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Alternating worked solutions.

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A while ago I came across this lovely little graphic taken from Learning About Learning: What Every New Teacher Needs to Know.

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Amidst the recommendations was one that caught me a little by surprise. That honour went to number four, ‘repeatedly alternating solved and unsolved problems’. Given that there was presumably quite a large variety of strategies that could have been chosen to form the ‘Fundamental Instruction Strategies’, it seemed somewhat unexpected that it had cemented a place in a selectively chosen top six. Particularly so given that I had never encountered it before. Even more so given that it featured in a text helpfully titled ‘What Every New Teacher Needs to Know’.

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The text largely focuses on the instructional design of textbooks and the mysterious lack of adherence to these recommendations. Let’s face it, for every twenty textbook questions, you might get a couple of examples in a blue box with three partially answered (but often in a totally different format) introductory questions for students to complete. Personally, I have never been fully convinced by the examples in Maths textbooks. Very rarely have I seen a student refer back to the example as a way of plugging a gap in their understanding. It certainly pales into insignificance when you compare it to how often I’ve seen a student copy out the example for the simple reason that they thought it was question one. But I am prepared to be convinced otherwise.

An excellent source of some of the research on alternating worked solutions is provided in the remarkably similar Organizing instruction and study to improve student learning: A practice guide, with the level of evidence described as ‘moderate’ (only asking deep explanatory questions and using quizzes to re-expose students to key content were found to have a ‘strong’ level of evidence).

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The ‘potential roadblocks’ section acknowledges some likely concerns teachers might have. Initially, I was concerned that it would reduce the level of thinking that was required, that it would simply result in remembering and regurgitating with little understanding of the process. However, alternating worked solutions isn’t the end game. It is more of an intermediate stage between teacher instruction and independent practice – guided practice. It is perhaps best thought of as the we do part of I do, we do, you do. The number of examples or the number of steps provided in the examples can easily be decreased (and should be) as students become more proficient. Indeed, much research unsurprisingly indicates that ‘gradually fading worked solution steps leads better learning outcomes than example-problem pairs’ (Renkl, 2011).

I have been testing alternating worked solutions in my own classroom for a little while. Initially, it began by taking a few worksheets and answering every other question myself. Here’s an example with some of my neatest handwriting ever that makes for an interesting comparison:

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And then…predictably…a blank one.

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To be clear, you still have to teach this stuff first. This isn’t about replacing teaching with sufficiently high quality examples that students can instead use to teach themselves. Often, teaching of such content can involve a lengthy teacher explanation followed by independent student practice without a sufficient bridge inbetween. This particular example, however, lends itself rather nicely to worked examples given the number of steps involved to reach the solution. It reduces the likelihood that students will encounter total overload once the teacher has finished demonstrating the task and they are faced with a set of problems of their own. Staring at a blank page with a vague recollection that there about seventeen lines of working out neccessary to reach the solution and not being able to recall the first is less than ideal.

It came as no surprise that the worked examples were studied closely by my students to aid with the problem that followed. One important thing I have definitely noticed: the quality and presentation of student work became a far closer approximation to my own as a result of the studying of the worked examples. Since the gap between teacher instruction and independent practice has been reduced and the format of the student response has been standardised, there is less opportunity for the student’s own work to bear little (sometimes zero) resemblance to that which was presented by the teacher. We’ve all been there…

However, in this particular example, I found that it would have been more productive to gradually fade the steps in the example, slowly increasing the demands for the student until the concept was thoroughly understood rather than simply alternating completed and unsolved problems. The amount of work required in the question required a lot of flicking back to the example, whereas if the question was to fill in a missing line of the working out it may have lead to a lower cognitive load at the beginning (and hence, less flicking) and more in-depth study of the solution in order to work out the missing line of working out.

The other ‘potential roadblock’ identified was that these sorts of teaching resources simply don’t exist. Obviously I wasn’t going to be writing too many solutions like that in my best handwriting. That would require me to work almost has hard as my students. Fortunately (and I suspect that the amount of space alternating worked examples take up is part of the reason they are so unpopular), such resources can easily be made by recycling previous students’ work. Rather than time consumingly creating these resources, inserting exemplar work of previous students is far quicker and provides the additional motivation of seeming achievable (the Goldilocks principle again). Admittedly, I benefit from the slight advantage of being able to disseminate resources to students digitally which means there is no requirement to limit the number of pages. Here is a copy of a student answer, which will now double as a worked solution the next time that particular set of questions is required:

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Another victory for laziness! Points to that student for following the example so closely, too. If the examples need gradually fading it’ll come in handy again. Throw away student work at your peril…

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Often, the gap between teacher instruction and independent practice is too large. This is a trap I’ve fallen into myself plenty of times. A combination of whole class alternating (teacher example, students answer problem, check, repeat) and the alternating of worked solutions at the beginning of independent practice might just be useful options to consider when looking to avoid that particular problem.


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