday, last period. My grade 9s always seem a bit too boisterous when I am the most exhausted. Sigh. All I wanted was to sit them down, get them to be quiet, and do some mindless, repetitive task. Unfortunately, I had done one thing at the beginning of the year that ruled out a boring lesson.

I told them never to bring their textbooks to class.

As a result, every lesson has to involve some kind of activity. We have to talk, sort, write, create, classify, or debate. I have no choice but to get them doing and moving. Even when I most want to sit down and supervise them, I rarely can.

We talked about the index laws, then did a quiz-quiz-trade activity to practice simplifying complicated expressions using indices. Then we wrote a few of the “best” examples (the hardest are the best, right?) in our notes. Then we solved a corny riddle using indices. “How do you write a song to knock over a cow?” “In beef flat.”

Not using textbooks in class can be tiring. But it helps me cause active learning.

Exam Preparation Activity: Collaborative Practice Questions

My students are preparing for winter exams. As a review activity we are making our own revision questions. I set up two shared Google Docs: one for revision questions, and one for the solutions. Each student had to make up at least one revision question for everyone and add it to the first document.

Writing the questions requires an understanding of the concepts behind our topic, so this activity let me see what my students understood and how well.

Students had to say which MYP level their question was. Our upcoming exam focuses on MYP Criterion A: Knowledge and Understanding, which is graded from 0 to 8. We regularly discuss the criteria in class and so students are beginning to become familiar with the descriptions of the levels.

We had a good variety of questions added, and I circulated to encourage some students to fill holes I thought I saw. I urged some students to write harder questions. In the end, the activity is self-differentiating: each student works on problems at the exact right level. And for each student there are harder problems to stretch them, with me providing the stretch for the topmost students.

As for the solutions document, some students chose to type their mathematics using the (minimally acceptable) equation editor on Google Docs’ Insert menu. Others worked on their mini whiteboards and then took photos to upload to the document.

After students had made at least one question, I encouraged them to go on to make another, harder one or to answer the questions of their classmates. I assigned as homework to work through all the questions in preparation for our exam.

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What exam preparation strategies work for you and your students?