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Interleaving. An introduction to the research. The effort to improve mathematics learning has primarily focused on how the material is taught rather than the practice students do and how they go about doing it.
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An introduction to the research • The effort to improve mathematics learning has primarily focused on how the material is taught rather than the practice students do and how they go about doing it. • As we know, students learn mathematics by doing, that is tackling problems rather than by reading examples so looking into the practice they do could be a worthwhile investigation. • Two experiments were conducted to investigate: • The temporal dispersion of practice (length of time between practice sessions) • The order in which the problems are solved
“Standard” textbooks – Massed and blocked practice • In most Mathematics textbooks, examples on a particular concept are followed by similar questions. In addition, problems of the same type are usually in blocks. • For example after some examples on adding and subtracting fractions the next exercise of the textbook is on adding and subtracting fractions. All of the adding fractions questions are grouped together followed by all of the subtraction questions (or the same denominator questions are together then questions which involve changing one denominator and then questions which involve changing both denominators). • Massed practice is when all of the questions following an example appear on the following exercise rather than being spaced across multiple exercises. • Blocked practice is when an exercise has questions on just one topic rather than a mix of questions from multiple topics.
“Shuffled” textbooks – Mixed and spaced practice • Only a small number of textbooks use a shuffled format. • Could have identical examples to a “standard” textbook and the number of exercises and questions per exercise could also be identical. • But the practice questions in a “shuffled” textbook are systematically arranged so that they are spaced and mixed. • For example, after some examples on using the quadratic formula the following exercise has a few quadratic formula questions but also some questions on completing the square and factorising quadratics. More quadratic formula questions are then spread across other exercises (the total number of quadratic formula questions remains unchanged though).
Interleaving seems to be two things • Spaced Practice • Temporal dispersion • Mixed practice • Lots of topics taught and questions are distributed randomly
Oh no – that sounds like the opposite of what I have planned this year • This just seems counter to everything that has been said about teaching Maths, how textbooks are written and how I wrote the scheme of work. • I bet the research on interleaving was a small study – pseudo science. • It will have been for some other type of subject not at all like Maths.
What is it that we were set up to do • Heavy blocked mass practice with overlearning • All practice takes place following the lesson on the topic • Questions follow a pattern: questions 1-5 are all complete the square for a=1, questions 6-10 a>1, questions 10-15 a-1 • Overlearning - students continue to do questions of the same type even after they have answered 2 questions correctly – this only has a marginal benefit over correctly answering two questions of the same type.
Experiment 1 • Investigated the effects of temporal distribution (massed vs spaced practice) and overlearning (massed practice vs light massed practice). • College students were taught how to calculate the number of permutations of a letter sequence with at least one repeated letter (for example aaabcc).
What does the research suggest?Spaced Practice Spaced Practice Massed Practice Students had a tutorial and did practice questions in week 1 In week 2 they were tested There were 2 of these groups. One did 2 questions (light practice) and the other 4 questions (overlearning) • Students had a tutorial and in week one did 2 questions • In week 2 the students did 2 further questions • In week 3 the students were tested
What does the research suggest?Spaced Practice • Spacers gained 74% in the experiment compared to 49% for the massed practice students – those who only had light massed practice (half the practice questions) scored 46%
Adjustment needed? • I don’t want to have to reteach every technique and/or fact. It might be okay to have a vague recollection and then reinforce it but I don’t have time to teach the syllabus twice. • Some forgetting is fine but completely forgetting – forget that. • The retention model suggests that after 3 days they have forgotten 40% of what they have learned. • I looked into these memory retention models online – there are a few and some that seem to fit this graph. • The problem is that they have a “strength of memory” variable which is individualised and could also be dependent on the learning method. • I took this graph and used it as a basis to calculate the numbers for the model and then thought this is not that practical as I am trying to be precise with something that isn’t. • I would suggest spacing the practice at a couple of days then a week then the “magic” 4 weeks and then approximately every 4 weeksthereafter.
Experiment 2 • Investigated the effects of blocked practice vs mixed practice. • College students were taught how to find the volume of four obscure geometric solids (wedge, spheroid, spherical cone and half cone) given the formulae for each shape.
What does the research suggest?Mixed practice – the experiment Blocked Mixed All 4 tutorials were delivered at the start. The students then attempted a mixture of problems in week 1 and week 2 Results were worst for this group in practice • Students had 4 tutorials (similar topics): 2 in the 1st week and 2 in the 2nd week • After each tutorial they attempted questions based on that tutorial • Results of the practice were best for this group
What does the research suggest?Mixed practice – the experiment • In week 3 they took a test and the results were dramatic. Blockers went from 89% correct answers to 20% and mixers went from 60% to 63%... • Does this sound familiar? “I/they can do it in class but not in a test/exam” • It is the style of practice that could be causing this – they can not identify which technique to apply to which question because they are not in discrete blocks. • This is the biggest headache to deal with – textbooks and websites are not set up like this. We need to create our own stuff. • We also need to ensure enough is covered in a piece of work that there is opportunity for students to identify what they need to do.
Addressing that last point (primarily for A-Level students) • Give them substantial problems to start with. • When they have solved a few and come up with techniques to use then give them a set of mixed practice questions. Homework is going to be important for enabling students to have enough time to put the techniques that have learned into practice. • This is a lot of work but if it has anywhere near the impact of the study it will be worth it.
Limitations of the research results • The experiments were done with college students so it is not known whether the results would be applicable to much younger students. • The experiments used the same type of questions in practice and in the tests so it is unclear whether similar results would be achieved if the questions required applying the same knowledge to a different type of question. • The experiments were done in a laboratory setting so future research would need to be done to see if the same effects apply inside the classroom. • The tasks done in the experiment were procedural rather than conceptual so it remains unknown whether the benefits of mixed and spaced practice would apply for more abstract questions. • To summarise, the experiments leave open the possibility that they do not generalise to different subjects, tasks or settings and yet there is no reason why they would not.
Additional advantages of the “shuffled” format • If practice questions related to a specific topic are spread out across multiple exercises then any student that fails to understand that topic (or misses the lesson) is still able to answer most of the questions within any given exercise whereas a “massed practice” style would result in those students having little success. • If these students gain a better understanding of the topic later on in the year then then a “shuffled” format will give them an opportunity to practice these skills in the future.
So how do we incorporate spaced practice into our schemes of learning? • Explicitly in homework’s and starters • Spaced at 1-2 days, 1 week then 4 weeks • Implicitly – spaced practice occurs naturally as one skill builds upon others in Maths so they will be practiced (revisited) regularly. • By looking at your schemes of work and grouping topics together which rely on similar skills and splitting these topics up, you can ensure that your schemes of learning are incorporating spaced practice. • Having regular tests which can cover any of the content on the syllabus (particularly for students retaking GCSE). • Having a revision lesson before a test (for AS / A-Level students) – exam question powerpoints are an easy way to give mixed up exam questions to students without needing to print anything out.
Practical ways of incorporating mixed practice into your lessons • Box lessons – you can cut up a bunch of exam questions on a particular topic or questions from a textbook or worksheet, mix them up and put them into a box and let the students pick the questions they do. You could also have more than 1 box so that the questions are differentiated. As a bonus, this also reduces printing. • When using a worksheet or textbook you can specify which questions you want them to do and the order they do them in. This way you can ensure that they are not “over-learning” and are doing questions which are mixed up. • Corbett Maths 5 a day is good for starters / plenaries (for GCSE - has both old and new spec): https://corbettmaths.com/5-a-day/gcse/ • Corbett Maths also has 5 a day for Core 1 on the old specification A-Level
Gap tasks • Take 2 different classes of a similar ability level and spend 1 lesson revising 2 different topics. For class 1 give them all of the questions for each topic blocked together and for class 2 do a box lesson. In the following week give them a test on just those topics to see if there is any effect on doing mixed practice vs blocked practice. • Take 2 different classes of a similar ability level and spend 1 lesson on a particular topic with the first class. With the second class spend half the time on that topic (with half as many questions), give them a starter in the following lesson and then a week later give them the second half of the questions to do in lesson (no further explanation given) or for homework. Then do a test on this topic with both classes in the following week to see the effect of spaced practice vs massed practice.
Preparing for our next event • In our next event we are going to look at best practice for developing schemes of learning including how to incorporate interleaving into our schemes of learning. • In preparation for our next event it would be useful if you could consider which topics are linked together and come up with a list of topics which could be grouped together (for example ratio can be linked with probability and similarity).
