In Pursuit of Great Mathematics Teaching

Productivity: Never Wait Again

Waiting is a great pain of my work life. In my day, I regularly wait: for the bus, for the photocopier to finish, at the printer for my documents, by the microwave for my lunch to heat, for a doctor’s appointment, for someone who is late, for my tea to brew, and many other things.

I have decided that I will wait no longer. No, I can’t hurry up the photocopier or tardy person, but I can use my time more usefully.

When the waiting is a bit longer (like for an appointment or the bus), pull some reading out of your briefcase. Could you always carry an article or book or your eReader? While at the hospital this week I made it though this article about promoting deep learning (pdf).

When the waiting is a minute or two (like for the microwave or tea to brew), do a little job. Could you refill the tea canister while you are waiting or wipe down the countertops? Is the shared work fridge in need of a little sorting out? While making hot beverages in the staff room I usually wash my hands and tidy the dishes.

When the waiting time is unknown (like waiting for a colleague or queuing), try to do a few discreet stretches. Calf raises are easy; while standing, push up onto your toes for a few seconds, then release down again. Tense and release muscles up and down your legs and arms. Or, if there is anyone nearby, use it as a chance to catch up and build a stronger relationship.

And even if there is nothing to do when waiting, thinking is something you can do! Plan and visualise how the rest of your day or week can be successful. You could think ahead to the next lesson you will teach, or mentally run through the next series of lessons to see if one will lead to the other successfully. I often think ahead to my next meeting in the week and make notes about how I want to prepare.

This also reminds me about research years ago that figetity people burn more calories, which is reported on here. I think doing something while waiting is a mental version of fidgeting. It keeps your mind healthy.

What do you do while waiting?

Completing the Square: Starter Question

Here’s an idea that worked for me last week. My A-level class learned how to complete the square earlier in the year and I wanted to know if they still remembered how to do it. Instead of asking them to just do a couple of completing the square questions, I put this up on the board.

This appeals to me as a starter question because it’s more broadly accessible than a couple of examples and also allows for more in-depth extension work.

1. For the student who sees this and thinks, “Oh no! I can’t remember how to complete the square,” there’s a quadratic expression there for them to have a go at. While I’m circulating I can give tips to these students.

2. For the student who thinks, “Oh yes! I know how to complete the square,” there is a quadratic expression there for them to try and also permission to make up any quadratic expression at all to try. While I’m circulating, I’ll encourage them to proceed to generalisation.

3. For the student who thinks, “Oh yes! It is possible no matter what the quadratic expression is,” there is the permission there to say why and give evidence. While I’m circulating I can ask questions and push them to generalise with justification or evidence.

When I was writing this question I was trying to create something that would push forward my students’ thinking no matter what they could remember about completing the square. I think this worked for me and my students and I hope I can think of more good questions like this one!

What maths question worked for you this week?

Learning: Spotting Pattern Everywhere

I like this for a maths department motto: “Spotting Pattern Everywhere”. (Thanks to a former workplace for that one. I still love it.) Something I was reading today mentions that a lot of learning, not just in maths, is about spotting patterns. People love making meaning out of the things they see around them. The consciousness researcher Daniel Bor was interviewed in Time magazine talking about this. He calls the human brain “ravenous” for solving problems and making patterns. He says, “We get streams of pleasure when we find something that can really help us understand some deep pattern.”

Bor mentioned people who want answers to the question, “Why?” “The way I approach my job, it’s like trying to solve a really big fuzzy crossword puzzle and when you do put in that new clue and see the deeper pattern, that’s incredibly pleasurable.” I took this as encouragement to help students approach learning maths like a puzzle to be fit together. This could make learning more pleasurable for my students’ brains.

Bor was asked, “If our brains are hungry for information, then why do we tend to see learning as a chore and fail to recognize it as a huge source of pleasure?” He replied, “I don’t know. Obviously, more intelligent people get more pleasure from spotting these patterns, but I think almost every normal person does this. I think it’s a pretty pervasive thing but it’s almost as if we can’t notice it because it’s so pervasive.”

Ways to Make Maths Learning Like a Puzzle

1. Show a diagram and ask, What does this tell you?

I was about to teach about area and volume scale factors of similar shapes. I put this on the board and asked students to discuss what it might mean.

2. Ask students to watch you do a maths procedure in silence, then explain to their partner what it was.

Here’s a great example from the nrich website where you watch a very short video of someone summing a sequence, then students are asked to explain what just happened.

Thanks to Larry Ferlazzo for tweeting about this today and making me think more about it.

How do you make learning into a puzzle?

Reading and Writing in Maths: a new version of maths homework

This week I will teach a lesson about estimating to my year 9s. The lesson will be a pretty standard one for me: a sequence of tasks and activities, some whole class items but mostly pair work. The lesson will involve some discussion about how and why someone might estimate the answer to a calculation. I’ll be using a few slides, a puzzle, some problems to solve (sourced from UKMT Intermediate contests), and a plenary about a poor guy whose calculator doesn’t display decimal points in answers.

But it’s the homework I want to talk about here. I plan to set a two part homework: read, then write.

1. Read

First students will read an article from earlier this year about two skiers who allegedly tied for first at the Sochi Olympics. Actually, their downhill skiing times were reported as identical due to rounding to two decimal places.

Two skiers tied for first (image: New York Times).

2. Write

Next, students will access a Google form that asks them three questions for which they need to write at least 300 characters (about 3 sentences). Here’s a copy of the form that you are welcome to answer “for fun”. (My students will be using a private version of this.)

I’m excited to see what students write in response to the third question: other examples (outside sport) where rounding of a measurement makes a crucial difference.

I’m interested to see what my students think about being asked to read and write for their maths homework. After reviewing literature and reflecting on my practice last school year, I decided to try a whole range of different homework options this year.

Our school offers the IB diploma for the final two years of secondary school. The maths courses each contain a 20% internal assessment that is a written report. So part of my interest in reading and writing in maths is to prepare students better for writing in maths during the IB diploma.

Have you tried out reading and writing activities in mathematics classes? Please tell me about it in the comments.

Reading Notes: The One Minute Manager

When I read I usually take some reading notes (in Evernote) to help me retain the main points and some learnings. I thought it might be a good idea to start publishing these in an occasional series, in case they may be helpful to anyone. They are unedited; just notes that I take while reading about what struck me. There won’t be any commentary, just lots of bullet pointed ideas. Let me know if they are useful, interesting, or both.

The One Minute Manger by Ken Blanchard and Spencer Johnson

“The best minute I spend is the one I invest in people” (63)

“People who feel good about themselves produce good results” (19)

One Minute Goals
1. Agree on your goals.
2. See what good behavior looks like.
3. Write out each of your goals on a single sheet of paper using less than 250 words.
4. Read and re-read each goal, which requires only a minute or so each time you do it
5. Take a minute every once in a while out of your day to look at your performance, and
6. See whether or not your behavior matches your goal.

One Minute Praisings
1. Tell people up front that you are going to let them know how they are doing.
2. Praise people immediately.
3. Tell people what they did right – be specific.
4. Tell people how good you feel about what they did right, and how it helps the organization and the other people who work there.
5. Stop for a moment of silence to let them “feel” how good you feel.
6. Encourage them to do more of the same.
7. Shake hands or touch people in a way that makes it clear that you support their success in the organization.

At first, praise things that are approximately right, and help people move towards desired behaviour.
If, instead, we leave people alone and then punish them when they don’t do exactly the right thing, then they start to do as little as possible.

One Minute Reprimands
1. Tell people beforehand that you are going to let them know how they are doing and in no uncertain terms.
the first half of the reprimand:
2. Reprimand them immediately. [reprimand the behavior only, not the person or their worth]
3. Tell people what they did wrong – be specific.
4. Tell people how you feel about what they did wrong – and in no uncertain terms.
5. Stop for a few seconds of uncomfortable silence to let them feel how you feel.
the second half of the reprimand:
6. Shake hands, or touch them in a way that lets them know you are honestly on their side.
7. Remind them how much you value them.
8. Reaffirm that you think well of them but not of their performance in this situation.
9. Realize that when the reprimand is over, it’s over.

Never save up negative feedback, always give it immediately and in very small doses.
Definitely don’t have surprises at performance management time.
“When you ____, I feel ___.” Then reaffirm their worth.

“We are not just our behaviour, we are the person managing our behaviour.” (93)

why does it work?
“the number one motivator of people is feedback on results” (67)

“feedback is the breakfast of champions” (67)
there would be no point playing golf in the dark; people need to know how successful they are being

“most companies spend 50-70% of their money on people’s salaries. And yet they spend less than 1% of their budget to train their people” (64)

Factors Using Multilink Cubes

My year 8s (twelve year olds) have been learning about multiples and now it was time to talk about factors. Some of them have not got all their times tables memorised, which presents some difficulty for our unit of work on factors, multiples, and primes. So my teaching assistant and I doled out the multilink cubes and I asked them to make rectangles. I wanted to know if any number of cubes could be made into a rectangle shape, fully filled with cubes (no gaps).

2014-09-02 11.53.39

After the students played with this for a while, I started making a list on the board of all the sizes of rectangles they had made. Then I asked, which numbers of cubes can be made into more than one shape of rectangle? Above are two rectangles with 8 cubes and below are three rectangles with 12 cubes.

Next I introduced the idea of a factor by talking about the sizes of the rectangles and after some discussion we listed all the factors of 12 using the rectangles in the picture above.

After that, students made their own lists of all the factors of some numbers of their choosing. My assistant and I went around to their tables, asking if they were noticing anything. Using their results, we made a table on the board of all the factors of the numbers from 1 to 15 and talked about what they noticed.

Top on their list of noticings were: 1 is always a factor of every number and the number itself is always a factor. These were not obvious statements to my students. We discussed why each was true by talking about making a long skinny rectangle of 1 x __ for any number.

Next lesson is going to be about prime numbers, so I hope that their next noticing is that some numbers have a lot more factors than others.

Do you use multilink cubes in your lessons?

Is Your Email Signature as Nerdy as This?

I was reading blogs and came across this wonderful idea for an email signature.

Even if you think you don’t love mathematics, mathematics loves you.
Don’t believe me? Solve for “i”.
9x – 7i > 3(3x – 7u)

How marvellously nerdy! This is from the blog Math = Love.

Is your email signature as nerdy as this?

Circle Theorems Choice Board (A Differentiated Lesson)

I had just two 80-minute lessons and their homeworks to help my students learn about the circle theorems. So I put together this choice board for them (here’s a file for you). I had a (not very great) powerpoint with an overview of the circle theorems that they would use as a resource; you could also use a text-based resource or a video.

I told students about the first theorem, then said they were going to learn about the rest of them. They had to choose a line of three items to demonstrate their learning, horizontally, vertically, or diagonally. They could work individually or in a pair.

It seems that my students preferred the diagonal line from bottom left to top right, since many pairs picked those three items.

The treasure hunt is an activity in the pod outside my classroom with questions and answers scattered around that they have to find and follow in order. (It’s a paid-for item that my school has purchased from Mathsloops.)

The sorting cards are a collection of diagrams that need to be sorted according to which circle theorem applies, then the missing angles calculated. (Embarrassingly I can longer find the file from which I made them years ago. Here’s a card sorting activity I found online that’s a bit different but would work well here.)

2014-06-09 17.44.32

I had two great posters made by students and even had one short story. Two groups made games and one student tried a comic. Read the short story below!

The Life of a Circle

Once upon a time, in a land far away (called England) there lived a circle. This circle’s name was Kevin, his life was endless (pun intended) but his life was not as exciting as he wanted it to be. One day, when Kevin was rolling along the road he didn’t realize that there was a slope in his way. He started rolling down the hill very fast and couldn’t stop himself. “CRASH!!!” When Kevin looked up after recovering from his dizziness, he was amazed by the figure standing in front of him. He looked up to find that a beautiful tangent had stopped him from crashing into the brick wall just meters away from them. He finally spoke, asking her who she was. She replied kindly and softly saying, “My name is Tangentina Tangent, how about you?” Quickly, Kevin pulled himself together and gathered the words to tell her that his name was Kevin Circle and thanked her for saving his life. They both looked down nervously and realized that they had met at a 90° angle. Tangentine said, “You know, it’s a well known fact that circles and tangents become the very best of friends because they meet at a 90° angle. I think that this could be the start of a very very very long and wonderful friendship.” Kevin said, “Tangentina, I am very pleased to meet you but, you have to get your facts straight, it’s tangents and radii not circles. But yes, I am excited a the thought of having a new friend.” Tangentina smiled and took Kevin’s hand and they rolled and bounced off into the sunset.

THE END

(They live happily ever after)

I’m contributing this post to #made4math, a way for maths teachers to share projects in the classroom. It’s hosted by the blog Teaching Statistics.

Do you have a differentiation activity you like?

Mathematics Homework: 26 Good Ideas

This week I have been reading, thinking, and chatting about homework. On Monday I posted some doubts about homework and mentioned that I set homework in accordance with the school policy, which at the moment means I assign it after every class and it is mostly routine practice. Today I participated in a Twitter chat about mathematics homework as part of the #eduread group. The idea of the group is that we read an article each week and then discuss it on Twitter and our blogs. This week’s article was “Homework: A Math Dilemma and What to Do About It” by Patricia Deubel. You can read our Twitter chat on the Storify summary.

Meaningful and Purposeful Homework

The main point that hit home for me was there is no use setting homework unless it is both meaningful and purposeful. I sometimes set homework mindlessly and don’t value it much. I am coming to think that homework should only be assigned if there is a clear academic purpose and the task is not just rote practice.

Differentiation of Homework

Also, homework should be differentiated, says our reading. This is a struggle and I would love to hear from teachers who have managed this. I think my main barrier is the time to do it, but I realised that one way would be a homework task with differentiated products. Students would choose their own method of demonstrating their understanding. This is still an idea in its infancy for me and needs more thought.

26 Good Ideas for Mathematics Homework

During my reading and thinking, I made a list of possible homework tasks.

read or outline a chapter (pre-learning)
complete an organizer of a chapter (pre-learning)
write down questions they have about a reading/activity
write/diagram all you know about [upcoming topic]
do a few sample questions and explain the steps
do practice questions (time-based)
answer journal questions about something done in class (ask students what was done and why)
two parts: 1. three problems to check understanding of a concept taught today; 2. ten problems to practice a concept previously learned
draw pictures/diagrams to illustrate a key word
create a concept map
write two problems for others to solve
list the four most important ideas about ….
read and write sticky notes for things you have questions about
design your own learning strategy for a topic covered in class (cards, song, poem, etc)
create a Q&A game
write directions that teach someone else
test corrections: must write why they missed that question, then answer the question correctly
find examples of … at home, in the news, etc; take pictures of … ; then use these in the next class
respond to a thread on Edmodo/other VLE asking a question or sharing an idea
“sandwich” homework: give students the problem and answer and they must fill in the middle
write a summary of today’s class
write a reflection of your work in today’s class
in class, do a notice/wonder activity and generate questions, then students pick one/two to investigate for homework
a project that shows your understanding
adaptive online software such as Khan Academy or MathXL
teacher chooses 3-5 problems, then student chooses another 3-5 from a set

Students’ questions (posted on Edmodo) after reading an article giving English Premier League football standings if only English players’ goals counted

Using Homework in the Next Lesson

Another thing that really struck me started with this quotation: “Homework in the best classrooms is not checked–it is shared.” I was inspired to try to use homework that would generate discussion in the next class. Or it would contribute to the next class’s learning experiences. (Also, I hate checking homework. If the assignment needs to be used for something in the next class then it is kind of “self-checking”.)

Students' sticky notes with a question they had based on their homework reading (pre-learning for an upcoming topic) — Students’ sticky notes with a question they had based on their homework reading (pre-learning for an upcoming topic)

Things to do with homework in class instead of collecting or checking it:

self-assess your homework in terms of effort, understanding, completeness, or accuracy
reflect on which questions were the easiest/hardest and why
find a peer who approached the problem differently than you and discuss our strategies
quiz with 2 randomly selected questions from the homework assignment
get your peer’s feedback on some aspect of your homework
work with a friend to get a best answer to one hard question
give a new problem and ask how it compares to the homework problem(s)
choose the two homework questions that were most alike or most different and explain why you picked them
gallery walk of the results of investigates into notice/wonder questions (see idea 23 above)

More Homework Ideas and Links

I am still learning and thinking over these things. (Sidebar: I love being a teacher because I am always learning. Ten years in and I am excited to be learning about good practice in my profession.) Please share your homework thoughts in the comments or tweet me @mathsfeedback.

Here is a list of the things I read this week while thinking about homework:

“Homework: A Math Dilemma and What to Do About It” by Patricia Deubel
“Rethinking Homework” by Alfie Kohn
Five articles by Cathy Vatterott, accessed from her website, Homework Lady
The homework articles from the archive of The Marshall Memo (subscription required)

Please share one good homework idea!

Mathematics Homework: My Dilemma

A few months ago I started compiling my viewpoints about mathematics education. What I mean is that I have been using sticky notes to record my beliefs about mathematics education and collecting them on a piece of flipchart paper on my back classroom wall. (Strangely, no students have asked about this. I wonder if they read the stuff on the walls at all??) The viewpoints poster has been a personal exercise to help me clarify my thoughts and see which issues I have strong feelings about.

viewpoints on maths ed 1

I used yellow sticky notes for beliefs I can justify and for which I can provide examples. I used pink sticky notes for issues about which I don’t yet know what I believe. One of these is homework. It seems to me that some people feel quite strongly that homework is bad and should be abolished. Others hold that homework is essential; for some it is even sacred. I have no idea where I sit on this issue, hence the pink sticky note.

viewpoints on maths ed 2

Most of the time, I feel as though my thoughts don’t matter too much because homework policy is determined by the school I work in. If my school says to assign homework, I do so. I usually follow the lead of my head of department in what types of homework I set and how much.

But I think it is time to do some reading around this subject of homework and come to some conclusions of my own. After all, I have strong views about all manner of other things (for example, setting students into classes by ability and acceleration of more able students), and my views have to submit to the policy of the school and my department. So why not form some views about homework?

As a first step, I will be participating in the #eduread discussion about mathematics homework. The plan is to read the assigned article by Patricia Deubel and write about it. Then there is a twitter chat about it on Thursday morning (June 4). (Or Wednesday night if you live in a North American time zone. Or the wee hours of Thursday morning for Europeans…. Maybe Europeans are better off reading the summary afterwards.) Update: The second post in this series about homework is: 26 Good Ideas.

Do you believe in homework?