Following on from last year’s successful series of blog articles on Five 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.

This is the third blog post of the series from maths guru, Alex Blanksby.

Five Ways To Use Self-Explanation Prompts

The Self-Explanation Effect

If "memory is the residue of thought", that is, we remember the aspect(s) of something that we think about, how do we ensure students are thinking about the things that will help them learn? (Willingham, 2009). Prompt them to explain and elaborate on those things.

The self-explanation effect describes how "upon discovering an unexplained step, some learners temporarily suspended their examination of the example in order to generate their own justification for the actions depicted in the step" (Atkinson et al, 2000). In particular, more successful self-explainers tend to relate a step in a worked solution or feature of an object to an underlying principle, whereas less successful self-explainers tend to give superficial explanations often due to studying the examples for very little time.

Self-explanations work better earlier in the learning process while students are trying to wrap their heads around a novel concept more so than later when students are already proficient at the concept.

Putting it into Practice

Five ways you can use self-explanation prompts are:

Prompt students to self-explain
Teach students to give better explanations
Learn more question types to prompt students with
Consider the constraints
Pick your battles

1) Prompt students to self-explain

Students may not generate their own explanations of what the steps of a worked example are, instead digesting only the surface-level features (features that are only relevant to this specific example such as the specific values). To ensure that students attempt to explain particular details of the worked example, we can ask them directly about those details.

Pershan (2021) provides a series of prompts that you could consider, in particular focusing on "why" and "what if" questions and targeting common misconceptions or calculator mistakes:

"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?"
"[x] answered this question incorrectly. Explain what they have done wrong. Correct their mistake and find the correct answer. Now try this [similar] question."
"Which parts of the solution will look different, and which will remain the same?"
"Which of the following best explains why _______ is true?" (all options should be reasonable in the context)
"[x] used a different method and got the same answer. Explain why their method works. Explain which would be preferred for this example? What about this example?"
"What is similar/different about _________ compared to the example?"

The prompts can be provided orally or be written down with the completed worked example and should be designed to get the most out of a provided worked example by guiding attention to the key parts or crucial parts that are easily overlooked.
Over time, students should start to internalise these questions such that they can ask those questions to themselves when studying a worked example.

***Examples of worked examples with prompts for correct and incorrect working from*** ***Karen Hancock***.

2) Teach students to give better explanations

Students get better at self-explaining if they regularly attempt to relate the specific details of the problem to the general principles at hand (Aktinson et al, 2000).  

Pershan (2021) says it is helpful to ensure enough time is given for students to carefully read, think about, and notice the relevant features of a particular example as well as prompting them beforehand with the required pre-requisite knowledge needed to get the most out of that example. However, we can go a step further by explicitly teaching students how to give better explanations. This includes:

Using sentence stems such as "because, but, so", for example:

To add these fractions, we need a common denominator,

because we need a common unit to combine numbers together.
but if it is not the lowest common denominator, then we will definitely need to simplify after.
so, we find equivalent fractions that share a common denominator.

Providing scaffolds for explanations such as a menu of reasonable explanations, for example:

To add these fractions we need a

common denominator: because we need a common unit.
common numerator: because we need a common unit.
common numerator: because we need a common number of parts.
common denominator: because we need a common number of parts.

  Alternatively, have students fill in the blanks of an otherwise correct explanation, for example:

To add these fractions, we need a _________ _________, because we need a _________ ________ to combine numbers together.

This can be further scaffolded by hinting at the required word or phrase in each space or providing a selection of possible words or phrases to use (but ensuring there are more provided than spaces available so they cannot use elimination to complete the final space).

3) Learn more question types to prompt students with

The hardest part of being told to "ask more open questions rather than closed questions" is not knowing what specific types of questions to ask instead. Likewise, it's hard to know what types of questions to ask when prompting students to explain more. Watson & Mason (1998) provided the list below of general questions and prompts that are great to add to your arsenal.

Find one or two questions or prompts to embed in your practice, then regularly add in a couple more. Don't overwhelm yourself by trying to memorise and use all of them at once. I suggest starting with:

"What is the same and different about [example A] and [example B]?"
"Alter an aspect of something to see the effect."
"Give me an example of [concept] such that …" (repeatedly getting more and more specific and narrow range of possible examples)

These can lead to discussions about which similarities are a general feature or were coincidentally in common (try to minimise this where possible) and which differences are an axis of variation or cause fundamental changes in the working out (i.e., no longer an example as it was originally intended).

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) Consider the constraints

As with any teaching concept, you need to consider its affordances and where it is and is not best used. Rittle-Johnson and Loehr (2017) provide four constraints on using self-explanation:

Use self-explanation to highlight general patterns, not specifics of the example. Otherwise, this can lead to overgeneralising, applying a non-existent pattern to a new problem, or an inability to recall the specific details if that was the intended focus.
"[S]elf-explanation is often more effective when learners are explaining content that they know is correct or incorrect rather than their own ideas." Whatever students explain is more likely to stay in their memory. If students only explain their own ideas, they might be reinforcing concepts or procedures that are not always the most helpful for learning. The focus on students' own explanations (correct or incorrect) "can fail to improve learning or even reduce learning and transfer to new information" by failing to move on from “their preexisting idea”.
"[E]xplanation prompts must be designed with care so that efforts to direct attention towards one type of information does not come at the expense of learning less about another important type of information." For example, where there are more prompts connecting steps to principles, there is a decrease in the number of calculations performed and when there are more prompts on solution procedures there is reduced comprehension.
Consider alternative activities that "would more easily or effectively achieve the same learning outcomes." That is, “generating self-explanation can take considerably more time than receiving instructional explanations” and “[b]oth self-explanation prompts and solving unfamiliar problems can provide opportunities for thinking about correct procedures, including when each is most appropriate, and noticing patterns across problems.”

5) Pick your battles

Trying to get students to explain repeatedly is exhausting for both teacher and student. So, be purposeful about when you are asking for explanations and what you are asking for explanations of.

A student can't explain what they don't know. Asking for an explanation when the student does not have the necessary prior knowledge (activated or at all) will devolve the process into playing the terrible "guess what's in my head" game where the student will end up guessing blindly because they don't know better. In this situation, consider either explaining it to them and check for understanding using a similar question, or provide a more detailed worked example that is easier to follow.  

A student can't give rapid responses to quick-fire questions while also providing long-winded explanations. If the aim is to get in a lot of practice quickly, focus on just answering the question to get or keep flow going. Otherwise, ensure the student has a short, snappy explanation they can fire back.

Conclusion

Making the most of self-explanation prompts involves:

prompting students to self-explain in the first place,
model how students can give a good explanation with practice,
learning more ways to prompt students,
considering when self-explanation prompts are or are not beneficial, and
backing off with the questioning when students don't have the knowledge to answer effectively or when wanting lots of rapid practice.

References

Watson, Anne, and Mason, John (1998). Questions and Prompts for Mathematical Thinking. United Kingdom, Association of Teachers of Mathematics.  

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

 Atkinson, Derry, Renkl, Wortham (2000). Learning from Examples: Instructional Principles from the Worked Examples. Review of Educational Research, vol 70, no. 2, pp. 181-214.

Willingham, Daniel T. (2009). Why Don't Students Like School? A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. Germany, Wiley.

 Rittle-Johnson, Bethany and Loehr, Abbey M. (2017). Eliciting explanations: Constraints on when self-explanation aids learning. Psychon Bull Rev 24. pp. 1501-1510.

Five Ways To Use Self-Explanation Prompts

Five Ways To Use Self-Explanation Prompts