Book Review: A Mind for Numbers by Barbara Oakley
For the Teaching and Learning reading group at Bedales, we have been reading Oakley's A Mind for Numbers . The book is mainly aimed at students, particularly those who have struggled with the concepts they are being taught in science and maths lessons, but there are some valuable tips and tricks for teachers dealing with students with low self-esteem and/or poor study skills in STEM subjects.
I am going to briefly discuss the following points from the point of view of teaching:
Chunking of information
The deeper meaning of equations
The hard-start-jump-to-easy technique
Chunking of information means "pieces of information bound together through meaning", essentially the building of a neural pathway that links together disparate pieces of information. This means that when students are working through a problem they can access the 'chunk' required to solve it, rather than having to have all of the underlying pieces of information in the working memory at the same time, making the brain work more efficiently. Oakley gives the following steps to making chunks:
Focus on the information and learn it - in order to be able to chunk the pieces of information, the student needs to have those pieces in their working memory, not be looking up the various pieces, e.g. the relative mass of protons, neutrons and electrons, every time they approach a task.
Building understanding of the relationship between the different pieces of information - this holds the pieces of information together for the student so that they know how to use the chunk and requires practice in as many different situations as possible.
Knowing when to use the chunk as well as how to use it - this is called interleaving, switching between different kinds of problems, and means the student needs to be able to recognise which problem solving strategy is required in a particular situation.
Stage 1 is familiar territory for teachers. What I think is important for teaching in the 21st century is not to get sucked in by the idea that "today's students can just Google the information" and they can do the learning-off-by-heart closer to the exam. Oakley reminds us that learning needs to be active and visualisation is crucial. The other lesson is that students need to be given the time to actively recall the pieces of information before they start on their practice stage. So if students are learning moles calculations, the students need the opportunity to learn the equation and be able to recall it without their notes before they start working on the problem set.
Equally Stage 2 is familiar to teachers; we tend to set practice questions on each new concept which tend to be fairly repetitive at the start, but build up to slightly different situations by the end. Oakley emphasises that for chunking understanding needs to come from bottom-up and top-down learning. Bottom-up learning is the 'practice makes permanent' route to learning, but top-down, big-picture learning is also essential. This means that students need to understand how new concepts link to their prior learning. In my experience, when my students are struggling with a concept, it is usually because I have over-emphasised either the top-down or the bottom-up approach, rather than effectively balancing the two. In order to achieve this, when they have taught a concept via the bottom-up approach, teachers could use a big-picture plenary exercise such as completing a thinking map linking ideas or evaluating a concept diagram. If instead a concept has been derived from other ideas at the start of the lesson and students have been given a chance to learn and understand all of these stages themselves, there should be an opportunity for students to practice examples before the end of the lesson.
It is Stage 3 that is often skipped in normal teaching and it is not until a test that students work out whether or not they can identify the appropriate problem solving strategy in a particular situation. Sometimes homework tasks require interleaving, but this is a challenge for students as they have not had a chance to develop the strategies they need for identifying the correct chunk while supported by a teacher. In a lesson a teacher can help a student to pick up on the kinds of terms or information present in a question which 'triggers' bringing the correct chunk into the student's working memory for problem solving. Teachers can ensure that problem sets that involve interleaving are carried out on a regular basis in class. In order to make time for this, teachers could set a flipped learning homework task before the lesson for the students to achieve Stages 1 and part of Stage 2 themselves. Setting practice tests in the week before a real test and offering formative feedback during the lesson, which could be achieved through peer-assessment, can really help to motivate students and direct their revision in the areas that need work, as well as helping to finalise the chunking of the concepts they have been learning during the topic.
"There are hidden meanings in equations, just as there are in poetry."
Oakley is very keen that students understand the deeper meaning of equations used in maths and science: this means no formula triangles! (I was encouraged in my anti-triangle zeal this week by Pritesh Raichura's blog on Equations in Science which also forbids the use of triangles: https://bunsenblue.wordpress.com/2017/02/14/equations-in-science/). For Oakley, "there are hidden meanings in equations, just as there are in poetry" which means that for genuine understanding, students need to be able to unravel the meaning behind each of the terms in an equation and explain why they are encoded the way they are. This means that students need to be able to visualise what the equation means using simile (Oakley uses the term metaphor). The role of the teacher is to ensure that the visual similes that students are using do not lead to misconceptions which could stand in the way of the student deepening their understanding. I highly recommend the RSC course on Developing and using models (https://www.rsc.org/cpd/resource/RES00001448/developing-and-using-models) for teachers looking to develop their skills in this area.
"Testing is itself an extraordinarily powerful learning experience."
The hard-start-jump-to-easy technique is an approach for students to take during tests which makes efficient use of the brain's two modes of thinking: the focused mode and the diffuse mode. We have all been in the situation where we are working on a difficult problem and then the answer comes to us suddenly later when we are taking a walk, or having a shower. This is because the diffuse mode of the brain is good at figuring out novel problems. In an exam when students are extremely focused, they can often get stuck trying to approach a problem one way, even if it not the correct method, and be unable to switch focus and try another method. This is called the Einstellung effect and is a serious problem in Chemistry exams; think about the number of students who cannot put together all the information from an NMR problem until they have left the exam hall, because they have gone down the wrong track and been unable to backtrack. The hard-start-jump-to-easy technique suggests that students focus on the most challenging problem first and once they have been stuck for two minutes they switch to the easiest problem so that the diffuse mode can continue working on the more challenging problem; when they return to it they will then find it easier to work through the steps. Many teachers encourage students to move on when they cannot do a problem and return to it later, what Oakley has added is psychological reasons why this technique works and suggested how to push it to the extreme by actively searching through the whole paper for the hardest problem to begin with.
Oakley's book is full of further wisdom which can be helpful for learners: memory tips and tricks, means of avoiding procrastination, etc., and I will be tempted to recommend it as summer reading for students who have found the AS content difficult before they approach the second year of A-level. Throughout the book Oakley practises what she preaches; interleaving different ideas, repeating the same concepts in different situations and providing vivid visual prompts to explain the underlying science. This means that the book is extremely accessible for students and teachers alike.
In conclusion, Oakley's book is a valuable resource for teachers to tweak what they are already doing to maximise its efficiency, and demonstrate that this is founded in modern neuroscience. However, as it has been written with the learner in mind, teachers will need to reflect on strategies for taking advantage of the lessons that Oakley has provided us with. But that is all about linking the big-picture thinking with the practice and is what makes us better teachers.
Usefulness for classroom practice: 3/5