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Paul’s Blog – Maths Matters…
Post #2 – 01st August 2022
The 7 deadly sins of Maths Teachers
Some years ago, I listened to a talk by Dr Tony Humphreys, a clinical psychologist, entitled “The 7 deadly sins of teachers”. It provided the inspiration for this blog post.
I have always been interested in what separates ordinary teachers from good ones and good ones from great ones. While I do not believe there is a prototype for the “model” Maths teacher, I do think there are some things one really needs to avoid if one is to be better than average. Here are the 7 deadly sins I have identified – things which should be avoided at all costs. Like me they are in no order:
- A lack of joy and passion. In my previous life I employed many teachers. I always believed in employing for attitude rather than aptitude. I think one can help a colleague fill gaps in his / her mathematical knowledge, but it is far harder to incite passion or joy in a person. Passion for one’s subject is contagious. So is apathy. We cannot expect our students to be passionate about our subject if we are not. Our youth of today face so many challenges. We need to inspire them and give them hope by being joyful and optimistic. Some days we will have to fake it and put on our “game face”. We owe it to our students to do so.
- A lack of preparedness. At a time, I got so far ahead of myself that I believed I was sufficiently competent to be able to “wing it” in lessons. I now know that my best lessons are those for which I am properly prepared. Great teachers understand that contact time is too valuable to be wasted so they plan carefully and they arrive early enough to be set up and ready to greet the students before the lesson starts.
- Adopting a “colour by numbers” approach to teaching Maths. Great maths teachers attach significant value to understanding and they are relentless in finding ways to promote understanding, as opposed to dumbing the subject down to a set of recipes. Students who learn recipes without understanding are exposed by questions requiring insight and problem-solving abilities. They do not know how to think and are therefore not future proof.
- Using bad language. Here I am not referring to swear words which obviously have no place in education, but rather to language which is mathematically problematic. By way of example, the word “cancel” is unhelpful as it gives no sense of what we are doing mathematically. If we can cancel the 3s in
then why shouldn’t we cancel them in
? Far better to say, “divide the numerator and the denominator by 3”. When solving an equation such as 3x – 2 = y many teachers will say “take the 2 to the other side and change the sign”. This is problematic as, on the next line, we take the 3 to the other side but do not change the sign. Far better to say, “add 2 to both sides and then divide both sides by 3”. - Perpetuating an unwelcome or, worse still, a hostile environment. Most kids approach a mathematics lesson with a degree of apprehension which might arise from inter-alia: unrealistic parental expectations, the stress of just not getting it or a bad experience in the past. It is well documented that one cannot think optimally when one is anxious. We need to counter our students’ anxieties by making our classrooms as welcoming, friendly and fun as we possibly can. They need to understand that we are on their team, that we are champions for their cause.
- Playing God. Too many teachers are aloof and arrogant, holding the view that they know it all and lauding it over their students. They believe in the love of power rather than the power of love. Great teachers are confident in themselves, comfortable with the fact that some of their students are far cleverer than they are even if they do not know as much maths yet. They see their work as an act of service and they seek to form productive partnerships with those in their care. They are intent on seeing and teaching the whole person, understanding that it is easier to build a strong child than to repair a broken adult.
- Ceasing to learn. The best teachers are continually striving to grow their knowledge and to improve their practice. They model the lifelong learning they hope to inspire in their students.
In 2013 I visited Canada. The Alberta Board of Education had the following commission on a billboard:
Act wisely
Care deeply
Take joy in every day
This resonated deeply with me and has inspired me ever since. I hope it may do the same for you.
Yours in learning
Paul’s Blog – Maths Matters…
Post #1 – 14th July 2022
“Konke kuhlangane nakonke ezibalweni” – “In maths everything is connected”
Welcome to the first edition of “Maths matters…”, Paul’s blog on all things mathematical. The aspiration is to write something every fortnight or so. Sometimes we will deal with very grounded, practical tips on teaching a particular topic, while on other occasions we will be more philosophical. This one falls in the latter category.
“Mathematics is all about relationships, teaching Mathematics all the more so”.
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
