3blue1brownThere's More Than One Way to Read Maths
Audience:
Tags: algebramathematics-educationsomepedagogy
What does an expression like (2+3-4+5) actually mean?
At first glance, this seems like a very simple question. But the expression can be represented by several different binary trees, corresponding to different ways of grouping and reading the operations. Our conventional notation hides much of this structure, and the usual rules of BIDMAS tell us how to evaluate an expression without necessarily telling us the most useful way to see it.
This video explores an alternative perspective on algebraic notation.
Starting from binary operations and syntax trees, I look at what happens when subtraction is rewritten as the addition of a negative, and division as multiplication by a reciprocal. Once these inverse operations are normalised, familiar algebraic properties such as associativity and commutativity become much easier to exploit.
Instead of reading
[ 2+3-4+5 ]
as a sequence of binary instructions, we can read it structurally as a sum whose terms are
[ +2,\quad +3,\quad -4,\quad +5. ]
Likewise, products can be understood as collections of factors rather than simply chains of multiplications and divisions.
This leads to a broader question: how much of the difficulty of algebra comes from the mathematics itself, and how much comes from learning to read the notation?
I am a secondary mathematics teacher in the UK, and this question has led me to develop an approach I call SPI Maths — Sums, Products and Indices. Towards the end of the video I briefly show some experimental tools I have been building to make mathematical structure more visible: a syntax-tree builder and an algebraic transformation tool based on command verbs.
These are not presented as a finished solution, but as examples of what mathematics education might look like if structural reading were given a more explicit role. https://spimaths.com/demo3
The video finishes with several pedagogical hypotheses arising from this perspective and an invitation to think differently about something we usually take for granted: https://spimaths.com
How should we teach people to read mathematics?
Analytics
Comments
Instead of just memorising the BIDMAS rule, this video shows the various possible syntax trees an expression may take and how the BIDMAS fails to expose the algebraic properties. Not only that, you showed other two ways which are better than the BIDMAS rule. Keep on doing the good work 👍
The topic is clearly explained and the plentiful of examples make it easy to understand. However the mic quality and the one-time recording make it more fitting for a conference than a proper video, as the rythm suffers a lot from it.
The slides are quite nice and easy to read. The conference must be really enjoyable.
Good presentation, but I really didn’t understand what the purpose of this video was until near the end. For most people familiar with math, the intro is trivial and it is not clear what we can learn from it. I think it would really help to state upfront what the purpose of the video is. As far as I understand, it’s about developing mental techniques to read math expressions and perform simplifications mentally. My apologies if I misunderstood the goal.
If this is the case, I would recommend ending the video with a moderately complex expression and an explanation how one could deal with it and simplify it mentally. The encore section does that in some way.
Some minor comments:
1:48: I would have expected that XOR(T, F) = T.
15:45: Indices - this confused me a bit, first, the hat operator is really just a hack used by some computer languages and, second, indices isn’t the term used in other countries, maybe that could be mentioned, also the alternative terms BODMAS and PEMDAS.
One other thing: If you want to make more such videos, I would recommend investing in a good microphone. The sometimes poor sound quality detracts a bit from the message of this video.
- I had trouble understanding what the point of this video was until the very end, so I think it could benefit a lot from moving some of the end discussion to the beginning. In particular, I thought that video might be about different possible orders of operations, rather than different ways one might parse an expression using the order of operations. So an intro along the general lines of:
“To avoid ambiguity in expressions, mathematicians have created a convention in the order of operations so that they don’t need to write a confusing number of brackets every time they create an equation. We currently teach this convention in a very procedural way, which doesn’t really give any insight into the algebraic manipulations that are valid for a given expression. What if there was a better way to teach students how to read algebraic expressions that follow the order of operations?”
would be very helpful.
- Who is your target audience for this video? It seems to be math educators - if so, you don’t need to spend much time evaluating basic expressions in 5 different ways. Similarly, you probably don’t need to spend as much time focusing on commutativity or associativity.
Motivation: 6 / 9 – The motivation was good, could be explained more precisely and quickly.
Clarity: 6 / 9 – Very thorough explanation, which lacked a bit of storytellingness and awarness of it’s target audience.
Novelty: 5 / 9 – Interesting topic, that many people CAN find relatable. But I’m not sure whether the video tried to make it relatable. You could, for example, try to relate the problem to the many online debates on what is the answet to some expressions, especially in the beginning of the video.
Memorability: 2 / 9 – Sadly the technical side of things was not great and the visuals are not memorable at all.
Overall: 4.75 / 9 – The premise was quite good. But what threw me off was that, in my opinion, this video is for… no one. I fell that, for students of middle school or high school it might be to hard to get through because of the academic language you use. But for undergraduates or academics it might be just too… simple? I feel like there were just too many examples – not everything must be calculated on screen, some things could be skipped to keep the pace.