New Maths Marking and Feedback Policy

We have recently updated our marking policy in the maths department. It felt like quite a bold update to our policy: it’s much less prescriptive than before and acknowledges the many ways in which feedback can be given to students.

As a school, we are in transition with our digital tools. We work in a one-to-one school in which students have iPads with a stylus or a laptop/tablet; we are encouraging them all to get a stylus. The main teaching software is now OneNote on an interactive whiteboard.

A lot of our students’ maths work still goes into their exercise books, but a lot is now being done on their personal areas of OneNote.

We discussed our marking and feedback ideas over several department meetings and devised the following new policy. I’m looking forward to the learning walk that will focus on feedback later in the school year.

Maths Department
Marking and Feedback Policy

The purpose of feedback is primarily to help students know and articulate their strengths and areas for improvement in the current topics of study. Teachers should share with students an idea of where their learning is heading. This can be done using learning objectives.

We recognise that good feedback happens in many ways which are chosen by the teacher in relation to the needs of the students and the teacher’s preferences. We agree that teachers can make their own choices and this policy does not require that teachers all give feedback in a certain way.

The maths department will have an annual learning walk focused on feedback in which teachers can share their methods of feedback and student reflection. This will allow leaders to verify good feedback is being given and promote sharing of feedback strategies.

We agree that a minimum frequency for recorded, specific, personal feedback is every three weeks.

Homework and classwork tasks should be marked, whether by teachers or students. Students should reflect on their work and record this in some way.

There are many ways teachers can share feedback with students. These include:
• Student self-marking of work as directed by teachers
• Student peer-marking of work as directed by teachers
• Marking by teachers in exercise books or digitally, can be done in or out of class time
• Verbal feedback, which does not need to be recorded
• Written feedback in OneNote
• Student reflections, written with guidance from teachers; can be in exercise books or digitally
• Small quizzes, online tasks, and exit tickets
• Strength and target sheets used after tests

 

 

Two Things I Learned in My Teacher Training that I Still Use Today

Fourteen years ago I was starting my teacher training course. I had already been teaching a little (two undergraduate courses at university) and been to a mathematics education conference and numerous seminars. But I felt as though I knew nothing about teaching as a discipline and a career. After leaving my teacher training there were still so many things I did not know!

It’s impossible for teacher training courses to completely prepare teachers for what lies ahead. Mostly I hear other teachers say that they learned nothing of use in their training. But there are two things I still remember and draw from daily or weekly, fourteen years later.

1. Students need to experience an idea in order to learn it.

I was greatly impacted by reading Experience and Education by John Dewey. (It was a set text in my Philosophy of Education class.) In it, I learned that students need to play around with ideas in their pursuit of understanding. They need to create, investigate, take apart, put together, describe, and explain. Students shouldn’t be told what to think but come to think it for themselves. Additionally, students need to come at a concept from many angles before they gain a deep understanding of it. (Some of the other ideas in Experience and Education don’t sit so well with me anymore. That’s another blog post.)

For several years after first becoming a teacher I was struggling to keep pace with my workload and found it really hard to help students experience ideas. But I decided to make sure I thought about this in detail during my planning time once a week, or twice a week, or more often as I became able. Now it is the founding statement of my education platform: students must experience mathematics in order to learn it. I ask myself, how could a student come across this idea? How could them come to investigate it?

Recently I’ve added a new mantra to this one: the first time students see an idea they don’t have to produce exam-ready solutions. Getting to grips with an idea is important first, then coming to a deeper understanding, then being able to apply that learning. Exam-ready answers to exam-style questions on that topic can be introduced along the way, but aren’t needed at first.

2. The context for a mathematics problem can be “real world” or just plain “intriguing”.

My mentor and biggest influence during teacher training was a professor who studied and celebrated problem solving in the classroom. He didn’t care if a problem was based on some real life context or just really interesting.

In fact, working with him led me to prefer a puzzling situation over an over-simplified, meaningless “context”.

Thanks, John Grant McLoughlin, for embedding puzzling problems into my teaching. (He is still at the University of New Brunswick. See him in action in this video.) John is a huge fan of problem solving and non-routine questions and caused me to not really worry about real world connections; my students are intrigued by the problems and they are engaged in solving them, so I’m happy they are thinking about the concepts that I want them to learn.

Recently, I’ve been revisiting this idea thanks to reading about Peter Liljedahl‘s work on Building Thinking Classrooms. One of his recommended practices is to start each lesson with a quality problem.

I reinvigorated my love for problems this year; for the last term I set myself a goal of using a problem in every year 8 lesson that I taught. Here’s one of the problems I used. (It’s from a collection of problems devised for AQA GCSE [the link is a pdf].)

My students loved this problem and we spent most of a lesson talking about it. I used my visualiser to show several of the students’ solutions on the interactive whiteboard. They had lots of ideas about how to show the required idea – and this was one of the first times they used algebra to prove something. It was thrilling!

Do you still use ideas that you first encountered in your teacher training?

Things I Learned about Marking, Students Collaborating, Diagnostic Questions, and Students’ Reasoning (Notes from the Maths TeachMeet in Kuala Lumpur)

The maths TeachMeet in I attended today at Alice Smith School in Kuala Lumpur was inspiring. I attended sessions about marking (by Denise Benson from Beacon House School), a collaborative KS3 scheme of work (Phil Welch from Alice Smith School), things learned from Craig Barton (John Cartwright from Garden International School), and ways to deepen students’ reasoning  talk (Simone Dixon from Tanglin Trust School).

I also led a workshop about career progression for teachers (materials here). One of the tips I give is about reflective journaling. I recommend starting a blog, for instance. Someone asked me about my own blog and I realise I should probably practice what I preach and write some posts. A good first goal would be once a month since I’m currently writing a lot less than that.

Smart Marking (by Denise Benson from Beacon House School)

My main takeaways were twofold, one of which is not even about marking. First, exit tickets are the way Denise marks because it provides her with feedback for the planning and teaching of the next lesson and because it’s immediately useful to the student as well to see if they understood the lesson. I was struck that it would make them easier to have a little pre-printed form that they use and then the next lesson they could stick it in their book (if I deem that it would be useful for them to keep them). The scribbles below are a mock-up from my notes; hope you can read them!

Secondly, teachers complain about marking because they don’t have time. But actually teachers give up lots of time for lots of things, for example, reading teaching blogs, writing worksheets, or going to a TeachMeet on a Saturday. The problem with marking is that it’s not always a good use of time. Denise was saying that we should find a way to make it quicker and worth the time it takes.

She left us with a brilliant question.

A collaborative KS3 scheme of work (Phil Welch from Alice Smith School)

We had some good background discussion first about what KS3 looks like in each of our schools (using a Padlet). But I shall skip forward to the thing that struck me: Phil has capitalised on a timetabling quirk in that the whole of year 7 has maths at the same time. Thus they can have one week a term in which the students do a collaborative project in mixed groups. Also, the school has pairs of big classrooms with sliding doors between so they can squish a whole year group in for introduction or closing sessions. They also have a lot of breakout/corridor space for groups to work on projects. Really, this is such a great idea that works thanks in part to the brilliant facilities they have at Alice Smith School. Collaborative projects would still work in my school but would be a bit messier.

Highlights of the things learned from Craig Barton (John Cartwright from Garden International School)

Craig Barton was already one of my maths heroes but this feeling was intensified. John attended some CPD with Barton recently and in this 45 minute workshop he shared some of the highlights. It’s clear that I have to spend more time getting to know and using the Diagnostic Questions and Mr Barton Maths websites. John even said that all the classroom examples he uses on the board are now taken (via screenshot) from Diagnostic Questions. They are so good because the four multiple choice answers each stem from a common misconception.

Ways to deepen students’ reasoning  talk (Simone Dixon from Tanglin Trust School)

I was intrigued by the idea of multiple representations today – something I thought that I had considered before, but maybe I had not! Ha. Why do the triangles I draw on the board always look the same? Why are my fraction drawings always of 2D shapes? She showed this lovely example (poor picture alert) of fractions of a cuboid.

I’ve been lucky enough to go to a similar session by Simone in the past (we work at the same school). She mentioned both times an idea that I think would work for me. Put up questions around the room and ask for ‘silly answers’ to be written on them. For my students, who are older than hers, I might rephrase this to be ‘answers which look plausible but are actually not right’. This connected in my mind with what John said earlier in the day about having his students create diagnostic questions – complete with three wrong answers that each stem from a single misconception. Students have to really think hard about a concept to understand the misconceptions that others might have about it.

 

Besides these four sessions, I also enjoyed good chats with lots of thoughtful maths teachers. I always feel more energised and encouraged after a day like this one. What has encouraged you lately?

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.

The Best Compliment I’ve Received this Year

I was in my year 13 A-level lesson. Students were working away on some integration questions that use partial fractions. One student called me over, asked for a hint, and after my explanation he nodded. “Lit,” he said. And went back to work.

That’s the best compliment I’ve had in ages. And, gosh, lingo changes so frequently, doesn’t it?

Association of Teachers of Mathematics – Singapore Branch meeting

Last year we set up a branch of the Association of Teachers of Mathematics (ATM) in Singapore. The ATM is a UK professional organisation and I used to go to their meetings in the UK and in Hong Kong.

Last year, one of my colleagues in the Infant school decided to set up an ATM branch in Singapore. (Setting up a branch is free and astoundingly easy.) So far the ATM branch has met four times with about 50 teachers each time.

img_3097

We had a meeting this week and it was brilliant to meet both primary and secondary colleagues from around Singapore to talk about maths teaching ideas. And eat cakes and fruit drink wine!

img_3098

The theme for our meeting was shape and geometry teaching ideas. One good idea that was shared in the Secondary teachers chat was a gift-wrapped sphere with a note that said, “This Secret Santa present is yours if you can estimate its volume to within 50 cm2.” I made an estimate (without a calculator) but apparently I was 52 cm2 off the correct answer! Doh. I never did find out what was in that package.

I shared an idea for teaching about the surface area of a sphere using an orange that you unpeel. It’s described on this wonderful blog by William Emeny. (His blog’s name is eerily similar to my own!)

If you work in Singapore, please come along to our next meeting! We meet once a term (three times a year) and email invites are sent out.

If you don’t live in Singapore, consider joining a professional group near you. I highly recommend it for meeting new friends, hearing about other schools, and sharing ideas.

Animal Algebra: General Terms of Sequences

Use linking cubes and make these three animals.

algebra animals

Ask students to discuss:

  • How do they see these animals growing?
  • What does the next animal look like? How many cubes will be needed for its {head/body/front leg/hind leg}?

Using these answers, bring students to see the general term for the number of cubes used. I’ve used this picture to help:

animal-expanded

Now do a similar exercise using growing L-shapes, growing T-shapes, growing Z-shapes, growing animals of their creation.

I have used this lesson countless times and it continues to be a favourite. I like that general terms are introduced without the method of making a table of values. I want to avoid reducing geometric sequences to a meaningless sequence of numbers.

Excellent similar ideas are found at visualpatterns.org and this nrich problem about cable bundles. (The nrich problem has sample student work, making it good for teacher workshops, as well.)

Does anyone know where this idea is from? It’s not original to me and the second image in this post is a screenshot from a long lost book. I have also seen that Colin Foster had an idea like this in his (amazingly free) book, Instant Maths Ideas for Key Stage 3 Teachers: Number and Algebra.

a mathematics lesson that worked

Update:

Thanks to Colin, I think the original idea is from Paul Andrews’ book, Linking Cubes and the Learning of Mathematics. It’s available for sale from the ATM and I highly recommend it. (I have just bought a new copy.)

Learning Names at the Beginning of the New School Year

I am a slow but persistent learner when it comes to names. This year I asked my students to use a sticky note and put it on their desk with their name. When I’m walking around I use their names as often as I can. Then at the end of the lesson, I asked them to put their sticky note on the wall near the door.

Then the next lesson they take down their name to use again on their desk.

wall of names

In between lessons, I’ve been using the wall of names as a self-test. I ask myself which class someone is from, or interrogate myself to say something about them. So far, so good.

I teach one pair of siblings and the older sister found her brother’s name and decided to put it really high up the wall where she thought he couldn’t reach it. Ha!

Do you find learning names easy? Do you have some tricks to share?

 

Which is More Valuable: Getting or Giving Feedback?

Students can get feedback from me, their teacher, when I mark their work or talk to them in class.

Students can give feedback to me when I ask them about their learning.

Which is more valuable for student learning? According to an article I was reading today, getting feedback from students is more powerful than giving students feedback. Cris Tovani argues that when teachers obtain feedback from students they can make changes easily to subsequent lessons, and this leads readily to improvement in the students’ performance. Tovani is a reading teacher, so here are a few ideas from me to help get feedback from students.

Three Minute Feedback

I do love this strategy and it’s the only thing I’ve continued doing since the very first teaching I did at university. (Sometimes I forget to use it, though, for months at a time. Does anyone else have this problem, even with great ideas?!) See here for an example when my IB SL students were preparing for a test; see here for an example when students were learning to expand brackets. Here’s one I prepared for my year 9s for tomorrow; they are revising for a test. Tovani says that after she spots the patterns in her exit tickets, she throws them out – it was freeing to read that.

3 minute feedback revision

Looking for Themes in Book Marking

When I take in a set of books (or tests), I jot notes to myself about commonalities among the students’ work to see on which topics they need help. If it’s just a small group of students who need help with factorising, for example, I might invite them all around one table when the class are working on something.

Quick Quizzes

Short quizzes with only a few questions that can be done at the end of class let me know if a concept from earlier is still secure. For example, I gave a four question trigonometry quiz to my year 10s a few weeks ago (and discovered that I need to refresh their memories about the difference between trigonometry and Pythagoras). I would like to get into the habit of using more mini quizzes.

 

Giving and Getting Feedback

Each of these examples allow students to give me feedback about their learning but they are also a means of the students getting feedback from me. In the case of the Three Minute Feedback, they have the chance to reflect on their learning and identify what they need to do next. With book marking, I have been challenging myself to only write questions to prompt thinking that will help students improve. In the case of quick quizzes, I provide detailed exemplar solutions afterwards for them to see and analyse. Thus students know where they are now and how to improve.

Which do you find easier in the classroom: giving students feedback or getting it? Tweet me (@mathsfeedback) or comment below.

Fingers, Maths, and Music

To return to Trninic’s idea about body movement in maths, I was reading an article by Jo Boaler which mentioned students who count on their fingers. Boaler quoted a study which said that students see a representation of their fingers in their minds when calculating. Another study looked at finger perception – whether students know what finger is being touched when they can’t see. Amazingly, young students who get training on how to perceive and represent their own fingers go on to do better in maths. In fact, in six-year-old students, finger perception was a better indicator of maths performance than tests of cognitive processing. Boaler posits that this is also why some people are who are good at music are also good at maths.

[Sidebar: I’ve certainly known a lot of people who are great at both maths and music. I have quite a few in my department now. I had a friend who did joint Honours at university in maths and music and wrote a thesis that connected both. Recently I’ve seen some good IB Internal Assessments about harmonics and chord structure. Does this ring true with you also? (Punny!)]

Boaler has developed some activities for young children (pre-kindergarten/nursery) to increase finger perception. I would love to hear from anyone who works with younger children and tries them.

finger perception Boaler