Learning is always based on an experience.
The amount of meaning we attach to it is a measure of the quality of the experience and possibly the learning. The most common way that most of us learn anything is by reading it, hearing it, watching it on a screen or actually doing something. There is an old proverb that talks about the superiority of the actual doing over just watching, hearing or reading. We will all have examples though of very powerful learning experiences that we heard or saw. This is because we have so many experiences to draw on. Learning that draws on our experience and either adds to or contradicts our convictions is likely to mean much more to us.
David Kolb described a cycle for experiential learning. It described a learning activity as a cycle where one stage ideally leads to the next. A concrete experience leads to reflection which leads integrating this experience into your world view (theorising) and planning/experimenting based on your new ideas. This of course leads to a new experience which leads to new reflection etc.
Interestingly, Kolb labels the concrete experience as the feeling part of the cycle. How we feel about the experience is likely to have the most profound impact on the learner. In maths, there are two common experiences that seem to stay with people for their whole life and can inform how they respond to maths from then on.
One common experience, or feeling that remains from a range of experiences, is exclusion. This can be in the form of "I'm not good at maths" or I'm not a maths person" and often is linked in the person's memories with an event or a person that made them feel that way. Of course, it is also possible that they experience many "events" that lead them to feel that way. Some people seem to identify with the "not maths minded" type which they are likely to hear others around them saying.
The other experience that people report as having a long term impact on them is known as an "Aha" moment. So many people who identify as "maths people" seem to have experienced a moment when maths or numbers started making sense. I had a moment of that kind when I was around 8 or 9. That moment marks the time from which I remember actually thinking about maths. I was clearly able to do arithmetic prior to that moment, but I have hardly any memories with meaning attached to them compared to after that moment when I remember so much.
Of course these "Aha" moments don't happen in isolation. There are lots of things that need to be in place for maths to make sense. These are things that can be put in place for everyone. Not with a one-size-fits-all approach but taking the time to drill down into a learners deepest ideas, questions and thoughts about numbers. My personal moment came after an hour or so thinking about how to demonstrate a solution to one of the simplest maths problems I can think of. It was thinking about the problem rather than coming up with an answer.
Our mission at Prime Colours is to enable more people to feel positively about maths. We use a model to evaluate an experiential learning called the tetrahedron. The full description of this model is fairly involved but it's basically an idea that takes the 4 vertices or nodes as a necessary part of an learning experience.
- The first node is connected with feeling, meaning or experience. It could be a physical experience that is linked to the concept or activity or at least something that creates a sense of meaning for the participant.
- The second node is to do with inquiry or curiosty. We feel that unless a person is asking a question, they are not really going to learn. A question in this case really represents a hole or inconsistency in their mental model of the topic rather than an articulated question. It means that there is a place for the new learning in their internal schema.
- A heuristic is a more traditional sense that something has been learned. It's the key take away, the rule that you can repeat or the transferable aspect of the skill or idea.
- The fourth aspect is practice. Making this new skill automatic takes practice. This is the musician practising the piece or the mathematician repeating a technique several times. All art forms involve some form of repetitive practice. If we don't make some parts of our craft automatic it becomes extremely difficult to build up more complicated formations. In maths, this would mean that addition of single digits is automatic so that you can do procedures for long arithmetic without having to spend too long on each individual addition that forms part of the long technique.