Six effective learning strategies from the science of learning

There is extensive research into how our brains work and how people learn most effectively, across all subjects and all ages. But these findings are not well-known by students or by educators! The Learning Scientists (a group of cognitive psychological scientists) are aiming to change this.

Here is a digest of their top six effective learning strategies and how I think students and lecturers in mathematics can use them.

Retrieval practice

Retrieval practice is the process of recalling ideas from memory rather than looking things up. The more you practice retrieving something from memory, the better you will able to retrieve it in future, that is, it strengthens your memory of it. Timing is important: retrieving something immediately after you have just learned it is pointless.

How students can use

Self-testing is the main way in which you can practice retrieval. You can test yourself in many ways. When you are working on problems, try to recall definitions and concepts before looking them up. You can set yourself quizzes, or use flashcards. But remember, in order to achieve deeper learning it’s important to practice retrieving more complex ideas, rather than just facts and definitions. I’m fond of the ‘five-minute paper’ exercise – put all your notes away, then spend five minutes writing down everything you know. Afterwards you should check your notes for accuracy.

Past papers are a useful resource – but I frequently see students use them incorrectly in the following two ways: (1) working through the solutions and fooling yourself that you understand; and (2) making erroneous conclusions about what will be on the exam based on what has been on previously.

How lecturers can use

Asking students to complete a few quick questions at the start of a lecture as a warm-up activity can work well. My first-year students also appreciated a mid-semester pop quiz in a workshop. However, I did allow students to use their notes and textbook – maybe it would be better to not allow this even though the final exam is open-book? 

Remember that retrieval practice should be part of the learning process, not just an assessment exercise. I would therefore suggest that any retrieval practice exercises do not count towards the final course grade, as this will change students’ behaviour.


This strategy, more than any others, leads to a deep understanding of concepts. Elaboration is the process of  expanding given information by interrogating the material and making connections with previous topics. I think this is one that students somehow know they should be doing when studying, but often don’t.

How students can use

When studying, really question the material. Why is this theorem important? In this proof, how does one line follow from the last? How might this method be applicable in other situations? How does this topic relate to this other topic? Write down your questions, and ask the lecturer or your classmates if you struggle to come up with answers yourself.

How lecturers can use

In order to be effective, it is necessary for the student to construct their own interrogation of the topic. But you can support this by giving ‘elaboration pointers’ – that is, questions that students should be asking themselves. You could also build this into class time – ask students to spend a few minutes writing down (or discussing in groups) more details about the concept.

Dual coding

Dual coding is the idea of representing information in a visual format alongside a written explanation (it is important to have both, hence the ‘dual’ bit).

How students can use

Draw your own diagrams or doodles to help you understand concepts better. When looking at visuals that your lecturer has provided, explain in your own words what they mean.

How lecturers can use

Ensure you have both written and visual explanations where possible.

Concrete examples

In order to understand abstract concepts, it is important to be able to bring concrete examples (and non-examples!) to mind.

How students can use

For every concept, collect examples – examples the lecturer has given, examples in other textbooks and think of your own examples too!

How lecturers can use

I think mathematicians have a good understanding of the importance of examples, but remember the importance of non-examples too. Also pay attention to whether your examples are illustrative – could a student be misled about the nature of a general principle by looking at one example only?

Spaced Practice

This is the study strategy with the most conclusive evidence behind it. Spacing out shorter study sessions over several weeks is more successful that one long study session even if  the long study session is immediately before the exam. This is because you will start to forget some things between the study sessions, and the process of recalling them actually strengthens the memory.

How students can use

Don’t leave revision until the last minute! Organise your time so that you are consistently practicing earlier topics throughout the semester. This strategy is about when you study, not how you study – so make sure you are using the other effective study strategies in conjunction with spaced practice. In particular, as you will have started to forget some things, it’s important to use retrieval practice rather than immediately looking things up.

How lecturers can use

The obvious answer is to build spaced practice into the course schedule. But how practical is this in the context of a fast-paced one semester course? I’m not sure I have a good answer to this yet – but setting problems on previous topics as well as current topics is a good option and ticks off the ‘interleaving’ strategy too (see below).


Say you have a set of integration problems to do. If they were grouped by method (by parts, by substitution etc.) you would not need to do the work of deciding which technique is appropriate for which problem. In order to practice this skill, it’s better to interleave different ideas during the same study session.

How students can use

You can create your own interleaved study sessions by choosing a set of problems from different sections of the same course. Note that the research is unclear as to whether the same benefits hold for interleaving ideas from different subjects. This will feel harder than ‘grouped practice’ (i.e. doing lots of similar problems back-to-back) but it will be more beneficial – after all, an exam paper is an interleaved assignment!

How lecturers can use

Set assignments /workshop sheets which vary the topics and styles of questions. This seems straightforward enough, but nearly all textbooks and worksheets are designed around the idea of ‘massed practice’.


Thanks to Toby Bailey whose presentation on the Learning Scientists’ work and its relevance to learning and teaching in maths has informed this blogpost.




Book: Understanding how we learn: a visual guide by Dr Yana Weinstein and Dr Megan Sumeracki


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