A while back, I found myself visiting my sister-in-law who had just moved to a new town. I decided to drive to the local supermarket and this seemed like a good chance to test out an idea that I had heard from David Didau and others. There is a debate about when feedback should be provided to students: should it be immediate or delayed? Feedback is a complicated issue in education and this is probably due to the fact that we are lumping together very different things under the one heading and so the answer to the question on timing is likely to be, ‘it depends.’
Didau had used the example of a car satellite navigation system or GPS in order to make a point against immediate feedback. GPS systems provide such feedback and it was Didau’s contention that this is why we don’t learn routes very well if we rely on them. Didau discussed using GPS to navigate a new city and not learning any of the routes.
So I decided to try and use GPS to learn the route to the supermarket. I paid attention to it on the way there and then tried to drive back without it. I learnt the route just fine and was even able to complete the same trip the next day from memory. I was using Google Maps on my iPhone – this shows a live map of the surrounding area and not only gives an indication of the next turn but the turn after that, something I find useful for ensuring that I turn into the correct lane if there are multiple lanes. I never suggested that this was the best way to learn a route – I was simply testing whether it was possible with GPS.
Dan Meyer picked up on my post about this and decided to relate GPS to explicit instruction in mathematics. He had found a study from 2006 where participants were given various different ways of navigating around a German zoo. Three conditions involved using handheld Personal Digital Assistants (PDAs) that gave various visual and auditory information (such as a picture of the intersection, an animation of the pathway and verbal instructions to ‘turn left’) when a participant reached an intersection. The fourth condition involved giving participants map ‘fragments’ i.e. maps of routes rather than of the whole zoo with photographs of the intersections in numbered order.
Participants were not told in advance that they would be tested on route and survey knowledge. When these tests were later carried out, all conditions showed learning but the map fragment condition demonstrated a statistically significant advantage over the others. Meyer then states:
“So your GPS does an excellent job transporting you efficiently from one point to another, but a poor job helping you acquire the survey knowledge to understand the terrain and adapt to changes.
Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.”
This is a non-sequitur. It is not clear that we can make any inferences from such a study and apply them to maths teaching. Even if we were able to, why does Meyer think the PDA conditions are more like explicit instruction than the map fragment one? In classic studies on the worked example effect the experimenters often made use of example-problem pairs. This is where a student is shown a whole worked example and then has to complete a similar problem themselves. You might argue that this has more structural similarity to the map fragment condition than the PDA conditions. I won’t be making such an argument because the link between German zoos and maths teaching seems tenuous and any conclusions we might draw about maths seem a little eccentric.
Many of those commenting on Meyer’s post made similar points to this and he added a coda in a comment of his own:
“The question that’s useless to us is “should we use [x] in helping students learn?” The answer for most values of x including worked examples is “yes.” The more interesting question to me is, “What kind of knowledge is easy and difficult to learn by way of worked examples?” And, “Under what preconditions are worked examples most helpful?”
The answer to those questions for some of the traditionalists whose blogs I tune into now and then seems to be “all knowledge for all novices” and “no preconditions are necessary.” That kind of maximalism is pretty easy to falsify. (See Greg Davies‘ comment for an example: “Surface structure always comes first.”) Even one datum falsifies a universal claim.” [sic]
I am not sure whether Meyer is referring to me here but I have certainly never claimed that “no preconditions are necessary” for learning from worked examples. To learn from a worked example you would need to understand a whole lot of prerequisite maths. If you don’t know multiplication tables, for instance, then algebra can be tricky – try factorising quadratics. This is a point I’ve made many times and actually sounds like a viewpoint that you might label as ‘traditional’.
I have also written about the expertise reversal effect where the usefulness of worked examples fades as students become more expert. This is why explicit instruction gradually moves from explicit examples, through guided examples to independent practice. Rosenshine provides an excellent explanation of this process which is one that also makes objectives clear and is highly interactive. It is odd that Meyer links to Rosenshine but then insists on such a weird interpretation of explicit teaching.
The idea – dismissed by Meyer – that surface structure comes first is pretty well known in the field of cognitive psychology. I note that Meyer has recently become a fan of Dan Willingham and so he should perhaps return to this piece that Willingham wrote about the subject. Sadly, there are no pedagogical magic beans that we can buy that will helps us accelerate students towards apprehending deep structure. I am deeply sceptical that problem-based learning (PBL) can do this. I note that Meyer links to some Bransford and Schwartz pieces to support his view. I haven’t had chance to read these yet because I’ve been focused on the GPS study but I would be surprised if they draw upon well-controlled experiments that test strong explicit approaches against strong PBL ones.
We all want students to apprehend the deep structure of problems but we must recognise that this is hard work. The methods used by explicit educators might be to highlight non-examples to prevent students overgeneralising principles – a key source of many maths misconceptions – and providing deep explanations. Some have even tried to turn this into a science. Indeed, I find it surprising that those who are so eager to promote problem-based approaches to mathematics are also keen to see explicit instruction as simply a set of step-by-step directions, ignoring the role of explanations altogether: “Do this. Now do that. Don’t ask why.”
Actually, it’s not that surprising because it is much easier to knock-out a straw man than a heavyweight boxer.