In the book Symmetry and the Monster (Amazon link), Mark Ronan recounted the amazing story of how John Conway successfully discovered 3 new mathematical objects within a 12 hour period. Having had little time to devote to the problem, he worked out a solution with his wife to work on it only on Wednesdays from 6pm to midnight and on Saturdays from noon to midnight:
“At noon on Saturday, as agreed with his wife, he started work, ‘I had a last cup of coffee, kissed the wife and kids goodbye, locked myself in the front room, and started to work.’ With a 12-hour period ahead of him he took a long sheet of white paper and wrote down everything he knew about the Leech Lattice. By 6 p.m. he calculated that it should yield a symmetry atom of the following size, or possibly half this:
222 × 39 × 54 × 72 × 11 × 13 × 23 = 8,315,553,613,086,720,000
He phoned Thompson.”
Thomson actually rang back 20 minutes later, and told him there were two other symmetry atoms associated with the Lattice. He continued working on it, and made significant progress, but by 10pm was feeling exhausted. He continued working on it, and realised all he needed to do was to do another 40 calculations. He did one in detail, and it worked out well, with 39 to go:
“It’s all going to work and so, really now, I’m going to bed.”
But he decided to press on, and eventually had the whole thing figured out:
“I just said, ‘Well, how bloody stupid to give up,’ and so I carried on. At a quarter past midnight, I telephoned Thompson again, saying it was all done. The group was there. It was absolutely fantastic – twelve hours had changed my life. Especially since I had envisioned it going on for months – every three days spending six or twelve hours on the damn thing.”
When we think of mathematicians, we usually envision them working solo in their room, like Conway, hammering out equations and working out the details to their new mathematical theories, but we sometimes neglect that just like all other research endeavours, mathematicians actually rely alot on actively discussing their ideas and research with each other, not to mention checking each others’ work.
The same is true for how we acquire knowledge in general. We actually learn new concepts and ideas two ways: we can either introspectively sit ourselves down alone with the learning material and process it by ourselves, and we also at the same time need to talk through what we are learning in order to understand it better.
Four dimensions of learning preferences
In this article, we explore four dimensions of learning that could be said to characterise our learning styles, based on research by Richard Felder and Linda Silverman.
In the above example, we can see that students can learn either actively, whereby learning takes place through discussion or engagement in some form of group activity, or reflectively, whereby learning takes place through solo introspection and thought. This can be described as a learner’s preferences for Processsing.
The other three dimensions of learning that we are looking at are along the lines of Perception (Sensing/Intuitive), Input (Visual/Verbal), and Understanding (Sequential/Global).
In this scale, students can lean either towards Active learning or towards Reflective learning. As described earlier, active learners prefer to learn through activity and discussions, while reflective learners would prefer introspection and self study. This scale is actually quite related to the MBTI’s scale of extravert-introvert, where extroverts are likely to prefer active learning while introverts prefer to process new information and ideas on their own.
At schools, homework can be seen as an activity which tends to reinforce reflective learning, while teachers can encourage active learning by assigning group projects or discussion. Educators which only focus on homework only may be doing a disservice to their students as active learning is also a very important component of how students learn. At the same time, it is important to find space for personal introspection even for group based projects, so as to allow for students to benefit more fully from having a well balanced way of teaching. For instance, in group work, it is also important for educators to assign individual deliverables, such as a personal report or diary.
Another scale which is related to the MBTI (the Sensing-Intuitive scale), Perception describes whether a student prefers to receive information that is either concrete, in great detail, and factual (Sensing), or on the other hand, more abstract information, where emphasis is placed on associations, concepts, and meaning.
It has been suggested that students with different learning preferences would gravitate towards different kinds of subjects, depending on whether the subject tends to be more “practical”, which would suit sensors, or more “theoretical”, which would suit intuitors.
This scale measures a learner’s preference for either a more “visual” style of learning or a more “verbal” style of learning. In our experience, more students are inclined towards a “visual” style of learning, but what does that actually mean?
Visual learners prefer to learn via pictures, diagrams, and flow charts, which have some form of illustration which can link either facts ore concepts together in a logical manner, while verbal learners are a lot more comfortable with just reading and hearing spoken explanations.
For much of history, the main way knowledge has been transmitted has been through the written word and oral histories, which would mean that verbal learners would have had an advantage at gaining new knowledge. However, the explosion of new media, such as online learning videos, rich interactive media aids to learning, and tools such as the iPad has led to more knowledge being easily accessible to students these days. However, it seems that not all educators have picked up on this trend, and there might still be an unneccessary bias against visual learning.
This scale describes whether a student prefers a step by step exposition of a concept (Sequential), or if they would find it much easier to pick up a new topic if they were provided at the start with an overview of how the different facts or concepts link together for a overall view (Global).
While Sequential learners may be able to function while possessing only a part of the material being taught, Global learners are unable to gain comprehension unless they are able to see how new concepts they learn can be linked to knowledge they already possess. However, once they are able to see the big picture, they find it easy to find solutions to problems which Sequential learners might find difficulty with.
For instance, in the macroeconomics syllabus, since schools have to introduce the concepts sequentially, Global learners may find themselves confused with how all the different concepts fit together until they are taught everything and can see how do the different concepts link with each other. However, once they are able to do so, they have a much stronger ability in applying what they have learnt to different contexts, countries, and economies.
Index of Learning Styles
These four dimensions or categories of preferences are captured by a psychometric test called the Index of Learning Styles, which can tell you whether you have a mild/balanced preference along each of the four scales, or if you are moderately even strongly leaning towards either side of the scales. You can take the test online here.
How can knowing your learning styles help you become a better student?
At Basecamp, we find that helping students find out more about their learning styles and preferences helps them alot in charting their own learning journey. As educators, we do not believe in teaching to a style, as students need to be able to adapt to learning in different styles, in order to prepare them for success in the complex workplace.
Active or Reflective?
Active learners tend to do best by discussing what they know or explaining what they have learnt with other people. If you are in a class where the focus is more on individual learning, with not many opportunities for discussion or activities, you may make up for it by joining study groups, or smaller tuition classes, where there are more opportunities for you to engage in discussion with other students and the teacher. At Basecamp, we encourage students to speak out during class, since all classes are in small groups. In a study group, you may try out explaining different topics with each other.
Reflective learners prefer to think quietly. In classes which are more discussion based, reflective learners should consider taking extra time to think about what has been taught and consolidate their learnings by writing them down.
Sensing or Intuitive?
Sensing learners tend to prefer learning facts, while intuitive learners prefer to discover new relationships and concepts.
If you are more of a sensing rather than intuitive learner, you are likely to find that you will dislike being tested on material not explicitly covered in class.
Sensors also tend to be more careful and meticulous in their work, while intuitors tend to be more innovative. If you are an intuitive learner, you will need to discipline yourself to check your work, for instance for O Levels Maths, you will need to set aside time to check through your working once through carefully. For sensing learners, the opposite is true: you need to learn how to move on quickly once you have checked your working so that you have more time to complete the entire exam.
To be an effective learner, you need to embrace both modes of learning. If you overemphasize intuition, you may miss out on important details or end up making careless mistakes; if you overemphasize sensing, you may miss out on reaching a true understanding of the subject and rely too much on memorisation.
If you are a sensing learner, you may have difficulty in classes requiring abstract thought and analysis. Try to link what you learn to concrete applications and work through sample questions to see how can the theory apply to a problem.
If you are an intutive learner, you will likely be very bored in classes which require more memory work. In other to learn better, keep asking your teacher for theories that can help you build connections and concepts between the various facts that you are learning.
Visual or verbal?
Visual learners learn well when they are able to see what they are learning in pictures, mindmaps, and diagrams, while verbal learners learn well through written and spoken words.
In school, the primary mode of teaching is through verbal, and this is more true the higher the level you are at. However, most learners prefer the visual to the verbal, and it can be a challenge to come up with visual learning tools that can optimise most learners’ learning.
However, while everyone can learn via both methods, everyone also gets to learn more when information is presented via both modes.
For visual learners, one good way to revise and consolidate your learning is to create flow charts, graphs, and mind maps of what you have learnt. Spend time replacing words with images or symbols, and to prepare for exams, practice writing out your exam answers and converting your mindmaps into written explanations.
For verbal learners, to learn better, remember to ask more questions during class. You may try clarifying what you have learnt with your teachers by rephrasing, in your own words, what your teachers are saying, and checking with them if you are correct. Verbal learners can gain alot from studying in groups and explaining what they have learnt with their friends. During exams, try speaking out the answers to questions in your head while you write them out, and imagine you are talking to the marker.
Sequential or Global?
If you are sequential learner, you learn best when the exposition is done in a linear, step by step manner. If you are a global learner, you may be confused until you can see how all the different facts and concepts fit together. Global learners may be better at grasping complex issues, and see how can seemingly different facts link together.
Strongly global learners may find it difficult to do anything at all with what they have learnt until they manage to synthesise everything into a global framework, but sequential learners are better at solving problems one step at a time even if they are unable to see the big picture.
In secondary school, topics are generally taught in a sequential manner, and this helps sequential learners. At higher levels, for example A Levels macroeconomics, there should be a stronger focus on global learning as it is not possible to apply concepts in isolation from each other, and there is an expectation that students are able to apply what they have learnt to different contexts.
If you are sequential learner, take the time to sort everything out in a linear sequence that makes sense to you. This can also help you find out gaps in your understanding. When your teachers skip steps when teaching, raise your hand to ask about what steps were skipped. You should also aim to be come a stronger global learner by drawing links between different topics to see how can concepts in one subject be applied to other topics in that subject or in other subjects. For instance, there are very deep links between everything you learn in the mathematics and sciences.
If you are a global learner, you may find it hard to grasp topics which are taught sequentially and you do not have enough knowledge to piece out the big picture. You may ask you teacher for an outline of the year/term’s syllabus so that you have a better idea of what’s coming up. You might also find it helpful to read ahead of what’s being taught in school, and spend more time trying to draw connections between what you already know and what is new. Do not be discouraged if you don’t immediately understand new topics, and be confident that you will eventually understand it even better than some of your classmates when you see how it all fits together.
At Basecamp, we find it very useful for our students to understand their learning styles. The Index of Learning Styles serves this purpose very well, as each of the dimensions are easy to understand, and students are given actionable tips that are easy to implement. To take the test and get a free report, click here.
This article is part of our series on Models of Learning Styles.
For the first article on the MBTI, go to http://www.basecamp.sg/blog/myer-briggs-test-mbti.
For the second article on the VARK, go to http://www.basecamp.sg/blog/5-common-mistakes-that-people-make-with-vark.