I previously attended #mathsconf5 in Sheffield back in 2015. The impact of which has been felt within my own classroom and our wider department ever since. It was a truly transformational day in my career. With a great deal of enthusiasm (not to mention tiredness- 4 working Saturdays out of 5) I attended #mathsconf9 in Bristol yesterday.
Drill and Thrill – Dani Quinn
I would encourage all maths teachers, and heads of department in particular, to read Dani’s blog. She is from the (in)famous Michaela school in North West London - the one where a focus on knowledge, hardworking kids and teachers actually teaching is the expectation. Within the first ten seconds of hearing Dani speak it was clear that any of the negative ideas about Michaela being evil oppressors of children were clearly not true. Her enthusiasm was contagious and she smiled and laughed the whole way through the presentation. She had the room in stitches as she told anecdotes about the kids in her classes. How much she really cares about those kids and how well she knew them was evident.
The main focus of the workshop was on the idea of using drill exercises. Drills can be used to improve fluency in several aspects of mathematics:
Decision Making – Similar to something I mentioned in my own talk at Stirling last week Dani used drills to help students make decisions about the mathematics, without necessarily doing the calculations. She described using a subtraction drill with a lower ability class. The task was simply to say if borrowing was required or not. The pupils didn’t have to do the calculations; the idea was simply to develop awareness of when it is required. Another example was a list of integer calculations where pupils had to state if the answer would be positive or negative, again without actually doing the calculation. The benefits of pupils having this awareness are obvious – particularly when checking their answers.
Speed – As unfashionable as it may be for the likes of Jo Boaler, speed is a metric worthy of measure. Dani argued that due to constraints in working memory we need to have these simple skills as sharp as possible. If a pupil has to spend 20 seconds working out a multiplication within the context of another question then this is a high cognitive load, which will impact negatively on the main question being tackled. Similarly speed in multiplying out single brackets, dealing with index laws and multiplication and division of negatives should be easy goals to attain. I liked how adding and subtracting negatives weren’t on this list. Although we’d like for speed in this, I personally feel the risks of failure are too high for many pupils if we ask them to do those questions quickly.
Recognising and Deciding - For adding fractions pupils can be asked to identify if LCM is required to be calculated for the denominators. Dani talked about those pupils who end up turning 2/5 + 3/5 into 10/25 + 15/25, something I’m sure we’ve all seen!
Muscle Memory – This struck me as being another name for procedural fluency. We obviously desire this for essential skills such as completing the square, finding the gradient of a line and rationalising the denominator. A quote I liked was “Practice doesn’t make perfect. Practice makes permanent.”
Drilling is most suitable for those topics which are high leverage (vital/advantageous for later learning and attainment) such as fraction to decimal conversions, rounding, adding and subtracting fractions, multiplying and dividing by powers of 10, adding and subtracting negative numbers etc. Drilling isn’t suitable for topics that are too complex. For example, metric conversions have too many decisions and steps required so don’t lend themselves well to drilling. Things that are too simple and allow pupils to go into autopilot such as multiplying fractions or multiplying 20 values by 0.1 aren’t ideal either. Furthermore, progressively harder questions shouldn’t be used in drilling either – that is normal work! As a guide, anything that will take over 15 seconds to do isn’t good for drilling.
Dani spent significant amount of time explaining the culture in her classroom and how she explains the relevance and importance of the regular drills to her pupils. She also spent time talking about the joy factor that the drills have brought into her class. I really didn’t expect this but as she described the way she has created a culture of kids taking it seriously, aiming to do the drills quickly (but not carelessly rushing) and being slightly competitive while having fun, I could see how it would engage them. Dani uses drills to help make improvement visible to her pupils via ‘repeat, improve’.
Interestingly Dani spent time explaining that they do explain things to the pupils. I’ve heard the “just tell them” soundbite often, but suspected this was more to do with emphasising that the pedagogy isn’t inquiry based. Controversially she suggested that we should just let the kids move the decimal point when multiplying and dividing by powers of ten. We do it ourselves as teachers, so why deny the kids this knowledge? If she’d suggested that we teach kids to ‘change side, change sign’ for solving equations I might have dismissed the idea out of hand, but I now find myself wondering if she has a point. If we’ve explained that the point doesn’t really move, but we can do it as a cheat – will it have any real negative impact later? Dani actually blogged about this yesterday (link above).
Finally, Dani explained the streaming approach at Michaela. An interesting reverse weighting approach is used. The bottom class is for those who have the poorest literacy, while the top class is for those who are the best mathematicians. Pupils stay in these classes for all lessons across the school. Different pupils have different needs – Dani explained that she uses drilling with all classes but less often for fundamentals with the top class, however, with a lower group it is highly effective in developing that procedural fluency in the essentials. Music to my ears again – I don’t like the promotion of the ideology that everyone should go mixed ability. As @mrgraymath pointed out at Stirling last weekend it depends on your context. The evidence also shows this, if you read below the surface.
Most of the Things you Think are Processes Probably Aren't – Kris Boulton
Kris is somebody I’ve followed closely on Twitter and on his blog over the past couple of years. I was delighted to finally get the chance to hear him speak. I’ve been reading bits and pieces about Direct Instruction over the past year or so, but Kris is the guy who really has his finger on the pulse. Direct instruction can have many meanings, as Greg Ashman pointed out in the latest Mr Barton podcast - however Kris’ focus is on Direct Instruction as explained by Siegfried Engelmann. A disclaimer: Kris said he has found reading this work very hard going and that he didn’t know all of the answers. What I’ve written up here serves two purposes – to share the ideas from yesterday with those who couldn’t be there and also to clarify my own thoughts on what I’ve learned. I may not have this completely accurately and will be happy to correct any misunderstandings.
Kris discussed the use of examples and non-examples. I’d first encountered this idea when Bruno Reddy came to visit our department back in October. Bruno had talked about illustrating what was a difference of two squares and what was not – but showing pupils examples of each. Kris discussed how in some cases comparison is more suitable to example/non example. For instance with gradient, to say that something is a gradient or is not a gradient is a bit of a mute point. However, to compare gradients has merit– this is steep/this is shallow, this is increasing/this is decreasing.
A myth to be exposed was that Direct Instruction is about teacher talk. A message I’ve taken away is that if the board work is good enough, then very little has to be said. Both Kris and Dani were vocal about the the fact we can’t listen and read at the same time. Talking over what we are writing on the board, or speaking as pupils copy notes down is ineffective. Giving pupils time to read before explaining is important. Kris demonstrated a lesson sequence where the only thing said was repetition of the question wording.
Kris explained the difference between a process and a transformation. Multiplying out a single bracket is a transformation while multiplying two binomials is a process. My understanding is that if the vast majority of pupils can infer the knowledge by simply seeing some questions and associated answers (no working) then it is a transformation. If the majority of learners would struggle to see the relationship between the question and answer (again with no working shown) then it is a process.
I’ll now attempt to do justice to Kris’ explanation.
The format is as follows:
Drill and Thrill – Dani Quinn
I would encourage all maths teachers, and heads of department in particular, to read Dani’s blog. She is from the (in)famous Michaela school in North West London - the one where a focus on knowledge, hardworking kids and teachers actually teaching is the expectation. Within the first ten seconds of hearing Dani speak it was clear that any of the negative ideas about Michaela being evil oppressors of children were clearly not true. Her enthusiasm was contagious and she smiled and laughed the whole way through the presentation. She had the room in stitches as she told anecdotes about the kids in her classes. How much she really cares about those kids and how well she knew them was evident.
The main focus of the workshop was on the idea of using drill exercises. Drills can be used to improve fluency in several aspects of mathematics:
Decision Making – Similar to something I mentioned in my own talk at Stirling last week Dani used drills to help students make decisions about the mathematics, without necessarily doing the calculations. She described using a subtraction drill with a lower ability class. The task was simply to say if borrowing was required or not. The pupils didn’t have to do the calculations; the idea was simply to develop awareness of when it is required. Another example was a list of integer calculations where pupils had to state if the answer would be positive or negative, again without actually doing the calculation. The benefits of pupils having this awareness are obvious – particularly when checking their answers.
Speed – As unfashionable as it may be for the likes of Jo Boaler, speed is a metric worthy of measure. Dani argued that due to constraints in working memory we need to have these simple skills as sharp as possible. If a pupil has to spend 20 seconds working out a multiplication within the context of another question then this is a high cognitive load, which will impact negatively on the main question being tackled. Similarly speed in multiplying out single brackets, dealing with index laws and multiplication and division of negatives should be easy goals to attain. I liked how adding and subtracting negatives weren’t on this list. Although we’d like for speed in this, I personally feel the risks of failure are too high for many pupils if we ask them to do those questions quickly.
Recognising and Deciding - For adding fractions pupils can be asked to identify if LCM is required to be calculated for the denominators. Dani talked about those pupils who end up turning 2/5 + 3/5 into 10/25 + 15/25, something I’m sure we’ve all seen!
Muscle Memory – This struck me as being another name for procedural fluency. We obviously desire this for essential skills such as completing the square, finding the gradient of a line and rationalising the denominator. A quote I liked was “Practice doesn’t make perfect. Practice makes permanent.”
Drilling is most suitable for those topics which are high leverage (vital/advantageous for later learning and attainment) such as fraction to decimal conversions, rounding, adding and subtracting fractions, multiplying and dividing by powers of 10, adding and subtracting negative numbers etc. Drilling isn’t suitable for topics that are too complex. For example, metric conversions have too many decisions and steps required so don’t lend themselves well to drilling. Things that are too simple and allow pupils to go into autopilot such as multiplying fractions or multiplying 20 values by 0.1 aren’t ideal either. Furthermore, progressively harder questions shouldn’t be used in drilling either – that is normal work! As a guide, anything that will take over 15 seconds to do isn’t good for drilling.
Dani spent significant amount of time explaining the culture in her classroom and how she explains the relevance and importance of the regular drills to her pupils. She also spent time talking about the joy factor that the drills have brought into her class. I really didn’t expect this but as she described the way she has created a culture of kids taking it seriously, aiming to do the drills quickly (but not carelessly rushing) and being slightly competitive while having fun, I could see how it would engage them. Dani uses drills to help make improvement visible to her pupils via ‘repeat, improve’.
Interestingly Dani spent time explaining that they do explain things to the pupils. I’ve heard the “just tell them” soundbite often, but suspected this was more to do with emphasising that the pedagogy isn’t inquiry based. Controversially she suggested that we should just let the kids move the decimal point when multiplying and dividing by powers of ten. We do it ourselves as teachers, so why deny the kids this knowledge? If she’d suggested that we teach kids to ‘change side, change sign’ for solving equations I might have dismissed the idea out of hand, but I now find myself wondering if she has a point. If we’ve explained that the point doesn’t really move, but we can do it as a cheat – will it have any real negative impact later? Dani actually blogged about this yesterday (link above).
Finally, Dani explained the streaming approach at Michaela. An interesting reverse weighting approach is used. The bottom class is for those who have the poorest literacy, while the top class is for those who are the best mathematicians. Pupils stay in these classes for all lessons across the school. Different pupils have different needs – Dani explained that she uses drilling with all classes but less often for fundamentals with the top class, however, with a lower group it is highly effective in developing that procedural fluency in the essentials. Music to my ears again – I don’t like the promotion of the ideology that everyone should go mixed ability. As @mrgraymath pointed out at Stirling last weekend it depends on your context. The evidence also shows this, if you read below the surface.
Most of the Things you Think are Processes Probably Aren't – Kris Boulton
Kris is somebody I’ve followed closely on Twitter and on his blog over the past couple of years. I was delighted to finally get the chance to hear him speak. I’ve been reading bits and pieces about Direct Instruction over the past year or so, but Kris is the guy who really has his finger on the pulse. Direct instruction can have many meanings, as Greg Ashman pointed out in the latest Mr Barton podcast - however Kris’ focus is on Direct Instruction as explained by Siegfried Engelmann. A disclaimer: Kris said he has found reading this work very hard going and that he didn’t know all of the answers. What I’ve written up here serves two purposes – to share the ideas from yesterday with those who couldn’t be there and also to clarify my own thoughts on what I’ve learned. I may not have this completely accurately and will be happy to correct any misunderstandings.
Kris discussed the use of examples and non-examples. I’d first encountered this idea when Bruno Reddy came to visit our department back in October. Bruno had talked about illustrating what was a difference of two squares and what was not – but showing pupils examples of each. Kris discussed how in some cases comparison is more suitable to example/non example. For instance with gradient, to say that something is a gradient or is not a gradient is a bit of a mute point. However, to compare gradients has merit– this is steep/this is shallow, this is increasing/this is decreasing.
A myth to be exposed was that Direct Instruction is about teacher talk. A message I’ve taken away is that if the board work is good enough, then very little has to be said. Both Kris and Dani were vocal about the the fact we can’t listen and read at the same time. Talking over what we are writing on the board, or speaking as pupils copy notes down is ineffective. Giving pupils time to read before explaining is important. Kris demonstrated a lesson sequence where the only thing said was repetition of the question wording.
Kris explained the difference between a process and a transformation. Multiplying out a single bracket is a transformation while multiplying two binomials is a process. My understanding is that if the vast majority of pupils can infer the knowledge by simply seeing some questions and associated answers (no working) then it is a transformation. If the majority of learners would struggle to see the relationship between the question and answer (again with no working shown) then it is a process.
I’ll now attempt to do justice to Kris’ explanation.
The format is as follows:
- Initial instruction
- Initial Assessment
- Expansion Sequence
During all of this, there is minimal/no teacher talk. At the end of the exposition the teacher will ask for any questions from the class. Next follows a short assessment sequence of questions, posted to check if pupils have gained the correct inference. If there are errors at this point (which from the sounds of it is actually unlikely) then these can be rectified.
Kris may then extend this knowledge further by posing the following as questions.
Further down the line, having taught pupils the distribution property for two binomials
(a + b)(c + d) = a(c + d) + b(c + d), which is also a transformation and can be taught this way, he may pull it all together by teaching a process. This would then be done as shown in the following slide:
I really liked how the vocabulary is precise. I liked how it pieced together with the previous learning being so explicit. The pupils would be very aware of how they were using the previous knowledge.
Now all of the above may sound a little bit like discovery/inquiry based learning but, as Kris explained to me after the presentation, it really isn’t. The examples have been so carefully chosen that it is very difficult to infer anything else. The difference with discovery-based learning is that it is too wide open in scope. It is possible for pupils for infer all sorts of incorrect ideas, which would then need to be explicitly unpicked in any case.
Kris discussed ‘the best lessons’ he’d ever taught, which were using this method for simultaneous equations. He had very low ability pupils solving by elimination with real fluency. Interestingly he had broken down simultaneous equations by elimination into, I think, 13 different transformations/processes. Kris showed several slides based upon just one of these elements – which was adding two equations together. This was not for the purpose of elimination, but as a skill in its own right. I spoke last week about breaking things down and then breaking them down again. Kris is a master of this. You’ll see how clever the examples below are and how the final examples lead to elimination of a variable:
Kris also showed some examples for calculating area of rectangles and then eventually area of a triangle.
Kris also showed some examples for calculating area of rectangles and then eventually area of a triangle.
A final important point to make is that, contrary to the beliefs about Direct Instruction being about the teacher lecturing and pupils being passive, the duty is on the pupil to be engaged and thinking. The cognitive demands are the on the pupil too, however, because the lessons are broken down so much there is no risk of cognitive overload. The teacher isn’t going to explain a process/transformation, until the pupil has considered it.
This all correlates with the idea of learning as being the residue of thought. As pupils really are being active then there is much more ‘active learning’ than when copying down notes and having a teacher do all the explaining and talking.
From a personal perspective I intend to reflect about where this pedagogy can appropriately be incorporated into my own teaching. I also need to be sure how it will fit with the existing pedagogies we are trailing such as visual representations for algebra. Rigorous lesson planning will be required. This isn’t something you can do in two minutes before the end of lunchbreak. Kris suggested that index laws, adding fractions with like denominators, multiplying fractions, simplifying by collecting like terms, angles on a straight line, adding and subtracting negatives etc could be taught this way.
Final Thoughts
Once again I found the conference to be thought provoking and challenging. I would encourage all of my colleagues in Scotland to support this event when it makes its Scottish debut in Dunfermline in August. I think there is a lot to be gained from cross-fertilising ideas from both sides of the border. At under £30 it is a bargain and a great addition to our conference calendar. Details can be found here: https://completemaths.com/events/scotland
Once again I found the conference to be thought provoking and challenging. I would encourage all of my colleagues in Scotland to support this event when it makes its Scottish debut in Dunfermline in August. I think there is a lot to be gained from cross-fertilising ideas from both sides of the border. At under £30 it is a bargain and a great addition to our conference calendar. Details can be found here: https://completemaths.com/events/scotland