In Pursuit of Great Mathematics Teaching

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.

How I Organise My Lesson Ideas

My favourite discussion with other teachers is to share lesson ideas. But remembering them at the right time is hard work. I think, “I’m sure someone told me a good idea recently about quadratic equations…!” I have probably forgotten more good ideas than I have remembered.

A few years ago I started a document to save good ideas. Below is a screenshot from the document, and you can see my whole Resources Listing document here.

resources listing screenshot 2

Every time someone tells me a good idea or I see one on the web, I write it down here. Then later I can look back through the doc to find it again.

how to get stuff done as a teacher

You can make a document like this, too. Recently I made a blank copy of this document so I could share it with others. Here it is – download a copy and enjoy! Please let me know if it’s useful to you.

Concept-Based Teaching in Mathematics

Some members of my team and I were recently meeting with the IB subject manager for Maths HL assessment. She mentioned that one possible development for the next round of curriculum updates (pdf, IB OCC login required) is a greater focus on inquiry learning and concept-based teaching. When we started discussing this I realised I don’t really know what concept-based teaching looks like in maths. Do you?

I am just beginning my learning journey about concept-based teaching in maths. Here are my stops so far.

Jennifer Wathall has written a book about concept-based maths – it will be published in 2016. I knew Jennifer when I worked in Hong Kong and I attended workshops given by her. I think it is likely to be a very practical and useful book.

This brief blog post by Jeff Hadad argues that the idea of concept-based learning is a good one for maths teachers.

The blog post above links to a book called How Children Learn: Mathematics in the Classroom by the US National Research Council (free download).

Here is a short article by an MYP teacher that gives an example of concept-based learning using simultaneous equations.

What can you share about concept-based mathematics?

What Went Well Bookmark for Peer Assessment

Peer assessment can be a bit bland if students don’t know what to look for in their friend’s maths work. Today @tesmaths tweeted a resource to help with this: these What Went Well bookmarks (free, sign-in required). I adapted them slightly since I’m more interested in students’ mathematical communication than in their neatness. Here is my version of the What Went Well bookmarks. These are in a PowerPoint file and are double sided, five to a sheet.

Reflection Time

Last week we had a professional development session with Andy Hind (@andyhind_es4s) about deep learning. One thing that stuck out to me was the value of reflection time in order to deepen learning.

Reflection time for students.
Reflection time for me.

Students need time to reflect on their learning in order to embed it and connect it to their existing knowledge. One strategy Andy used which I will use in lessons was a small picture of a nutshell that popped up about twenty minutes into a session. Andy said, “Tell your partner everything that has happened so far, in a nutshell.”

He said that students should reflect at four points in a one hour lesson. At the beginning (thinking back to the last lesson), after twenty minutes, after forty minutes, and at the end. I scribbled down this time line.

Tomorrow I’m incorporating reflection time into my lessons twice. At the beginning of one of my lessons we are going to recap the last session with the instructions on this slide.

In another lesson I want to ask students to reflect at the end of the lesson, and I’m using the slide below. It’s a feedback structure I have used since even before I was a school teacher. I learned about it from my colleague Richard Hoshino while I was a lecturer at university.

The “3 Minute Feedback” questions always follow the same pattern. The first question is related to today’s lesson and allows me to see if students have succeeded with the objective of the lesson. The second question relates to my teaching. The third one is always worded exactly as above and gives students a chance to share anything on their minds. I like to respond to these via Edmodo after the lesson by giving the class an idea of the proportion of responses of each type and by answering the questions.

I need reflection time in order to become a better teacher. I used to blog more regularly and this was a good method of reflection for me. But Andy also suggested a private reflection journal and I’ve started one this week. I set an alarm for half an hour before I want to go home. I use 15 minutes for writing reflection and 15 minutes for tidying up my desk. I haven’t managed to do it every day this week, but I’m pleased that I have done it three out of the last six workdays. I’m going to either write in my journal or on this blog during my afternoon reflection times this year.

How do you include student reflection into your lessons?

Stereotypes of Mathematicians

I was reading on the NCETM website about films that have mathematicians as main characters. Four films are mentioned:

A Beautiful Mind
Pi
Good Will Hunting
Enigma

One commentator says that these films contribute to stereotypes about maths since they are all about men, men who are unsociable, and that they are uncomfortable in their roles. Films are not helping maths break away from a “nerdy” stereotype.

Then another commentator goes on to lambaste this idea by saying:

“the first three films are about mental illness, not mathematics: the characters happen to be mathematicians, their profession is incidental to the drama that arises from their malfunctioning brain chemistry. The negative, frightening “nutter” stereotype they perpetrate is far more reprehensible, and dangerous, than any “nerd” stereotype.”

The article ends by asking what schools are doing to counteract these stereotypes. For our part, we have named ten of our classrooms after mathematicians. Nine are men and one is a woman: Sophie Germain. They are not all dysfunctional; though Georg Cantor did go insane.

My classroom is named after Paul Erdös but I have yet to capitalise on this with my students. There are so many good stories about Uncle Paul and his love of maths. He was a bit nutty though, so I am not sure he refutes any stereotypes. The truth is, a lot of mathematicians are men and a lot of them are a little odd. (I say this as a proud nerd.) I think this is especially true among academics, and perhaps less so among mathematicians in industry.

Does your school do anything to counter stereotypes of mathematicians?

A Nice Activity for Statistics and Data Representation: Estimating One Minute

My year 8 students were learning about working with grouped data. I used an activity that I pull out regularly when I need some data from the class to work with.

Students worked in pairs and one partner timed the other while they were estimating one minute. I asked the estimating student to close their eyes, say start, and then sit quietly until they think one minute has passed. Their partner used a stopwatch (on their iPad this time) to time the duration.

We recorded on the board all the times from the class. In my year 8s we took several tries each since we are a small group and I wanted to have about 30 pieces of data.

Then we went on with our lesson about grouped data. We got to consider all the authentic data questions–for example, how broad should the classes be to record this data? Also our outliers seemed very far from the bulk of the data. Students naturally asked about these. I like it when there is less need for me to point out these things. Because it is their own data they are intensely interested.

a mathematics lesson that worked

We had a bit of a giggle because one of the biggest outliers was my own estimate for one minute. Once I had my eyes closed I vastly overestimated how long one minute was. My teaching & learning assistant said I was having a nap for 114 seconds!

What data collection activities do you like to use?

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?