This is a summary of the presentation from Maths Conf 9, held in Bristol on 11/3/2017. Thanks to everyone who came and who asked questions!
What is a drill?
A drill is narrow. It should be focused on a single thing, such as:
- Which fraction ‘rule’ to use for a mix of fraction operations (i.e. choose the rule, don’t complete the operation)
- Do I need to borrow? Write ‘B’ above each calculation where this is the case
- Will the answer be positive or negative? Write + or -, nothing more (a mix such as -2-6, 9-12, -4+8, etc)
- Times tables
- Expanding single brackets
- Simplifying indices
- Multiplying and dividing directed numbers
- Improving accuracy (and fine motor skills!)
- Multiplying and dividing by powers of 10 (e.g. practising simply ‘moving the point’ correctly!)
- Rounding (underlining to the correct digit, circling the correct digit)
- Recognising and deciding
- A drill to with a mix of questions that are either rounding OR multiplying/dividing by powers of 10 (confusing for a small number of pupils!)
- Do I need an LCD? A mix of questions: some multiplication, some fractions which already have an LCD, some fractions without
- Improving muscle memory (automation of multiple steps)
- Completing the square
- Calculating the gradient of straight line
- Rationalising the denominator
- Many operations require a level of deep understanding that overwhelms pupils. We need to build proficiency in every exception; first separately, then together. As teachers our expertise and knowledge can blind us how challenging this is for pupils. We a fluent in exceptions in how we speak; we must help pupils become fluent in the exceptions of maths (which is, in this respect, much like a language).
- It’s important to recognise that progress doesn’t happen in a lesson, but over time. It isn’t seen in their books from that lesson, but in long-term memory and the speed of subsequent connections…drills are an investment in their long-term memory!
- Drills allow you to build motivation, as they can manufacture the sense of having lots of success
- Drills offer quick wins: automaticity, confidence, buy-in
- In the long-term, drills strengthen vital links that allow maths to feel less laborious and confusing.
Here are three examples that Hin-Tai Ting used over several months with 7 Zeus (fourth quartile group in Y7). He has described the design process in fascinating detail here.
Let’s focus on the first column. In the first example, pupils are completing a simple procedure, focusing on a single decision (i.e. what happens when multiplying by 10). This is focusing on accuracy and motor skills, and automatising the many ‘weird’ things that seem to happen with the decimal point…
In the second example, we can see that they now have mastered ‘moving the decimal point’ and are focusing on fluency with moving 1/2/3 decimal places in either direction.
7Zeus are now very competent with multiplying and dividing by powers of 10. This drill is now focused on fluency with varied representations: using powers and decimals (e.g. recalling what happens when multiplying by 0.01).
What can be drilled? And what should be?
- The aim is effortlessness. If it feels effortless for you, as a maths teacher, you want it to feel effortless for them.
- Focus on:
- High leverage (topics that reap benefits across the curriculum, such as fraction-decimal conversions)
- High frequency (topics that you KNOW they will need in many exam questions, such as rounding)
- High complexity (topics that have mutiple and confusing steps that need to be chunked and automated, such as adding and subtracting fractions)
- Error prone (topics where they know roughly what they should do, but tend to mess up, such as multiplying and dividing by powers of 10)
- Confusion prone (topics where pupils are easily confused and tend to eventually guess, such as adding and subtracting directed numbers)
- System 1 override! (topics where their first instinct is often wrong, such as dividing fractions and index laws)
The drill above is an example of a decision-making drill: pupils need only to decide if the answer will be positive or negative.
This is an example of a speed drill: the focus is on getting some (very weak!) pupils to be faster and more accurate with very simple mental calculations, both to move them away from finger counting and to improve accuracy in column addition and subtraction. Each day was +-2/3/4/5/6/7/8/9, cycling back until all were speedy at all of them. They seemed to really love it, and it was very quick each lesson.
Ones that didn’t work, or don’t suit
- Some topics are too complex to be suitable (e.g. metric conversions)
- Some Too simple (multiplying proper fractions – it quickly becomes a times tables exercise, and doesn’t make them better at multiplying fractions)
- Questions that allow them to go on autopilot (20 values multiplied by 0.1…they’ll quickly switch to ‘divide by 10’ in their minds, and will not have strengthened their recall of what happens when multiplying by 0.1)
- More fiddly does not mean more challenging (making the numbers longer or more annoying is just….more annoying)
- Progressively harder questions (that’s normal work!)
- Varied questions (Corbett Maths and Numeracy Ninjas are both AMAZING but they are revision and varied practice, not drills)
- Questions that make you stop and go ‘hmmm’…these are part of a (nice:) normal lesson
- Take more than ~15 seconds per question…possibly even more than 5 seconds per question, although it depends. If it takes too long, urgency will be lost and it will feel flat.
This is a drill I used for ~a month last year with 7Poseidon (first quartile). Some worked well, but some were a nightmare. Writing those numbers in base 2 is not suitable as a drill unless you are the Rainman! And they inevitably dawdled when they got to x2.5 – it would have been better as part of mixed practice each day, with space for working, not as mental maths.
Rolling out: when and how
- Drills are not a teaching tool! They are for automating procedures/connections already in place.
- During a teaching sequence: to practice a specific and isolated decision (e.g. What is the LCD?)
- AFTER the content has been grasped and foundations are in place: to improve speed and accuracy (and confidence)
- To get more ‘bang for your buck’ (and check they are ready), you can complete it orally (Line 1…Sarah…Line 2…Thomas…Line 3…Abdi…etc), then in writing (In your books…go!)
- WATCH OUT: Practice makes permanent! If they are not secure with the content, they will be practising and automating getting it wrong. This is the nightmare situation! To avoid this, check the whole class on whiteboards first, possibly many times (to allow for false positives) and for the first few days you do a new drill (they’ve slept since the last one so may well have forgotten………)
- Narrate the why (To build up our confidence, To improve your accuracy, So we can test ourselves and push ourselves to improve, To see how much we can improve as a team)
This may be hard to understand if you weren’t there! You’ll have to visit our school to see the kids in action…
- Make it quick and short: race per column, or even per 10 (think: spinning! It’s unbearable to push for 3 minutes, but manageable if the trainer breaks it into 30second bursts)
- Raise the tension: music and timers (Youtube’s ‘tension music’ is surprisingly good!)
- Raise the stakes: Check for cheaters! Just before it starts, they ‘check for cheaters’ (peering at each others’ like little meerkats; if a cheater is ‘caught’ writing before the ‘go!’ they have their hands up (as if they were a robber in a cartoon(!)) for 5 seconds before they get to join in)
- Feel like a team: All have pens poised, the teacher calls “Ready!” [everyone bangs the table with their other hand] “Get set!” [two bangs] “Go!” [everyone writes furiosly, and there is a crisp start to raise excitement]
- Celebrate together:
- Mexican Wave (“Mexican wave if you got…10 or more! 15 or more!”;
- “Show me one hand if you managed ______! Show me both hands if you managed _______, wave your hands like crazy if you managed _____!”;
- “If you managed to do [really difficult thing], then 3,2,1….” [pupils who succeeded go ‘yussssss’ and a fist pump together] [less mad than it sounds, they seem to love it]
- Celebrate individuals: top rockstar wears ‘rock glasses’, Queen of Quadratics gets a crown
- Make improvement visible: tick (Mark column 1), target (write a target for the next one), repeat (do column 2…possibly give them more time, but don’t tell them!), improve (give yourself a pat on the back if you improved).
- Patterns in the answers: If you are a masochist with time to spare, pupils LOVE if there is a pattern in the answers (e.g. Fibonacci fractions)
- Use Excel for speed/fluency questions
- Even if written ‘by hand’ on your computer, use Excel to shuffle them and reuse day-by-day
- Think about what the pupils will need to produce…this affects spacing
- Make sure rows and columns have names!
- Write on or read from? A drill sheet can be reused many times if they have to write in their books, using a ruler to go down the rows and keep track.
- Make it re-usable (lists of numbers, some 1-20, some integers, some decimals, some fractions), then give different instructions, depending on the class
- Add 1
- add 10
- write as words
- round to 2s.f.
- find lower bound if they were rounded to the nearest 10
- x or ÷ by 10
- Very mixed ability groups can lead to dead time for the quickest
- UKMT question on board
- ‘tough nut’ at end of column
- pattern in answers
- Privileges speed: it is essential that pupils understand that being fast is a route to fluency, and NOT the same thing as being good at maths. I talk about athletes doing drills for fitness and speed. Being a quick runner won’t guarantee you are a good footballer, but being slow and unfit guarantees you won’t be (i.e. the causality goes in one direction)
- Speeding leads to errors (pre-emptive crossness! Talk about how annoyed you’ll be if it is a sloppy rush, and how they won’t improve. List physical signs of care you expect to see – underlining, decimal point moved, etc)
- Inaccurate marking (focus on improvement when you narrate, do random checks, narrate ‘lying to yourself’)
- Practising making mistakes: as discussed, you MUST check they are competent before they do a drill for speed/accuracy
- No deep thought – that’s the point! Deep thought is for the rest of the lesson.
- Workload and proliferation of paper: use Excel, reuse sheets
- Get in touch with what works and what doesn’t!
- Obviously photos and videos of kids feeling proud will make my day… 🙂
- Easiest way is @danicquinn
- Or come say hi and see our kids! We’re in Wembley: MCS Brent
And to make use of Hin-Tai’s super drills for Y7, go to http://tes.com/teaching-resource/maths-drills-generator-rounding-halving-doubling-multiplying-by-powers-of-10-11536042