God and Nature 2026 #2
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Making it Add Up
By Mike Clifford
In my previous column, I reflected on my new secondment as part of the University of Nottingham’s Educator Academy. It’s been good to spend more time thinking about teaching and learning. Recently, my thoughts have turned to mathematics education, particularly since my engineering department is planning to take our mathematics teaching in-house rather than have it delivered by the maths department in the very near future.
Mathematics was my first love. When I was at school, I wanted to be a maths teacher, right up until I spoke to my (excellent) maths teachers about their careers. Not wanting to be an accountant, and unable to think of any other jobs for mathematicians, I drifted into an engineering degree. Although my technical knowledge was somewhat limited—I didn’t know the difference between a nut and a bolt—I found the course easy but unsatisfying. I remember lectures where after deriving some beautiful equations, the lecturer would say that because we were engineers, we’ll cross out any terms that we don’t understand, we’ll make a series of crude assumptions, apply a sizable safety factor, cross our fingers, and hope for the best. I wept frequently at this pragmatic approach.
After my undergraduate degree, I eloped with mathematics and had a passionate and chaotic three-year affair with nonlinear dynamics, culminating in a PhD entitled "The parametrically excited pendulum: a paradigm of nonlinear systems”. It was wonderful to play with bifurcations, delve into braid and knot theory, and to see chaotic blue sky catastrophes appear on my computer screen. But all good things must come to an end, and after several postdoc positions and a near-miss at getting a lectureship in mathematics, somehow, I found myself back in the greasy world of engineering.
Don’t get me wrong, I’ve grown to love my job; I’ve come to terms with the pragmatism of engineering and only occasionally am I tempted by particularly sexy differential equations which leap off the page with those “come up and integrate me sometime” double dots. Beauty is in the eye of the beholder, and I’m aware that many of my students do not share my passion, and, for some, mathematics is a foreign language. My eldest child describes maths as a box of tricks and mathematical techniques as a bunch of keys. If you use the right technique, a problem can be unlocked in seconds, but if you use the wrong one, then it’ll just lead to frustration and a C minus. Working out which technique to use is vital, but then so is knowing how to use it. Perhaps we need to look at different ways to teach mathematics.
My three stepchildren were educated at a Steiner School. The teaching methodology was based on the educational philosophy of Rudolf Steiner, with a holistic style of education intended to develop pupils' intellectual, artistic, and practical skills, and a focus on imagination and creativity. Unfortunately, there aren’t many of these schools left in the UK. I found the treatment of mathematics particularly interesting. Take multiplication, for example. The times table was introduced with a story about a miller grinding corn into flour in his windmill, which was drawn in chalk on a traditional blackboard. The miller had to deliver six bags of flour to the castle, but his cart could only take three bags at once. So, the question was posed, “How many times does the miller have to make the journey?” By using this visual, narrative approach, the word “times” was given a concrete meaning, and the sails of the windmill formed the X symbol.
All well and good for primary education, but the method can be extended into secondary school and beyond. For the interested reader, there’s lots to get your teeth into here: https://www.waldorf-resources.org/en/vertical-curriculum/mathematics
I’m particularly impressed that at twelfth grade, “Narrow mathematical matters should be followed up by philosophical reflection upon the foundations of mathematics. This could involve consideration of the relative merits of, e.g. formalism, Platonism, constructivism, intuitionism.” Amongst the reference list, there is a curious book, Science Without Numbers: A Defence of Nominalism, which I managed to locate in my university library. This book on the philosophy of mathematics by Hartry Field is a defence of Nominalism, the view that mathematical objects such as numbers do not exist, and stands against Platonism, the view that mathematical objects exist in their own right. I confess that I didn’t get very far in my attempt to read the book. With his emphasis on concrete examples, I suspect that Steiner would line up with Plato rather than with Field, which led me to wonder if there might be a market for a book entitled
“Engineering Without Equations”. If I get around to writing such a book, I’ll certainly need some co-authors. First port of call will be the current President of Western New England University, and former professor of industrial and systems engineering, Joseph Hartman, who makes a strong case for Waldorf education here: https://www.waldorftoday.com/2017/12/why-this-skeptical-engineer-trusts-waldorf-education. In the article, Hartman quotes G. Wayne Clough, who realized that some of the best engineers “might not be the ones who could solve the calculus equation better, but they could define the problem that the calculus had to solve better than anyone else.”
What does our faith have to say about all of this? I could dodge the question and sign off with the witticism that there are no equations in the Bible, but there is a whole book of Numbers… But I think I owe you, dear readers, a bit more than that. In my life, I’ve often felt the tension between having a practical job in engineering and the more creative field of mathematics. Blame it on my Protestant work ethic if you like; the need to do something of worth rather than to sit and wonder at the mysteries of the universe and creation. I’ve always been a Martha rather than a Mary, but in Luke 10:42 Jesus says, “Mary has chosen the better part and it will not be taken from her .” So when I write that book, maybe I’ll hide a few equations away in an appendix, just for the interested reader.
Mike Clifford is an Associate Professor in the Faculty of Engineering at the University of Nottingham. His research interests are in combustion, biomass briquetting, cookstove design, and other appropriate technologies. He has published over 80 refereed conference and journal publications and has contributed chapters to books on composites processing and on appropriate and sustainable technologies.
By Mike Clifford
In my previous column, I reflected on my new secondment as part of the University of Nottingham’s Educator Academy. It’s been good to spend more time thinking about teaching and learning. Recently, my thoughts have turned to mathematics education, particularly since my engineering department is planning to take our mathematics teaching in-house rather than have it delivered by the maths department in the very near future.
Mathematics was my first love. When I was at school, I wanted to be a maths teacher, right up until I spoke to my (excellent) maths teachers about their careers. Not wanting to be an accountant, and unable to think of any other jobs for mathematicians, I drifted into an engineering degree. Although my technical knowledge was somewhat limited—I didn’t know the difference between a nut and a bolt—I found the course easy but unsatisfying. I remember lectures where after deriving some beautiful equations, the lecturer would say that because we were engineers, we’ll cross out any terms that we don’t understand, we’ll make a series of crude assumptions, apply a sizable safety factor, cross our fingers, and hope for the best. I wept frequently at this pragmatic approach.
After my undergraduate degree, I eloped with mathematics and had a passionate and chaotic three-year affair with nonlinear dynamics, culminating in a PhD entitled "The parametrically excited pendulum: a paradigm of nonlinear systems”. It was wonderful to play with bifurcations, delve into braid and knot theory, and to see chaotic blue sky catastrophes appear on my computer screen. But all good things must come to an end, and after several postdoc positions and a near-miss at getting a lectureship in mathematics, somehow, I found myself back in the greasy world of engineering.
Don’t get me wrong, I’ve grown to love my job; I’ve come to terms with the pragmatism of engineering and only occasionally am I tempted by particularly sexy differential equations which leap off the page with those “come up and integrate me sometime” double dots. Beauty is in the eye of the beholder, and I’m aware that many of my students do not share my passion, and, for some, mathematics is a foreign language. My eldest child describes maths as a box of tricks and mathematical techniques as a bunch of keys. If you use the right technique, a problem can be unlocked in seconds, but if you use the wrong one, then it’ll just lead to frustration and a C minus. Working out which technique to use is vital, but then so is knowing how to use it. Perhaps we need to look at different ways to teach mathematics.
My three stepchildren were educated at a Steiner School. The teaching methodology was based on the educational philosophy of Rudolf Steiner, with a holistic style of education intended to develop pupils' intellectual, artistic, and practical skills, and a focus on imagination and creativity. Unfortunately, there aren’t many of these schools left in the UK. I found the treatment of mathematics particularly interesting. Take multiplication, for example. The times table was introduced with a story about a miller grinding corn into flour in his windmill, which was drawn in chalk on a traditional blackboard. The miller had to deliver six bags of flour to the castle, but his cart could only take three bags at once. So, the question was posed, “How many times does the miller have to make the journey?” By using this visual, narrative approach, the word “times” was given a concrete meaning, and the sails of the windmill formed the X symbol.
All well and good for primary education, but the method can be extended into secondary school and beyond. For the interested reader, there’s lots to get your teeth into here: https://www.waldorf-resources.org/en/vertical-curriculum/mathematics
I’m particularly impressed that at twelfth grade, “Narrow mathematical matters should be followed up by philosophical reflection upon the foundations of mathematics. This could involve consideration of the relative merits of, e.g. formalism, Platonism, constructivism, intuitionism.” Amongst the reference list, there is a curious book, Science Without Numbers: A Defence of Nominalism, which I managed to locate in my university library. This book on the philosophy of mathematics by Hartry Field is a defence of Nominalism, the view that mathematical objects such as numbers do not exist, and stands against Platonism, the view that mathematical objects exist in their own right. I confess that I didn’t get very far in my attempt to read the book. With his emphasis on concrete examples, I suspect that Steiner would line up with Plato rather than with Field, which led me to wonder if there might be a market for a book entitled
“Engineering Without Equations”. If I get around to writing such a book, I’ll certainly need some co-authors. First port of call will be the current President of Western New England University, and former professor of industrial and systems engineering, Joseph Hartman, who makes a strong case for Waldorf education here: https://www.waldorftoday.com/2017/12/why-this-skeptical-engineer-trusts-waldorf-education. In the article, Hartman quotes G. Wayne Clough, who realized that some of the best engineers “might not be the ones who could solve the calculus equation better, but they could define the problem that the calculus had to solve better than anyone else.”
What does our faith have to say about all of this? I could dodge the question and sign off with the witticism that there are no equations in the Bible, but there is a whole book of Numbers… But I think I owe you, dear readers, a bit more than that. In my life, I’ve often felt the tension between having a practical job in engineering and the more creative field of mathematics. Blame it on my Protestant work ethic if you like; the need to do something of worth rather than to sit and wonder at the mysteries of the universe and creation. I’ve always been a Martha rather than a Mary, but in Luke 10:42 Jesus says, “Mary has chosen the better part and it will not be taken from her .” So when I write that book, maybe I’ll hide a few equations away in an appendix, just for the interested reader.
Mike Clifford is an Associate Professor in the Faculty of Engineering at the University of Nottingham. His research interests are in combustion, biomass briquetting, cookstove design, and other appropriate technologies. He has published over 80 refereed conference and journal publications and has contributed chapters to books on composites processing and on appropriate and sustainable technologies.