Following on from last year’s successful series of blog articles on 5 ways to implement the science of reading, this year experts will be providing ready-to-use tips on the science of maths.

Inspired by Tom Sherrington’s Five Ways Collection, the posts have been edited and curated by Brendan Lee and Dr Nathaniel Swain.

The first blog post of the series comes from maths guru, Alex Blanksby.

Catch Alex’s Talk on May 17!

Five Ways to use Worked Examples

If you put a problem in front of a student that involves new or novel information or skills that you are not expecting them to know, either they solve the problem (with time and lots of effort, by chance, or because they have been shown by someone else), or they can't solve the problem in which case we need to show them how.

We can streamline this process by showing students how to solve the problem from the get-go. This avoids, in extreme cases, the "guess what's in my head" game where students blindly guess because they have no clue, as well as possibly disenfranchising students who (rightly) don't know how to complete the problem when other students are able to (possibly from being shown beforehand). We can show students how to do something by using a worked example.

The worked example effect describes how we can enhance learning "by studying worked examples to problems rather than by trying to solve the original problems" (Ayres, 2012). We use worked examples to leverage the knowledge of experts who know how to complete a skill to show a novice how to complete that same skill. This involves breaking the skill into clear, concise, and manageable steps that the novice can then replicate.

Putting it into Practice

Five ways you can use worked examples are:

Provide pre-completed worked examples
Model worked examples
Pair the worked example with a related problem
Optimise the examples you use
Simplify the conditions

1) Provide Pre-Completed Worked Examples

Much of the cognitive load theory literature uses pre-completed worked examples. The benefit is that students can read and interrogate the worked example at their own pace as well as refer to it as necessary.

Before displaying the worked example, tell students "Explain each step to yourself as best as you can. Put a thumb up when you've finished reading, even if there are parts of the solution you don't fully understand yet." (Pershan, 2021)

If you display the worked example (whiteboard, projector) rather than providing it on paper (loose-leaf, booklet, textbook), before displaying the worked example, be clear to students whether they should copy it down or not. If you are using the third way below, it should not be necessary to copy it down at this stage.

Once you have provided or displayed the worked example, give students time to notice and wonder about the specifics of the question and solution (consider two stages: digest the question, then the solution). You can nudge their thinking by providing specific questions (written or verbally) or annotations to draw their attention to specific parts of the worked solution. Give students a chance to verbalise their understanding of the worked example by using pair-share.

To check for understanding, warm call students to explain the different parts of the example. Ask a variety of students a variety of questions about parts of the example:

Avoid asking 'what' questions... Focus on asking 'why' questions
Push students to think harder by asking them to explain "why" and "what if" questions.
"What did [x] do as his first step?"
"Would it have been OK to write __________? Why or why not?"
"Why did [x] combine ______ and ________?"
"Would [x] have gotten the same answer if they _______ first?"
"Explain why _______ would have been an unreasonable answer?" (Pershan, 2021)

2) Model Worked Examples

While pre-completed worked examples have their own benefits, sometimes you need to see the steps in action to understand how moving parts connect to each other. We model the worked example by showing students, step by step, how to solve the problem.

While planning, consider the mode you will model your example in. Do you lose any important physicality by presenting it click-by-click on a slideshow compared to doing it by hand on paper under a visualiser or on the whiteboard?

Before starting, tell students they should be sitting and watching to focus on the details as they appear and not trying to copy it down at the same time where they may miss important pen strokes or gestures.

Ensure students see you model in real-time (at least once) so they get an accurate sense of how long it should take them once fluent. That way a 30-second skill won't come across as being a 2-minute skill. You can do this by modelling in silence and then narrate and talk to the working out afterwards (like with a pre-completed worked example).

Where the completed worked example disguises the individual steps and their order, reduce the transience by repeating the working out and adding to it.

3) Pair the Worked Example with a Related Problem

To make the most of a worked example, provide students with a similar problem to solve immediately after studying the worked example. We call this an 'example problem pair'. The paired problem should keep the key features of the question the same but vary the specifics to check if students are able to, at the bare minimum, mimic the steps.

  For example:

Find the mean of 1, 4, 7, 5, 9
Find the mean of 8, 1, 3, 6, 2

  But not:

Find the mean of 1, 4, 7, 5, 9
Consider the set of data: 8, 1, 3, 6, 2. Find the mean of that set.

  While the second paired problem is asking the same thing, the change of wording of the question decreases its effectiveness. Englemann and Carnine (1991) call this the wording principle: "To make the sequence of examples as clear as possible, use the same wording on juxtaposed examples (or wording that is as similar as possible)."

A common format for example problem pairs is putting the example on the left half of your display (whiteboard, PowerPoint, visualiser), and the problem on the right like this:

Example-problem pairs | Filling the pail (wordpress.com)

The expectation should be that all students complete the paired problem. To increase the likelihood of this, consider using mini-whiteboards, think-pair-share, or warm or cold calling.  

If you are not providing a copy of the example to each student, after displaying a correct solution, you can have students write the paired problems working into a reference book (or edit theirs to be correct if it was written directly into the book, not on a mini whiteboard).

4) Optimise the Examples you Use

Engelmann & Carnine (1991) lay out a set of facts about presenting examples:

We cannot assume that from one example or even one example problem pair students will necessarily notice all of the necessary key features and the range of variation that is allowed beyond that. The magic number appears to be about 7 examples or exposures to get the idea across:

Engelmann and Carnine (1991): "Make sure that the single interpretation emerges as early as possible. It must occur by example 7; however, it should be strongly implied by example 5."
Koedinger, et al. (2023): "a typical student learning a typical KC tends to require seven additional practice opportunities to reach mastery after noninteractive verbal instruction (i.e., text or lecture)"

  To make the most of those examples, we want to make sure they encompass as much of the concept as possible by

1. using maximally different positive examples to show the broad range it applies to:

2. using minimally different negative examples to show where the boundaries are:

3. avoiding examples that can lead to methods that do not generalise:

4. increase pace and apply variation theory by transforming one example into the next by erasing and replacing or using digital tools:

What changes from one example to the next? How does that affect the working and solution?

5) Simplify the Conditions

Sometimes a particular skill has too much to handle at once, so we can simplify the conditions in which students need to be able to perform it. Some ways that we can do that include:  

1. model the worked example, then have students progressively complete more of the working (faded worked example). You can fade from the last step to the first, the first step to the last, or even the most difficult step to the easiest.

https://taylorda01.weebly.com/uploads/4/2/3/8/42387051/backward_faded_sketching_quadratic_graphs.pdf

2. complete the enabling features for students while they complete the essential features (or vice-versa).

For example, in a long multiplication problem, you provide the value of each product (enabling feature), but students dictate what needs to be multiplied and considerations of place value (essential feature).

3. where applicable, teach the steps in isolation, then bring them back together later.

For example, solving simultaneous equations involves the separable skills:

Identifying a method: graphical, substitution, elimination

For graphical: being able to (a) accurately sketch both graphs, and then (b) read off the coordinates of the point where the graphs intersect
For substitution: being able to (a) substitute an expression and value into another equation, and (b) solve the related equations
For elimination: being able to (a) identify the variable to eliminate, (b) add or subtract equations, (c) solve the related equations, (d) substitute a value into an equation to solve.

Conclusion

In summary, make the most of your worked examples by:

carefully planning a series of examples or how you will modify one to the next,
providing or modelling the examples to students,
having students explain to themselves what the steps are,
pairing an example with a related problem to check for understanding, and

simplifying the conditions for students to practice as necessary.

Come to Alex’s Talk on May 17!

References

Ayres, P. (2012). Worked Example Effect. In: Seel, N.M. (eds) Encyclopedia of the Sciences of Learning. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1428-6_20

Engelmann, S., & Carnine, D. (1991). Theory of Instruction: Principles and Applications (Rev. Ed.), Eugene, OR: ADI Press.

Pershan, Michael (2021). Teaching Math with Examples. United Kingdom, John Catt Educational, Limited.

Koedinger, K.R., Carvalho, P.F., Liu, R. and McLaughlin, E.A. (2023). An astonishing regularity in student learning rate. Proceedings of the National Academy of Sciences, 120(13).

Five Ways to use Worked Examples

Catch Alex’s Talk on May 17!

Five Ways to use Worked Examples