What I've Learned About Teaching... Negative/Directed Numbers with Counters

Two sided counters have massively impacted how I teach negative numbers. Below are suggestions of what I found beneficial for students and reflections on what to improve on in the future. I hope it may be of help to anyone wanting to try this approach to negative numbers. Before I dive in, I must thank Bernie Westacott and Craig Barton whose discussion on negatives I watched avidly in lockdown and ultimately inspired me to be brave and try this.

1. Allow time for students to look at counters and work out the number they represent.

1. Starting with a negative counter (red) representing negative one and a positive counter (yellow) representing positive one respectively, ensuring students know which side is which.
   
   

      positive  or    +1

   
      negative    or   -1

   
   

   Then have small numbers of positives, small numbers of negatives. Then have examples of some of each. This is great on mini whiteboards as a show me the value of this diagram, draw a diagram that represents -2 etc.

2. Develop the concept of "zero pairs" and use the term often

At this point you need to introduce the concept of zero pairs - students will likely be familiar with it but naming it and being clear with what they are helps adding and subtracting later. If you have the same number of negatives and positives you have a total of zero. The negatives negate the positives - you can discuss the word negate and how it relates to negatives in this context. Show some diagrams and ask are they equal to zero?

Why is it easier to spot with some than with others (when they line up)? Might it be helpful to arrange our counters in a certain way? What do I need to change to make the diagram equal to zero?

Make sure at this stage you show how a number can be represented in numerous ways depending on the number of zero pairs (e.g. start with 2 then add zero pairs, is it still 2?). Make sure they understand you can add as many zero pairs as they like to a number and the value doesn't change.

Again this is another important concept particularly when subtracting, so worth spending time with doing some questions with the counters and mini whiteboard work with diagrams. When students are fluent in evaluating counters from looking, confident with zero pairs and happy moving and arranging the physical counters and drawing diagrams of the counters they are ready for the next step!

3. Focus just on ADDING

Adding is when you start with a line of counters and then add more counters, collecting them together and evaluating the result. It is the easier concept for students to work with. Start with positives add positives using the counters to get used to the process.
e.g. 3 + 2

There are five positives altogether so (+3) + (+2) = (+5)  (read as positive 3 add positive 2 equals positive 5).

As students are confident doing this WITHOUT the counters, it is good to get them modelling with what they are confident with before moving on to the concepts they tend to find more difficult. Do a couple and explain to the students it is an important part of the process even though you know that they know the answer.

Next introduce negatives add negatives.
e.g.(-3) + (-4)

There are seven negatives altogether so (-3) + (-4) = (-7)  (read as negative 3 add negative 4 equals negative 7).

Which follows the same kind of logic as above with all positives. Practice a couple of these. Students will start to notice positives add positives mean you have more positives so you have a greater positive pile and that negatives add negatives mean you have more negatives so you have a greater negative pile. Avoid you making these generalisations and try not to encourage them yet, we want them using counters until they are ready to move on from them, so don't rush the counters/diagrams stage.

Then look at a mixture of adding negatives and positives in various forms. Make sure they see that the sum can either have a positive or negative answer.

e.g. 3 + (-2)

There are two zero pairs, and one extra positive counter, so (+3) + (-2) = (+1)   (read as positive 3 add negative 2 equals positive 1)

At this point, instead of continuing in the line, if you are adding a different type of counter (negative/positive) then start a new line underneath. You can ask the students why we might do it like that instead (so they can line up zero pairs and see the answer easily). With counters you can actively take a positive and negative and say they are a zero pair and physically remove them out of the way. But when drawing them, you don't have that luxury - you could cross them out instead like this:

It depends which you/your team prefer.

You also need some examples where the answer is negative. You may need more examples when you are adding two different kinds of counters.

e.g. (-4) + 2

There are two zero pairs, two extra negative counters, so (-4) + (+2) = (-2)   (read as negative 4 add positive 2 equals negative 2)

At this stage avoid generalising – students will want to find a quick and easy way to do it or find a phrase like “a negative and a negative make a negative” which can lead to misconceptions. Allowing students to perform the calculations in a structured way with the concrete manipulatives and/or diagrams will build a concrete understanding. With time and enough practice they will build these connections in a way that means they do not need the counters and can perform bigger calculations more fluently, but do not rush to remove the counters!

Think carefully about the types of question you are asking when practicing too - help guide the students to identify important relationships, such as (-3) + 4 will be the same as 4 + (-3) and ask students why (the diagrams are essentially the same but the other way around).

4. Think of subtraction as removing

Students find subtracting a bit trickier. Start what they should be comfortable with with bigger positives take away smaller positives.

e.g. 4 - 2   which is (+4) - (+2) and you might want to phrase it that way. You might also want to phrase it as "positive 4 REMOVE positive 2"

There are two positives, so (+4) - (+2) = (+2)   (read as positive 4 subtract/remove positive 2 equals positive 2)

Start with one line of counters, remove the ones you are subtracting from the line, and see what is left. With physical counters you can literally remove them, but with the diagrams it is worth crossing them out (as above). Again reinforce we know they can do that calculation without the counters but it is important for the next stages to practice a couple of these.

Then look at more negative numbers subtract less negative numbers

e.g. (-5) - (-2)  which you could rephrase as "negative 5 REMOVE negative 2"

There are three negatives, so (-5) - (-2) = (-3)   (read as negative 5 subtract negative 2 equals negative 3)

The most difficult subtractions to do are when you do not have the counters needed to remove them.

e.g. 2 - (-3)   or (+2) - (-3)  which you could phrase as "positive 2 remove negative 3"

There are five positives, so (+2) - (-3) = (+5)   (read as positive 2 subtract negative 3 equals positive 5)

Highlight the fact that the middle diagram is still worth positive 2, that we can add as many zero pairs as we like and the value doesn't change. To enable you to have the negatives there to remove, you need to use zero pairs. You will likely need quite a few examples of these for students to get their heads around.

e.g. (-3) - 2  or (-3) - (+2)  which you could phrase as "negative 3 remove positive 2"

There are five negatives, so (-3) - (+2) = (-5)   (read as negative 3 subtract positive 2 equals negative 5)

Give loads of opportunity to practice just subtraction, again ensuring you select questions you know they need to see, for example (-3) - 3 is NOT zero.

6.  Practice a mixture of adding and subtracting

Develop some fluency by practicing a mixture of questions, using the counters physically, drawing the counters and when students are ready you can remove the scaffold for them. You can also relate questions to number lines and the direction in which you are moving along the number line. To lead them into generalising when they are ready, questions such as these brilliant ones from Chris McCrane: Alternative representation of Integers – Starting Points Maths 

Top Tips:

·        A visualiser will be your best friend! I love writing the question on the desk with a whiteboard pen and then modelling with the counters. Students can build the question with you and if you can, get students to sit under the visualiser to show how they would do the calculations.

• Don't be afraid of using them with older students - it is a bit more difficult but actually I've done with Y9 and Y10 and found it to have great impact.

·        Use lots of mini whiteboards to check for understanding! Get them drawing the diagrams on their boards, pick out some to discuss. The next step after the counters is diagrams so they need to develop that skill too (they can always draw a diagram in an exam, but they can't use counters!).

·        Don’t rush – it is worth investing time in this early on so you can reap the benefits for years to come. How many Y11/sixth formers still make silly errors due to negatives? Worth the time investment early on.

·        If possible, use this approach as a whole department so there is consensus and students have the consistency of using the same method.

·        Avoid ‘simplifying’ with phrases like ‘adding a negative is the same as subtracting’ but instead encourage students to either do the calculation with counters or diagrams instead. They will then build their own connections based on understanding.

·        Play with the counters yourself first, completing questions and getting really used to them.

• The counters are a scaffold, so they do need removing eventually (imagine doing (-345) + (-45) with counters) but they can always go back to a diagram as a bit of a nudge by doing something smaller like (-4) + (-3) what happens? How can we apply that to (-345) + (-45)?

Problem Solving Chunking Resources

In a previous blog post, I discussed some problem solving strategies. This included some problems chunked into steps, which I initially called 'Baby Steps' and then changed the name as students didn't like the name 'Baby Steps'. I've called them 'Chunking Resources' for now as they take a bit problem and break them into manageable chunks, but if you think of a better name let me know! You can call them whatever you like with your students!

At the time, people asked if I had a bank of them to share, I didn't but said I would when I had time (ha!). Over a year later, and a pandemic giving me a bit of time to sort them out, here we are! I'm hoping to release more and more gradually when they are ready, so let me know if you have any requests.

What They Look Like:

They do look a little confusing so you need to know the intent behind them before using them (don't just photocopy and hand out blindly).

Intended Uses:

The resource is supposed to be flexible for your classes/students. I've purposefully added them as a PowerPoint document as well as PDF so you can adapt them as needed.

Some students may be able to answer the question in the top third with no extra help, job done - you must be a fantastic teacher as they can do it without the scaffold!

For those who are unsure what Maths they need to use, the middle third is questions which enable students to first practice the skills needed to attempt the question - note the numbers are purposefully different so they can practice the skill then try to apply it to the original question in the top third.

Finally, for those who still can't access the question, the original question is broken down into three steps in the bottom third (in the same vein as the middle section but with the correct numbers for the original question), so some students will complete all three sections before answering the initial question whereas others may be able to answer it independently without the bottom two thirds, but they could use the bottom third to check their working and maybe see a different method. Some students may start answering in the stages and then have the 'penny drop' moment where they can see how to answer the question themselves and abandon the scaffold to freestyle the problem.

You can also use them as a we do, you do resource, where you look at the problem, discuss the Maths needed and look at the middle section as a class together to practice the skills, then students can attempt the question themselves from the top third, or if they need the structure try the bottom third to get the answer to the original problem without being teacher led.

I have previously used these by printing double sided, so the question is on one side, and the question broken down into stages is on the back, giving students the option of turning over if they need the scaffold. This way I can put three problems on one sheet, with the scaffolding on the back, so that students can choose for each question whether they need the scaffold or not. See example below:

These resources are free for you to use and adapt for however suits you and your classes, all I ask is that you give them a share on Twitter and tag me @mathspeptalk to tell me how you've adapted them or used them successfully - sharing is caring. Also if you have any recommendations for improving them or recommendations, please get in touch.

SET ONE: From White Rose Maths Hub GCSE Problem Solving
Circle Problem:
PDF
PowerPoint

Pythagoras' Theorem Problem:
PDF
PowerPoint

Trapezium and Pythagoras' Theorem Problem:
PDF
PowerPoint

Pythagoras' Theorem and Trigonometry Problem:
PDF
PowerPoint

Mean with Surds Problem:
PDF
PowerPoint

Volume (Backwards Cylinder) Problem:
PDF
PowerPoint

Front and Back Worksheets of Set One:
PDF
PowerPoint

SET TWO: From Edexcel Specimen Papers
Area and Proportion:
PDF
PowerPoint

Combining Ratio:
PDF
PowerPoint

Complex Speed:
PDF
PowerPoint

Complex Volume:
PDF
PowerPoint

Coordinate Geometry:
PDF
PowerPoint

Coordinate Geometry Circle:
PDF
PowerPoint

Front and Back Worksheets of Set Two:
PDF
PowerPoint

How Being 10% Braver Saved My Career

I'm going to be brutally honest. This year I fell out of love with teaching. I felt worn out, I felt my hard work wasn't paying off and my "spark" for my vocation had almost completely ebbed away. It started feeling like 'just a job' to me when before it had been something I was incredibly passionate about. I would read avidly, discuss new ideas, try loads of stuff out in the classroom, I was an active and reflective teacher who always strived for better and absolutely lived for my job (hence my nerdy blog!). It was therefore incredibly unusual for me to lack motivation (given I thrive when I have something else on my plate whilst teaching, for example doing my masters project and writing my dissertation all during my NQT year). This year (my fifth year in teaching) it has felt like the fire has gone out and I've just been going through the motions - a passive teacher. It honestly filled me with shame and I felt I wasn't good enough for the students who deserved better than a lacklustre teacher. Several times over the course of six or so months I considered leaving the profession. I even signed up for a job site that wasn’t TES! (Keep reading – I promise there is a happy ending!)

Then something changed. I had tried to read books this year (as I have enjoyed doing in previous years) but I found it difficult to motivate myself which tended to result in me reading the first chapter, putting it down and then forgetting about it. However, when I started reading 10% Braver Inspiring Women to Lead Education by the fabulous WomenEd group something finally resonated with me for the first time in a long time. It reminded me of what element of teaching I was passionate about: action research and trying different ideas in my classroom, fine-tuning my practice to be the best possible teacher for the young people I teach and a hope that I can inspire young teachers in the future. It reminded me of those staff in school who I look up to as they are also passionate about teaching and learning (and I am very lucky to have many inspirational women in my school!). I decided I needed to change something to get myself out of the pit I'd somehow created for myself. So feeling 10% Braver I asked our inspirational Assistant Head Lyn Lawton if I could shadow her and see what it was like to be so influential in terms of teaching and learning around school. Lyn is fantastic and immediately showed interest in supporting me with this idea. Even before doing any shadowing I felt immediately more inspired and my enthusiasm for teaching came back like a bolt.

The first thing I shadowed Lyn doing in her role was observing and giving feedback to a colleague in a lesson. I've been blessed to have the insight of Lyn's feedback in several of my lessons in the past and have always valued her suggestions for ways forward. This time I was invited to be a part of the observation process. Beforehand she talked me through the documents she uses to help structure her feedback, what she tends to look out for, that she takes it in initially and then goes to talk to students and look in their books. We watched the lesson and I watched her give feedback to the staff, highlighting all the positives and having a discussion about the lesson, suggesting a couple of things which may help move the class forward. It was incredibly useful for me to see and I was overwhelmed by how lucky I was to have this experience - I don't think many teachers get the opportunity to watch an experienced member of staff give a lesson observation and feedback. I think it is something staff should do more of - then you have a better idea of what to look for in lessons and how to hold the discussion afterwards. Also, the impact on my teaching was more significant than if I had just popped into the lesson myself to watch.

The next experience I was invited to be part of was a training session Lyn held with some members of the senior leadership team regarding lesson observation feedback. Some of the documentation she had used when I shadowed her observation was her trialling something new (which she mentioned at the time). The purpose of this training session was to feedback to her teaching and learning team the successes of this new document and discuss ways forward with the aim of ensuring continuity and consistency of giving feedback across the school. Lyn organised a presentation with her research all done, calling on the experienced members of the team to share their observation and feedback experiences. I was invited to take part in the group tasks and discussions and made to feel welcome (despite initially feeling a bit nervous and having to overcome an initial feeling that I didn't belong with all these important people!).

At the end of the training session, some of the lead teachers discussed trying the document by observing each other and giving feedback, so I arranged to do a reciprocal observation with one of our Lead Teachers Emma Higgins who is also a Maths teacher. Emma has also been a member of staff who has been a great support to me (and continues to be) and initially helped me find my feet at school. I really enjoyed going to watch her and filling in the documentation, then later meeting with her to discuss the lesson. After awkwardly discussing whether we should pretend it is an actual observation or whether we just discuss the helpfulness of the new document, I decided to be 10% braver again and tried to give proper feedback! Then the experience of watching Lyn give feedback came in really helpful, we had a discussion about the lesson and I found some useful tips and suggestions on ways forward from the new documents Lyn had designed. I now feel if someone asked me to do an observation I would feel much more confident and I am assuming you only get more confident the more you do! It is something I hope to be involved with more in the future.

Emma then came to watch me teach, and keen to impress I definitely raised my teaching game! I tried something new and really enjoyed the lesson. Emma gave me feedback using the sheet and we both felt more inspired by the experience. If anything, I feel staff should observe each other like this more. Not formally, but with a clear and consistent structure as Lyn intended from her document. I definitely got more from the experience than any of the previous observations where I have just dropped in and watched for part or whole lessons. It forced me to think and reflect more deeply on what I was watching and it had a more significant and long term impact on me.

I've been so lucky to be given the chance to be involved in shadowing Lyn and as a result working with Emma. It has totally reignited my spark for teaching and reminded me how much I love this job. But my main message to anyone reading is - if you don't ask then don't get. I was fully aware in asking if I could shadow Lyn she could say no as she is very busy, but I wanted to at least try. Thankfully Lyn has been incredibly accommodating and I've learned a great deal from her and Emma. I also want to tell anybody reading this feeling disengaged from teaching as I did that you can do something about it. The power is in your hands to change something. You don't have to drift from lesson to lesson debating every evening whether or not you still enjoy teaching, stuck in a pit of complacency. Remind yourself why you got into teaching, think what it is you enjoy the most about this wonderful profession and see what you can do to reignite that spark. I've come out the other side of my teaching identity crisis and you can too. Just be 10% braver.

GUEST BLOG: Transferring Language of Exams into Language use in the classroom by Emma Higgins, Maddie McClure and Tara Hall

Transferring Language of Exams into Language use in the classroom



Over the past 6 months we have been working in a Triad to overcome the problem that some pupils transfer of language in Maths exams is a concept they find difficult.


In our experience, when the Mathematics examination specification change in 2015 particularly, we have found that pupils are required to understand not only the method but the more technical language used. We chose this topic as our research for this academic year as we wanted to see how much of an impact the language barrier had compared to ‘teacher’ language. We tried to find research to help us with our study. However, we only found articles that related to pupils being able to understand problem solving questions. At this point we realised we needed to look deeper into this
topic and do our own research.


We began by pre testing different groups on ‘teacher language’ and ‘exam language’. The questions were exactly the same but instead of writing ‘expand’ (the bracket), we wrote ‘using Noddy’s hat….’ as this is the language we would use in Maths lessons to help them remember how to answer the question.

The words we find the pupils struggle with the most were, expand, solve, descending, estimate, equidistant, factorise, hence, identify, index form and evaluate. We found that in our lessons, as we taught the pupils these topics, we used simplified language to help their understanding. However, this has meant that pupils don’t always recognise the more formal mathematical language that is found in exam questions. This has a big impact on their assessment marks because they understand the method but don’t understand which method to use for each question.



When we did the pre-test, we found that pupils did significantly better at the ‘teacher’ language questions compared with the ‘exam’ language. The words factorise and equidistant had the largest score difference with a 51% and 46% increase!



We quickly realised that by simplifying the language we used in lessons, we were holding back our pupils understanding of mathematical language. The biggest adjustment we made in our classrooms was to focus more on ensuring that pupils knew which method to use with which word. For example, the teacher only using the word expand and the pupils having to work out the method they needed to use. We ensured that we broke down the mathematical language into smaller chunks that they would understand, rather than using simplified words. We also focused on exam questions more, encouraging pupils to highlight the mathematical words they were unable to understand.

Developing Problem Solving (4 Practical Strategies to Develop Resilience)

Developing Problem Solving (4 Practical Strategies to Develop Resilience)

I wanted to help my students develop their problem solving skills as I was frustrated (as I assume they were too) with questions being left empty on their assessments.
You know the type of question I am talking about? Where they shoehorn area of circles, volumes of cones and cylinders, density and converting units all in one question often worth a hefty 5 or 6 marks? Where students feel overwhelmed before they start and just turn over the page?

So here are some of the things I tried and a brief evaluation of each:


1. Baby Steps (soon to be renamed Stepping Stones)

My class and I have started referring to these as ‘baby steps’ as it breaks the question down into each of the single Maths skills required, gives them the opportunity to practice these skills separately before then applying them to a question. These are not an original idea, there have been different variations of this around including structuring exam questions. I first found them on Mathsbox and have since developed my own to help students access some of the questions they couldn’t access before.
For example, this exam question:

Taken from JustMaths Area & Perimeter (H & F) - Version 1 2016 (originally from Edexcel)

Would be supported with these baby steps:

NOTE: The baby steps are NOT directly related to the question (the numbers are different). The reason being they can practice the skills separately before then applying them to the question. They can then complete the entire question independently.


Pros:

• The students really liked them – it gave them a way into the problem when they didn’t know how to start.
• It meant everyone could do something.
• They didn’t have to use them unless they needed to, and the more we did the less they relied on them.

Cons:

• Need rebranding – some of the students felt put off by the name baby steps (implying they are babies maybe?) and I think I will rebrand them as “stepping stones” instead.
• Are they really learning how to independently answer the questions if I am chunking the problem down?
• These will only work if the students are confident with the basic skills needed to access the problems.
• Do they help students develop metacognitive problem solving skills and resilience if we tell them the Maths they need?

       2. Forcing silence

This is a strategy I use not just for problem solving, but I do really like doing in a problem solving session. When I first give the students a problem, I tell them they must work in silence for 2 minutes. Then I let them speak for 30 seconds before then finishing the question in silence for 2 minutes. I offer sentence stems so if students aren’t sure what to do then they can write down good questions they want to ask myself or each other when they get their discussion time. I walk around and check students are either trying to find a way into the problem, attempting the question or writing down any questions they have. I refuse to answer any questions or let students discuss the problem with each other in those first two minutes (note: these timings are a guide, I will vary them depending on the question). During their discussion time, I sometimes offer the “baby steps” to those who are unsure where to start.

Pros:

• It gives students the opportunity to practice in a safe environment, but in replicated exam conditions giving them the chance to see what they are truly capable of – often surprising themselves how much they can achieve independently
• It creates a quiet and focused environment in which to practice which removes temptations to ask others
• Gives students time to think of better questions to ask rather than just giving up and saying “I don’t get it!”

Cons:

• Students can find it frustrating but if you highlight the relevance in terms of a safe place to try in exam conditions and let them pleasantly surprise themselves with how far they can get independently, they soon appreciate why you are doing it.

       3. Open Goal Problems

You’ve likely seen these before! Miss Banks has a healthy supply on her Goal Free Problems site: https://www.missbanks.co.uk/goal-free-problems which I love to use. There are other sites too, so there are loads of already made goal free problems out there. I also like to use exam questions and take the last line out, then give them four boxes to write their own questions and find their own solutions. Then I can give them the exam question further down the line and see how they cope (normally better as they’ve explored the question before). I’ve used these as group discussion tasks, given one student the job of listening and making notes to then feedback their most interesting question to the entire class.

Adapted from Edexcel Higher Paper 1, June 2017 series

Pros:

• Encourages students to find any additional information which could in an exam get them marks, students often struggle to find a way in but using goal free problems encourages them to see what else they can find.
• Using as a discussion task helped me incorporate oracy in my lessons.

Cons:

• Students lacking imagination and picking easy questions – I found this happened to start off with but then as I praised those who had an interesting and complex question and answer the whole class seemed to up their game!
  

       4. Bookmarks

We have discussed as a group what strategies we use and collected a list in the back of our books (draw a diagram, underline/highlight key information, write down any formula, identify the Maths needed, try working out more information not given in question, etc.). We refer to this regularly. I then saw a colleague use a bookmark for exam questions with helpful hints on them, I thought it was a great idea to highlight the strategies and make them easily referred to. It means the students don’t have to flick between the front and back of their books and if they get a new book they just transfer their bookmark across. I imagine they will use them as a crutch initially and slowly ‘wean’ themselves off when they have the strategies committed to memory or the practice of checking the strategies off like a checklist will hopefully become automatic. I had four different categories for the students, how to help: at home (on homework), in lessons, if they don’t know how to start and if they get stuck (two on each side).

Pros:

• Students get to choose their own strategies so each is unique and personalised to them.
• They are mobile and can be used in different scenarios.
• Encourage students to be independent (with the crutch of the bookmark initially, hoping to need it less and less as confidence grows).

Cons:

• Some lack the metacognition to recognise what strategies work well for them, or even to establish what they are doing are strategies that could help them in other questions. A lot of discussion is needed to ensure those students with weaker metacognition get the same benefit as those who are very good at self-regulating their thought processes.

Oracy in Maths

I looked at how to develop oracy strategies in Maths after some feedback from a colleague after a peer observation. I always thought that letting students discuss their work and careful questioning was enough oracy in my class, so I began to think how I could provide more opportunities for discussion. I tried a couple of things, like putting a question up on the board and asking them to think, pair, share their ideas on how to start the problem. This was a simple way to introduce opportunities for talking and worked relatively well. I got a bit stuck as to what else I could do, but then I went on a course specifically on oracy in Maths and Science by Voice 21 and now have several more tried and tested strategies that I have used in my classroom. Here are some of my favourites:

1. Discussion Points

Before starting a topic, make a statement for the students to decide whether they agree. If you carefully design the statement it can cause disagreements and bring out misconceptions. You must go round and listen to what the groups are saying during their discussion time. Assign groups and roles if you wish, my favourite role is a silent summariser who sits and listens to the discussion making notes and then feeds back to the class what their group thought. You can then go on to teach and practice a skill before coming back to the discussion point as a plenary to see whether they have changed their mind.

EXAMPLE: An example might be “angles in any polygon add to 360” which I would ask at the beginning of the lesson, give students no more than 5 minutes to discuss and get general feedback with a vote of who agrees before any teaching. I would then teach angle sums in polygons, investigate and generalise a formula which the students then practice. I would then come back to the discussion point at the end of the lesson to conclude the learning as a plenary.

2. Cartoon Concepts

Similar to discussion points, you can have a question or statement in the middle with several different opinions around the outside (including misconceptions). The students can then identify which opinions they disagree with quite easily. It is more difficult for them to pick out and justify who they agree with the most. You can provide a grid to help structure their discussion and help them develop their arguments by collecting evidence.

EXAMPLE: This is an example of a concept cartoon I created to introduce order of operations. The table was used to help guide and focus the students discussion and help them structure their feedback to the class. I gave one table per group and whoever had the table was the ‘silent summariser’ who made notes on the group discussion but was not allowed to contribute. The silent summariser was then the only student in the group allowed to feed back at the end.

3. Grouping Statements or Questions

Give students statements in groups to organise into three piles: always true, sometimes true and never true. This promotes discussion and encourages students to challenge each other’s thinking. Giving a range of different questions and asking them to group them without giving categories promotes discussion of what is similar and what is different, sometimes different groups will categorise them differently. Also asking students to rank statements or questions based on importance, difficulty, how many marks they think it is out of. Alternatively asking which is the odd one out, using questions, diagrams or graphs - https://nonexamples.com/ contains some examples of spotting the odd one out.

4. Non-questions

If you give students a diagram and ask them “what could the question be?” first, the students list as many different questions as they can based on the diagram ranging from obvious up to extremely challenging maybe even impossible questions. I'm always amazed by some of the questions the students ask, often asking some you would never have thought of yourself. Working in groups they can then explore whether their questions can be answered. If their question is impossible (or they don't yet have the knowledge or skills to answer it), they can explore what else they would need to know to be able to answer it. They can then feedback one of their questions (and potentially their answer to that question) to the rest of the class. There are lots of freely available resources, such as http://goalfreeproblems.blogspot.com/ the home of Goal Free Problems but any question can be adapted to be open ended for discussion by removing the question (usually the final line in an exam question) and instead writing “what other information can you tell me?”. I’ve done this as homework, where students have time in lesson to discuss the problem and make some notes. Then they independently write up on no more than one side of A4 what their group discussed and what conclusions they came to – this ensured everyone was part of the discussion and they had to listen to each other.

4 Top Tips on Hinge Questions

4 Top Tips on Hinge Questions

I studied hinge questions for my masters dissertation project. Most of my project was based on Dylan Wiliams work on hinge questions. A hinge question is asked before half way through a lesson, every student answers and the teacher responds within 30 seconds. As the name suggests, the response to the question determines the direction of the lesson, reteach, practice or move on. You can use them at several stages to determine understanding and you can layer them to separate students into various levels of understanding.

Here are my 4 top tips for an effective hinge question:

1.      Use multiple choice

You need to gather answers and process them quickly. The best way is to offer multiple choice. It also means that you can give students options with common misconceptions, which leads me on to…

2.      Plan carefully

Hinge questions shouldn’t just be flung into a lesson unplanned. You need to design them carefully to check student understanding. Think about what pitfalls students might fall into and incorporate that into your question. You need to see and process every students answer, mini whiteboards would be good if you want to see their working (remembering that responding quickly is important, so you might not want all their working). Multiple choice can be collected quickly using voting pads, raising hands, showing coloured cards (I get mine to use their planners as they have red, orange, green, purple and white pages meaning I can have up to five multiple choice answers).

3.      Timing

Ask at the start of the lesson to determine what they already know, ask after teaching to see how much they have understood, ask at the end of the lesson to see if you can move on next lesson. When you ask your question, it must tell you something about where the students are and how you can get them to where you want them to be. The timing will depend on the purpose, and although initially research suggested before half way through the lesson, if your purpose is to determine prior learning why not ask before any teaching? Or if to summarise learning, why not at the end of the lesson? It goes back to tip #2.

4.      Have options to follow up the hinge question

So, my students understand. Do I waste time practicing the skills or do I move on to some problem solving? They don’t understand at all, do I reteach it or expect them to be able to practice? Or what if some get it, some sort of understand and some of them are totally stuck? You need to know how you are going to move forward with the lesson and prepare for every possible outcome. One thing I did was prepare a single resource based on every outcome, with a modelled solution followed by structured questions for those who are still struggling at the top of the worksheet, unstructured questions with scrambled answers at the side for those still a bit unsure, and problem-solving questions and rich open-ended tasks or investigations for those who have a good understanding of the topic. This meant students could move down the sheet as their confidence grew. You can also get a small group or individuals to work with you at the front if they are still really struggling.