My wife Win is a master teacher of Economics and the leader of Academics in our school, a role I held in another school in another life. We often end up in discussions about what makes one teacher more effective than another.

We inevitably come to the same conclusion: “It is all about relationships”.

Those teachers who know their students well, who care deeply and who can communicate that care are by far the most effective. Such teachers seldom struggle with discipline in the classroom and they are inherently optimistic about life in general and about each student’s prospects in particular. They compare each student with where they were the week before rather than with where someone else is today. They focus on potential and continuous improvement, their own and their learners’. Their narrative becomes a self-fulfilling prophecy, a rising tide which floats all ships. Parents ask for their sons and daughters to be moved to those teachers’ classes. While such teachers’ subject mastery is typically in place, it is almost never cited as the reason for their effectiveness.

Of course, Mathematics is all about relationships too and great teachers work hard at helping make explicit the connections between the various branches of the subject. Those who try to teach and learn maths as a set of disparate skills applicable at different times to different contexts are not nearly as effective as those who understand the connected nature of the subject. A skill in one area is almost always transferable to another. For example, completing the square is a skill which is useful in:

• Solving quadratic equations
• Optimising a quadratic expression
• Determining the turning point of a parabola
• Determining the range of a parabola
• Establishing how a parabola has been translated
• Finding the centre and radius of a circle
• Amongst others

Providing our learners understand the underlying rationale they can make sense of all of them. Completing the square becomes just another tool in our arsenal of techniques. We need surprisingly few tools for success in Mathematics – perhaps the subject for a future blog.

Relationships and connections are everywhere! A number sequence exists only due to some relationship between its terms. Every graph is related to the equation which defines it and vice versa. A trigonometric identity is merely an equality of different representations of a fact. As mathematicians we love to look for patterns, relationships and structures, recognising them as one would old friends.

Of course, the connections and relationships also exist between subjects. We miss opportunities when we fail to point these out. Making a the subject of the formula in Science given v = u + at is an identical process to solving 2x + 3 = 11 in Mathematics.

Do we help our students make the link? Science teachers tell their students that the gradient of a displacement-time graph gives velocity and we tell the same students that a derivative is a rate of change. However, does either of us take the time to show them that differentiating the formula they have for displacement  with respect to time gives u + at , the formula they use for velocity? Differentiating this in turn gives acceleration – the rate of change of the rate of velocity or the rate of change of the rate of change of displacement.

Finally, making connections outside of mathematics to analogous situations is a powerful way of helping our students make sense of mathematical processes. Just two examples:

• a balance scale is a powerful metaphor for solving equations. The equals sign in an equation denotes equality, sameness and balance, all of which will be preserved providing we apply the same operation to both sides.

• before we add fractions, we first need to ensure they are of the same type. The same requirement is in place when we add amounts of money in different currencies or lengths expressed with different units.

I hope this short blog post has given food for thought about the remarkably connected nature of our subject. I hope too that it might challenge us to be more intentional in looking for connections and in helping our learners appreciate them.

Finally, I hope that you have a wonderfully connected week.

Until next time