Tag Archives: spacing

3 Teaching Techniques That Made My 2014

At the start of this academic year I wanted to really put the theory of learning I knew into practice. Here are three teaching techniques I tried that I’ll be taking with me into 2015.

1. Interleave *everything*
We know that interleaving concepts and procedures is a desirable difficulty that improves learning. This year I’ve made it my aim to never teach a lesson that uses only one topic. Extension work doesn’t count: the main bulk of a lesson must include multiple concepts. In practice, this means that every set of questions I write includes applying the new thing being learned to problems involving other concepts from earlier on in the curriculum. I found this daunting at first – “what if it’s just too confusing for them?” – but I’ve found there’s tremendous power in my expectation that they can competently use everything I have taught them, at any time.

It’s also made the questions I write far richer and more interesting than ever before. A standard lesson on volume would include questions on upper and lower bounds, percentage changes to a volume when dimensions change by different percentages, and using Pythagoras’ Theorem to find missing dimensions before calculating a volume. These questions were the normal ones. They were not special extension questions for the top 20%. The message this broadcast, combined with the cognitive effect of the desirable difficulty, has made a noticeable impact on learning.

2. Spaced testing of *everything*
We know that spacing and testing are two of the most powerful tools in an educators’ arsenal. So I’ve made it my aim to space out testing of everything my students’ have studied.

The first way I’ve done this is through our departmental testing. This year we’re using low-stakes quizzes on each topic as a replacement for the half-termly summative tests we used to use. What’s great about this is that each topic has two quizzes: one taken at the end of the topic, and one a month later. This reinforces the expectation that students should remember what they’ve been taught, and incentivises them to revisit their learning when it will have the biggest impact on longer-term retention.

The second way is through an idea I’ve borrowed from Bruno Reddy and Kris Boulton at KSA. Long ago they told me about “Only 100% Will Do” starters: simple recall or procedural questions that had been previously studied. Now my students begin every lesson with an “Only 100% Will Do” sheet where they work on concepts from any past year group to make sure that they’re keeping up to speed.

3. Lightning fast in-class assessment
I have always been a big fan of in-class assessment, and could singlehandedly prop up the UK’s mini-whiteboard industry. But over the summer Joe Kirby introduced me to an app called Quick Key. It changed my life!

Quick Key is an optical scanning app for mobile phones. It works by scanning in students’ answer sheets to a multiple choice quiz, and giving you great analytics on your phone or in a spreadsheet. It’s also fast – I can scan a whole class set of answers in about a minute. This lets me pinpoint exactly how well the class are doing at a topic. I will discover that one mistake or misconception is bogging down the whole room, or that four students need extra support in this lesson and another five need extra challenge. The laser-like precision with which I can adapt during a lesson or plan the next one is having a big impact on my teaching.


 
Like any technique, these three have worked well because I made them a habit. We do these three things in the same way in almost every lesson. This consistency no doubt aids their effectiveness. But they are nonetheless three real techniques that put into practice the theory of learning we know to be so powerful.

Why homework is bad for you

Laura McInerny’s third touchpaper problem is:

“If you want a student to remember 20 chunks of knowledge from one lesson to the next, what is the most effective homework to set?”

After a day of research at the problem-solving party, I came to this worrying conclusion:

Setting homework to remember knowledge from one lesson to the next could actually be bad for their memory.

So stop setting homework on what you did in that lesson – at least until you’ve read this post.

Components of Memory

Bjork says that memories have two characteristics – their storage strength and their retrieval strength. Storage strength describes how well embedded a piece of information is in the long-term memory, while retrieval strength describes how easily it can be accessed and brought into the working memory. The most remarkable implication of Bjork’s research surrounds how storage strength is built.


Storage and Retrieval strength – courtesy of Kris Boulton


Retrieval as a ‘memory modifier’

Good teaching of a piece of information can get it into the top left hand quadrant, where retrieval strength is high but storage strength is low. Once a chunk of knowledge is known (in the high retrieval sense of knowing), its storage strength is not developed by thinking on it further. Rather storage strength is enhanced by the act of retrieving that chunk from the long-term memory. This is really important. Extra studying doesn’t improve retention. Memory is improved by the act of retrieval.

The ‘Spacing Effect’

Recalling a chunk of knowledge from the long-term memory strengthens its storage strength. However for this to be effective, the chunk’s retrieval strength must have diminished. ‘Recalling’ a chunk ten minutes after you’ve studied isn’t going to be very effective, as your brain doesn’t have to search around for such a recent memory. Only when a memory’s retrieval strength is low will the act of recall increase storage strength. This gives rise to the spacing effect – the well-established phenomenon that distributing practice across time builds stronger memories than massing practice together. 

Rohrer & Taylor (2006) go a step further and compare overlearning (additional practice at the time of first learning) with distributed practice. They find no effect of over learning, and ‘extremely large’ effects of distributed practice on future retention.

Optimal intervals

There is an optimal point for recalling a memory, in order to maximise its storage strength. At this point, the memory’s retrieval strength has dropped enough for the act of retrieval to significantly increase storage strength, but not so much to prevent it from being accurately recalled. Choosing the correct point can improve future recall by up to 150% (Cepeda, et al., 2009).

There has been a common design of most studies into optimal spacing. Subjects learn a set of information at a first study session. There is then a gap before a second study session where they retrieve learned information. Before a final test there is a retrieval interval (RI) of a fixed time period. Studies such as Cepeda, et al (2008) show that the optimal gap is a function of the length of the RI, and that longer RIs demand longer gaps between study periods. However this function is not a linear one – shorter RIs have optimal gaps of 20-40%, whereas longer RIs have optimal gaps of 5-10%.

Better too long than not long enough

Cepeda et al’s 2008 study looks at four RIs: 7, 35, 70, and 350 days. The optimal gaps for maximising future recall were 1, 11, 21 and 21 days respectively, and these gaps improved recall by 10%, 59%, 111% and 77%.



Perhaps their most important finding is the shape of the curves relating the gap to the future retention. For all RIs these curves begin climbing steeply, reach a maximum, and then decline very slowly or plateau. The implication is that when setting a gap between study periods it is better to err on the side of making it too long than risk making it too short. Too long an interval will have only small negative effects. Too short an interval is catastrophic for storage strength.

Why homework could be bad

Homework is usually set as a continuation of classwork, where students complete exercises that evening on what they learned in school that day. This constitutes a short gap between study sessions of less than a day. We know that where information is to be retained for a week, the optimal gap is a day, and that where this is not possible it is better to leave a longer gap than a shorter one. For longer RIs, the sort of periods we want students to remember knowledge for, the optimal gap can be longer than a week.

Therefore, if you want students to remember information twenty chunks of knowledge for longer than just one lesson to the next, the best homework to set is no homework!

Setting homework prematurely actually harms the storage strength of the information learned that day by stopping students reaching the optimal retrieval interval. In this case, students who don’t do their homework are better off than ones who do!

Why I might be wrong, and what we need to do next

There is not enough good evidence of how to stagger multiple study sessions with multiple gaps. For example, we do not know where it would be best to place a third study session, only a second. However we do know that retrieval is a memory modifier, and so additional retrieval should strengthen memories as long as the gap is sufficiently large for retrieval strength to have diminished. Given we know that retrieving newly learned information after a gap of one day is good for storage strength, it may be that studying with gaps of say 1, 3, 10 and 21 days are better for storage strength than a solitary study session after 21 days, where the RI is long (350 days or greater). In this case for teachers who only have one or two lessons a week, homework could help them make up the optimal gaps by providing for study sessions between lessons.

The optimal arrangement of multiple gaps is a priority for research. We need to better understand how these should be staged, so that we can begin to set homework schedules that support memory rather than undermine it. Until then, only set homework on previously learned knowledge, and better to err on the side of longer delays. My students will be getting homework on old topics only from now on.

Bibliography

Joe Kirby on memory this weekend
EEF Neuroscience Literature Review
Dunlosky, et al., 2013. Improving Students’ Learning With Effective Learning Techniques: Promising Directions From Cognitive and Educational Psychology
Rohrer & Taylor, 2006. The Effects of Overlearning and Distributed Practise on the Retention of Mathematics Knowledge
Cepeda, et al., 2009. Optimizing Distributed Practice
Cepeda, et al., 2008. Spacing Effects in Learning: A Temporal Ridgeline of Optimal Retention
Everything Kris Boulton writes

A Curriculum That Works

This is the second of three posts reflecting on my first term as Curriculum Lead for Maths. The last post, on our new post-levels assessment system, can be found here.

A New Key Stage 3 Curriculum

The curriculum is so much more than a statement of what is to be taught and when. It embodies a school’s vision for its students and its philosophy of learning. I can look at a school’s mathematics curriculum and tell you all about the person who wrote it – their expectations of students, their hopes for their futures, their beliefs about how to get there. The curriculum is the embodiment of all these things, and it is crucial to get it right.

To begin writing a curriculum you must begin from a vision of the mathematicians you want your students to become. Mine is that I want our students to become “knowledgeable problem-solvers who relish the challenge Mathematics offers”. They should, at the end of their time with us, be able to independently tackle an unfamiliar mathematical problem and create a meaningful solution to it.

I am mindful when considering this vision, of the roaring debate around discovery and project-based learning, and how it can fall foul of WIllingham’s novice-expert distinction. My view is this:

Knowing that students begin novices, the purpose of education is to make them into experts.

The curriculum needs to train students. We cannot assume expert qualities of them from the start, and plunge them into investigations where most or many students will fail to learn. Similarly, we cannot dogmatically write off any activity involving discovery, investigation or project work. I emphatically want my students to be capable of expert investigation when they leave school, and so our curriculum must explicitly prepare them for that. The last part of this post in particular looks at how we manage this in a practical way.

From the vision of how students should leave school, I drew up three design principles:

1) The curriculum must develop fluency.
2) The curriculum must develop conceptual understanding.
3) The curriculum must teach students to solve problems.

Principle 1: Developing Fluency

When I wrote the last iteration of our school’s KS3 Maths curriculum, I abandoned the traditional spiral structure and opted for a depth before breadth approach. We probably halved the amount of content covered in a year, as we wanted to give students the time they needed to develop fluency. This year we’ve cut it again. Each of the six terms covers a maximum of three ‘topics’, most of which are closely linked. Terms in Year 7, for example, look like this:

  1. Mental addition and subtraction; Decimal addition and subtraction; Rounding
  2. Mental multiplication and division; Decimal multiplication and addition; Factors and multiples
  3. Understanding fractions; Operations with fractions
  4. Generalising with algebra (expressions and functions only)
  5. Properties of 2D shapes; Angle rules
  6. Equivalence between fractions, decimals and percentages

Smaller concepts that tie closely with big ones above are taught alongside them. For example, perimeter is taught in Term 1 alongside addition, area and the mean average are both taught in Term 2 alongside multiplication and division.

Since we give so much time to teaching each mathematical skill, we expect a high degree of fluency. To take the National Curriculum’s definition, fluency is students’ ability to “recall and apply their knowledge rapidly and accurately to problems“. It means not just being able to do something, but being able to reliably do it well and quickly. I would add that a necessary condition for fluency in a skill or operation is that it is embedded in your long-term memory.

This is exceptionally valuable in mathematics. A student may learn to be able to multiply decimals, but not become fluent in it. When multiplying decimals they have to slow down, to stop and think, and may make mistakes. This means that in Term 4 when they are learning to substitute into formulae with decimal numbers, they will face two severe problems. Firstly, their working memory will be occupied thinking about multiplying decimal numbers together, and not about substituting into formulae. Secondly, their reduced pace will mean that they have less exposure to substituting into formulae in each lesson. Overall they will spend less time thinking about the new concept they are supposed to be learning, and will learn it less well as a consequence (after all, “memory is the residue of thought”).

Developing fluency then, means lots of practice time with well thought out problems. Practice has got a bad reputation in mathematics, with too many people having been turned off maths by pages of repetitive textbook questions. My response is that practice need not mean making maths dry or uninspiring. Our students appreciate the value of practice as something that gives them the skills to do fun maths, and to achieve things they are proud of. Practice is invaluable, but can be dangerous if not used alongside meaningful and motivating problems.

If fluency is about rapid and accurate recall of knowledge, then Kris Boulton will tell you that fluency depends on high storage strength and high retrieval strength. A depth-focused curriculum gives us storage strength, but could easily sabotage retrieval strength if knowledge is not revisited. This is probably our biggest area to work on. The curriculum includes notes about what content to revisit when (thanks Kris!) and our assessments presume previous content as mastered prior knowledge. However we haven’t yet found a more structured way of revisiting content consistently across classes.

Principle 2: Conceptual Understanding

It is not enough for a curriculum to say what to teach. A meaningful curriculum also says how to teach it. At WA we’re big believers in the Singaporean approach of concrete-pictorial-abstract (CPA), and use this to structure our teaching. One of the reasons mathematical understanding in Britain is historically so poor is because students have been immediately confronted with abstract representations, representations that are well separated from any concrete reality, and not been given enough support in understanding them. 

A favourite example of mine is ratio. I meet strikingly few students who can answer a question of the following type correctly:

“Bill and Ben share sunflower seeds in the ratio 3:2. If Ben has 20 sunflower seeds, how many does Bill have?”

I’d love to do some research and rely on less anecdotal evidence, but I’d guess that more British 16 year olds would say 12 than would say the correct answer of 30. Why? Because they were taught ratio in a completely abstract way, where they learned to apply a method but didn’t every receive the support needed to understand the concept of ratio.

In our curriculum, however, the pictorial bar model is central to teaching ratio. In fact I don’t teach my students an abstract method (they’re perfectly capable of coming up with it for themselves by doing the bars mentally and writing down calculations). For the unfamiliar, a bar model to represent the above problem would look like this:



Students draw the ratio, label what they know, work out the size of each block and then the size of Bill’s bar. I am yet to find a student who doesn’t understand this method, and who can’t do considerably harder problems using it. This is the benefit of having a pictorial representation to help students understand the concept they are learning, and to soften the jump into pure abstract. Every topic in our curriculum comes with CPA guidance to develop strong conceptual understanding in all students.

Also key to developing conceptual understanding are links between areas of mathematics. I am eternally frustrated by how students see maths as broken down into small discrete chunks that have little or no relationship with one another. Even when we have topics that are just different representations of identical concepts (sequences and linear graphs, for example), few British students will ever see them as linked. At the core of our curriculum then is a sequence carefully designed to make every concept learned useful to a later one. More than this, it guides teachers to make links, and uses assessment to make sure students are comfortable making these.

Principle 3: Problem-solving

Mathematics is essentially the study of problem-solving. The process of mathematical abstraction has been followed for millennia because it is so useful for generalising and solving what the National Curriculum calls “some of history’s most intriguing problems”. If our students are to become the experts we want them to be when they leave, we need to train them in problem-solving now.

For me, problem-solving is a skill to be taught, and it should be taught like any other. Adept problem-solvers have not come to be so through innate talent, but because they have seen the solutions to many problems before and are able to spot similarities and apply familiar techniques. Our curriculum aims to teach students the most powerful problem-solving techniques by exposing them to a carefully selected sequence of problems, some of which are taught and some of which are independently worked on.

Each term has a problem-solving focus. For example, Term 1 was “Working systematically”. Students began with a problem where they had to work out how many different possible orders there were for a two course and then a three course set menu at a restaurant. They began using ordered lists to write out combinations, before speculating on general rules and checking them on new possibilities. Through a range of different problems in the term students learned (a) how to work systematically in different contexts, and (b) the value of doing so. 

Conclusion

Our curriculum has definitely met the three design principles set out, and is working well for our students. Depth before breadth has given them time to become fluent, to develop conceptual understanding and to solve problems. They see the value in mathematics as they’re exposed to interesting and meaningful problems. However this is done in a deliberate and structured way to make sure they are learning throughout. By applying the concrete-pictorial-abstract principle throughout we make sure that all students can interact with the concepts they’re learning and develop their understanding to a deeper level.

For me, we have two key things to work on after Christmas. Firstly, the revisiting of prior knowledge. We need to keep retrieval strength high, and must find a more structured way of doing this. Secondly, developing the guidance we give for teaching, particularly around drawing links between areas of maths. Although this happens well it is not yet a big enough part of our formal curriculum documents, which risks it slipping away in future.