activity – In Pursuit of Great Mathematics Teaching

Test Champions

This idea was recently shared with me by Pietro Tozzi. He is a maths teacher at Gumley House Convent School in London but also works for Pearson (the Edexcel exam board) two days a week. We brought him out to our school in Singapore to do some training for the team about the new A-level in maths. (By the way, if you are looking for an A-level or (I)GCSE trainer, I would highly recommend him.)

During our time together, he shared this idea for returning a test. As it turns out I had a KS3 test to return and tried it out. As I marked I kept track of the best answers for each question. When I compiled this table, I made sure that every student was listed and students with lower scores were listed more frequently, if possible.

The class score represents the fact that among all the students, they have expertise to get full marks; someone scored perfectly on every question.

After passing back the tests, I asked students to get up and meet others who could help them correct their errors. They spent about half an hour up and about, talking over their questions, and making notations (using a coloured pen) on their tests. While they were doing this, I was able to work around the room and talk to those who had difficulties that their peers couldn’t solve.

They enjoyed this way of reviewing and correcting their tests. Having them out of their seats was good for their concentration (for the most part!) and also allowed me to be less conspicuous in my help of a few key students.

I completed this lesson with a fill-in sheet for students to list strengths and targets based on the test. We spent the following lesson with students doing individual work on their targets before we moved on to our next topics of study.

Five Superb Maths Lesson Ideas #2

1. Pythagoras and Trigonometry Revision Activity

I love activities that get students out of their seats. This task (designed by Steel1989) asks students to distinguish between Pythagoras and trig questions. Yet instead of a worksheet, the questions are designed to be printed out and stuck around the room on sheets of paper. Students get one to work on, answer it (in their book or on a mini whiteboard) and then write the answer on the back of the sheet. Then they put the sheet back up on the wall. When another student answers the same question, they check their answer with the one already written there. If the answers differ, they students need to talk to each other to discover which is correct.

pythagoras or trig

2. Polygraph Desmos Activity

Oh, wow, I’ve discovered a great one here and maybe you’ve heard the hype already. Desmos has introduced a teacher section that allows you to run class-side activities. I tired out the Polygraph: Lines activity with one of my classes. Have a look at the teacher guidance to learn more. Only you as the teacher needs to create an account; you give students a code to join the game. One student chooses a linear graph and their assigned partner has to ask yes/no questions to guess which graph it is. Meanwhile, as a teacher you can see all the questions and answer being given, who has been successful with the task (or not). I called one of my students over when I saw that she had typed “Does your graph go through the point y = 2x?”. I was able to clear up a misconception I didn’t even know she had until then.

The student’s view is shown in the screenshot below. Desmos is adding to the collection of class activities and I’m sure I’ll use them all in time!

polygraph lines.PNG

3. Tree Diagrams Challenge

A few of my year 11 students are ready to take on the challenge of those nasty tree diagrams questions that lead to quadratics. Fortunately, tonycarter45 has created this lovely sheet with probability extension questions. The sheet includes the answers.

Tony (who works at my school) has produced quite a few nice worksheets and you can see them on TES Resources. He specialises in thought-provoking questions. I like that his investigative worksheets often remove scaffolding parts as the questions progress.

tree diagrams

4. Two is the Magic Number worksheets

Three activities called “Two is the Magic Number” from Just Maths. Each one is a collection of cards solving a short problem, only two of which are done correctly. The rest show common errors and misconceptions. The cards generally cover number and algebra skills such as simplifying terms, using indices, and calculating with fractions. Depending on what you have taught your students, there may be a few topics that they haven’t learned, so check first. (My bottom set year 8 need to practice like terms, but they can’t do a conversion between meters squared and millimeters squared.) These sheets are great for checking students’ misconceptions.

two-is-the-magic-number-1

5. IB DP Maths Resource Collection

I have a former colleague, Andrew Clarke, who is a brilliant resource collector. He has now started three curated collections of maths teaching ideas for IB teachers. The one that is most relevant to me is Teaching Diploma Program Mathematics. He has collected all kinds of teaching ideas for Maths HL, SL, and Studies SL. One item that caught my eye is an investigation about using calculus to describe concavity, which is one topic I have never found a good way of introducing.

Andrew’s other two sites may interest you: Teaching MYP Maths and Teaching PYP Mathematics.

What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Five Superb Maths Lesson Ideas #1

1. Areas of Flags

Areas of Flags (from Owen134866 on TES Resources). One of my colleagues introduced me to this brilliant series of worksheets (and powerpoints) that use flags as a context for finding areas of rectangles, triangles, parallelograms, and trapeziums. There is also a further activity with circles.

areas of flags

2. BC Numeracy Tasks

I was browsing on the website of Peter Liljedahl from Simon Fraser University, Canada. (I was reading a paper of his about task design.) I discovered that he was on a team to develop tasks to assess students’ numeracy in British Columbia. They look as though they are lovely, well thought out tasks. However, there aren’t any solutions that I can see, likely because these are in use as assessment tasks in BC. I note that some of them are too Canadian, though! “Last week I went out crabbing with a friend. We took my canoe and paddled out to a point just off Belcarra Park and threw in our trap.” I’m not sure my city-dwelling, mostly expat students would know what to make of this. However, there are lots of great tasks here and I reckon I will try some of them out soon.

crab trap.PNG

3. GCSE Five a Day Sheets

These GCSE starter sheets, Five a Day, by Corbett Maths. Each sheet has five questions. They are available for numeracy, Foundation, and Higher, and answers are provided. One sheet for every day of the year. I have asked some of my students to use them at home on weekends, too.

4. “Think of a Number” Lesson for HCF and LCM

I’m planning to use this lovely lesson about highest common factor (HCF) and lowest common multiple (LCM) from the Mathematics Assessment Project. I like that it provides a pre-test (which could be used as homework) to help me plan the lesson. The main tasks are really well explained in the teacher notes and include a whole class discussion with mini whiteboard responses, and a card sorting activity. Then there’s a post-test to see what students have learned. All 100 of the lessons in this series are designed with a pre-test and a post-test; I love that it makes it easy to see how students have improved.

The only downside of this lesson is its American vocabulary. I am going to need to use white-out to correct greatest common factor (GCF) to HCF throughout!

hcf lcm shell map

5. Shakespeare and Numbers

Our Head of English has started talking about upcoming celebrations for the 400th anniversary of Shakespeare’s death (23 April 2016). I have been thinking about what we might do in maths to celebrate. So far I found this Numberphile video about the numbers in Shakespeare’s sonnets. I will continue hunting for some other things to use in lessons but this video (duration 4:36) will be a nice ender for lessons on that day.

What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Numerical and Algebraic Integration Cards

I made this set of eight cards about areas found by integration for my IB Standard Level students. The graphs are taken from a textbook exercise. (Screenshot below. Follow the link to get the cards.)

numerical integrals cards 1

I wanted them to use several methods for finding the areas, including numeric and algebraic integration, so I presented these instructions.

numerical integrals cards

With the answers displayed on the board, students could feel confident as they went through the cards.

a mathematics lesson that worked

This lesson really worked for my students. It was less boring than a textbook exercise, and allowed them to discriminate between methods for finding areas. It also provided good practice of integration and GDC skills.

Logarithm Questions Around the Room

Here’s a lesson that worked for me recently. I had six logarithms questions posted around the room. I gave each pair of students some sticky notes and asked them to go around adding to the posters. Each answer had to be different, clearly. (I didn’t even specify that each answer had to be different, actually, the students just assumed that.)

I love that students were out of their seats and talking to each other. They were more energetic about these questions than they would have been about a worksheet. And they were automatically noticing generalisations as the activity went on and more answers got added to the posters. Also the group feel to an activity like this spurs lots of students to try creating an example that is a bit harder than they would suggest normally.

This was a good activity just like this, but I added a little more. Later in the lesson I took pictures of the posters with my iPad. They are set to automatically upload to iCloud, so I accessed them on my classroom computer and could show them on the screen. We talked about a couple of interesting sticky notes and students noted the ones they thought were incorrect. A few of my students like notetaking more than others, so they copied a few examples.

Since posting one of these pictures on twitter, I have been featured by another teaching blog: Resourceaholic. My idea is one of five “gems”; the other four are (also) amazing ideas!

I am glad that the sticky notes idea seems to work for others; a few others have tweeted to say they liked it. Thanks for the feedback, Emma Cox and MathSparkles! Here’s the file I used with the logarithms questions (make a copy to save it to your own Google Drive or download it); the questions are based on a resource by Susan Wall.

What worked for you recently?

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?

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).

Screen Shot 2014-01-23 at 1.46.14 PM

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.