The worked example is a key mechanism for drawing attention to important mathematical ideas, building confidence and displaying a desired layout.
The word’s etymology has many different sources. From Old French essemple “sample, model, example, precedent, cautionary tale,” and from Latin exemplum “a sample, specimen; image, portrait; pattern, model, precedent; a warning example, one that serves as a warning,”
It helps us to show our thought process, sequence of steps and draw attention to essential details, as well as making an example of what not to do – “a cautionary tale” perhaps.
If student attention is the coin of the realm, the worked example is the bureau de change and it’s quality is the exchange rate.
I have only recently really paid attention to this process in any where near the detail that I needed to. I am by no means an expert yet but I definitely feel that my refined process is most certainly better than ever before.
I would previously, in my earlier years, not really take the time to think about what I was going to say or show, even though I knew that I had a large influence on what students were thinking about.
I was more concerned with what students were “doing.” If they weren’t doing, they weren’t learning. If they weren’t discussing, they weren’t learning. This was not true. Even when students are “doing” and “discussing” they aren’t technically learning. Learning happens over time, not in the moment. What students are doing is performing. They are showing that they “get it”, not that they can “keep it” or “transfer it.” “Getting it” is an essential pre-requisite to “keeping it.” If we don’t “get it”, we can’t “keep it.”
If I want my students to have a chance of transferring and keeping … I need to do my best to make sure they “get it.”
7 steps to teaching metacognitive strategies from the Metacognition EEF Guidance Report (2018)
Quick example and off you go.,..
In my first years, I would show an example or two and then try my best, as quickly as possible, to get students onto a set of questions, some of which may or may not have been well matched to the examples I had shown. I could just help students when they got stuck as I circulated right? This was in the days when teacher talk was a no-no. When we were to be the facilitators of learning. It certainly wasn’t the way to go.
Sage n Scribe
I then went through a phase of wanting my students to collaborate ALL the time. I wanted to show them an example and then ask them to do some questions together by taking turns. This method is called Sage n Scribe.
Partner A takes control of the pen and plays dumb – she is not allowed to think, only scribe. Partner B is the brains of the operation. Partner B has to explain exactly what they want partner A to write in minute detail. They would swap roles for each question. The advantage of this was that it got students talking and supporting each other in a structured way, once this routine was practiced and students got used to it. The major disadvantage was the cost this imposed on developing misconceptions. Students were still relative novices at this stage and could often confuse one another during their explanations to each other. It also meant that the questions took AGES and actually created a false impression of how long problems took to solve. This method does seem to tick the two essentials for effective group work (Slavin): groups goals and individual accountability, but it certainly isn’t the most efficient.
Example Problem Pairs – My Turn, Your Turn
This was the next phase of the evolution and definitely a step in the right direction. Students are presented with a problem and then given a problem to try themselves.
There are decisions to make here:
- How difficult is the example that I am trying to show and what do I want students to think about. Is it simple? Give them a problem that is just beyond their current knowledge. Is it complex? Give them one that is similar to build confidence.
- Model the example live with my pen – quickly so that students can see in real time how long the problem takes – or pre-write the example in a different colour so that students can see the big picture in front of them and talk through it. I find the latter of these works best when we are reviewing an example that has already been modelled, maybe at the start of the next lesson – “here is what we looked at last lesson.”
- Is it simple enough to write metacognitive thoughts or label the steps? Is there enough space? Will these be understood in the future if a student looked back?
This idea has come from studying the work of John Sweller in more detail.
In Efficiency in Learning Sweller et al describe something called A Completion Example, essentially a hybrid strategy where ‘some of the steps are demonstrated as in a worked example and the other steps are completed by the learner as in a practice problem’.
I have now moved towards a 4 pronged approach to worked examples – live modelling, metacognitive questioning, backwards fading/guidance fading then your turn.
The decisions above still apply but there are now other decisions that I have been playing with in the “faded guidance” section:
Type 1 – Finish off the problem
In Efficiency in Learning, Sweller et al talk about the idea that ‘completion examples reduce cognitive load because schemas can be acquired by studying the worked-out portion. Requiring the learner to finish the worked example ensures that she will process the example deeply.’
Type 2 – Here is the end of the problem… how did I get there?
Type 3 – Here is an incorrect example. “The student has made a mistake in the first 3 lines of working- can you spot and correct the mistake and re-write the resulting lines?”
To make this even better, we can print this template for the students in it’s current form and provide self-explanation prompts either for the worked example or the incorrect example:
Has the equation been re-arranged correctly?
If the lines are parallel, what should we compare in line A and line B? Has this been done correctly?
Have the values of x and y been identified correctly?
Have they been substituted correctly?
I much prefer using type 1 and 2 in the initial skill acquisition phase. I prefer using type 3 later in the instruction phase – maybe when we have already introduced this idea in a previous lesson and just want to review.
I have also considered changing the faded guidance section to non-examples. Examples of questions where the worked example would not work. This may work particularly well for things like difference of two squares. Where positive example(s) are shown on the left and negative boundary examples are shown in the other box (y^2 – 25y or y^2 + 25y).
This is something that I am still exploring so thoughts are welcome about how I can improve this process.