Here are some resources I’ve developed along the way while homeschooling my kids:

Clock: Learning to Tell the Time on an Analogue Clock

Multiply: Learn the multiplication tables

# Process, Concept and Context

The resources listed above are designed to help develop concepts.

Students who know how to run up and down the number line from -50 to +100, who can recognize and recall multiplication tables alongside a small handful of other math concepts are able to follow explanations in grades 10 to 12 math much more easily.

Following processes alone doesn’t really contribute to further learning in math. The difference between process and concept is whether or not we’re teaching to ‘get an answer correct’ (eg multi column adding and subtracting – which adults do using their phone) or understand a new concept such as place value which is critical later when for example, a student might wish to represent an exceedingly large number or an exceedingly small number using scientific notation.

Concepts are difficult to teach in a classroom context. Processes are easier – they can be demonstrated and copied and students can do worksheets on processes. A lot of employment is based on learning processes so processes have their place.

Concepts depend on dialogue – especially dialogue amongst people who already have a respectful relationship with each other so that trust is more evident than fear of being judged or ranked amongst ones peers which assessment inevitably does. It isn’t fair to expect a teacher with 25+ students of vastly different needs to engage in this dialogue with their students.

Concepts require time, a lot of time. Time to be new, time to be built and tested, time for the dust to settle, time for the concept to find its place relative to other concepts. Concepts require a sounding board to develop, a place where ideas can be bounced off of each other. Conceptual understanding is ideally exposed to diversity of experience and ways of thinking. The more introverted learner might test a concept while at play with characters, bouncing on the trampoline or repetitively throwing a ball against a wall.

When a learner notices a ‘short cut’ or a ‘pattern’ or an ‘efficiency’ in the calculations that they are doing because they have a reason for them (such as adding up their score in a game) – they feel genius. On the other hand when the learner is *told* a short cut or pattern when they are not ready for it when doing math that they have little relationship with by an adult it can be anywhere between annoying and defeating – it doesn’t make sense and they have no ownership of it.

From my observations in the high school classrooms it seems that about 10 to 20 percent of our children in school are ready for the pace of the K-12 math curriculum. They can make enough sense of patterns and methods to feel confident. Unavoidably, knowing that others make less sense of math boosts their confidence further. We see one group being promoted at the cost of another group.

Then there are a further 10 to 20 percent of our children who have sufficient faith in math to do as they are told by the processes and have enough numerical sense to keep getting the answers reasonably correct even if they don’t feel so good about it. I can do math, they say, but I don’t like it. As long as they bring home a B in high school parents are satisfied. They are a big enough group and do well enough for us, the educators, to continue to perpetuate this system. They don’t have a lot of confidence and might express frustration with having to learn it and don’t really see why.

Then there are a lot of students who by grade 11 or 12 have a fairly negative relationship with math as they’ve experienced it in school. Fortunately for them, the next time they have to interact with math in this way is not until their children are going through school. They can push the whole experience out of mind for the time being.

On the other hand, if we can create regular time for kids to interact with number and measurement in positive ways such as playing games and working with relatives in the kitchen and workshop then we can be providing them with the concepts necessary for further learning in math – regardless of all the details of the curriculum. Note that the BC curriculum implores teachers to pay more attention to the core competencies than the details of the curriculum. It makes very little sense that the places where students get to use math in context are often cut for budget reasons. Make the math classroom an extension of the sewing classroom, not vice versa.

Let’s be grateful to Indigenous peers in education who challenge us to consider our curriculum when they tell us that from their perspective “Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).” Is it possible for the great machinery of modern Euro-American decontextualized mathematics education to disengage, reassess and rebuild to serve our whole population in a more fair and relevant way?