David Butler
@DavidKButlerUoA
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Lecturer at Maths Learning Centre, Uni Adelaide (my views here). Grad Dip Ed & PhD Finite Geom. Love maths and helping people learn. he/him #MTBoS #100factorial
Adelaide, Australia
Joined September 2014
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I’ve never seen this geometric proof of the difference of two squares factorisation before and I love it.
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I am (possibly unreasonably) proud of this problem I wrote for the course:
Find the length of the path shown by the solid black line in this picture.
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All maths is advanced maths. From the perspective of your student, the maths they are learning now is the most advanced maths they have ever learned. Give them the credit they deserve.
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I am not ok.
I don’t have the energy to explain why, and I don’t have the energy to listen to advice.
But I just want you to know I am not ok.
Because I also don’t have the energy to pretend I am ok.
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My 10-year-old daughter was rolled in a ball crying yesterday, because of her maths homework. Maths class should not make people feel this way.
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Me: My feelings about a maths thing.
Mathematicians of Twitter: Oh that’s easy when you realise it’s just a floximatic arklesquish.
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Good teaching isn't spectacular. It's hundreds of little things each day, each moment. Most of it nobody can tell is there unless you point it out. Mostly you can't take a photo of it, or even describe it in a tweet. But it's there, it's real work, and it does make a difference.
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Daughter C (11 years old) just finished the whole set of Jenga Views puzzles. She was very proud of herself. (Puzzles can be found here )
#Homeschooling2020
😍
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I am rather surprised that the red path is the same length as the blue path. cc
@blatherwick_sam
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It is a common belief that a person who is “good at maths” will understand everything without putting in any effort, and this belief hurts everyone.
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Just thought of this problem:
How long is the red path?
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Hi
@jk_rowling
. My daughter is 14 and absolutely loves the film Fantastic Beasts that you wrote. She keeps rewatching it. Thank you.
I am also reading The Half-Blood Prince aloud to her right now and she is thoroughly enjoying that too. Thank you again.
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Separating rational functions using partial fractions is a change of basis. I knew this already.
But today I realised that this means you can do it WITH A MATRIX.
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Don't count the dots!
Show me how you know the number of dots is a multiple of three.
#MTBoS
#MathsChat
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Apparently, “logarithm” can be translated as “thought number” (“log” as in geoLOGy or psychoLOGy, and “arithm” as in ARITHMetic).
This appeals to me. I am imagining that log_a(b) is the number that a thinks of when it sees b.
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Mathematicians: 0^0 is undefined!
Also mathematicians: Σ_{n=0}^∞ x^n/n! = e^x for all x.
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I’ve just thought of a reason why corresponding angles (when a transversal cuts parallel lines) have to be equal: it’s so that the angle you turn away from the direction of the path matches the angle you turn back.
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The students in a couple of courses are studying probability, and the issue of independent versus disjoint events came up.
#MTBoS
Only now did I remember I wrote something about that:
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How would you explain why the red rectangle and the blue parallelogram have the same area?
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You don’t need anything too fancy to show students how scatterplots work. Here are some ideas.
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The word "prime" means "first".
The number 6 appears third when you count by 2's, second when you count by 3's and first when you count by 6's.
The number 7 appears first when you count by 7's. That's it.
Prime numbers never appear anywhere but first when you count by something.
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I learned this very cool thing today…
If the areas of the two triangles in a crossed trapezium are A and B, then the areas of the two triangles that complete it to an ordinary trapezium are both √(AB).
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I found this book my wife and I once made on the shelf today. I’m rather proud of it. It’s called “Building Numbers”.
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Teacher wants my daughter to learn her times tables up to 12 this term. So I’ve extended the Prime Climb cards up to 144 and included a colour for 11.
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Don't count the dots!
Show me how you know the number of dots is NOT divisible by five.
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I had this idea during the night for showing a particular quadratic equation has no solution. I think it’s fun.
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I learned how to use left_join() in
#RStats
. Also group_by() followed by summarise(). I'm very proud of myself.
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People ask me, as a mathematician, if I see numbers everywhere and in everything. The answer is no. Absolutely not.
I see shape and colour and pattern and beauty and awe. The world is an amazing place full of wonder. That’s what I see, and it makes me the mathematician I am.
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Maths suffers a lot from being useful. Because it often is useful, people insist that it must always be. And it’s a big burden to bear.
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My understanding of the slope of a line got so much better when I realised it was a multiplier rather than just a random number produced by a division.
That is, it's not so much that
[slope] = (change in y) / (change in x)
but that
(change in y) = [slope] * (change in x).
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While riffing off a problem from
@Cshearer41
at
#100factorial
yesterday, we (including me and
@zithral
) discovered this fact about equilateral triangles, which I think is very cool.
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I just realised the "=" sign means the OPPOSITE of what many students think it means. Many students think it's an indicator that something was done, but when you write an "=" sign, the point it is making is that actually you haven't actually done ANYTHING because it's THE SAME.
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It's official and I can share it publicly:
The two MLC lecturer positions are to remain as they are with no need to reapply. My job is saved.
The Uni of Adelaide has proposed that the two Maths Learning Centre lecturer positions (me and my colleague) be made redundant to be replaced with one professional position we would apply for in competition. There is a risk of losing many of the things I do.
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Serious question about indefinite integrals:
Is ∫x dx + 5 = ∫x dx?
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Don't count the dots!
Show me how you know the number of dots is divisible by four.
#MTBoS
#MathsChat
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I’ve been angry all day and I can’t turn it off. Angry to the point of tears.
Because there are still people who believe their job as a university lecturer is to weed out those who don’t belong.
They never say it in those words, but that’s what they mean.
And so I’m angry.
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This photo I took years ago shows water flowing in a channel that splits into two and then rejoins. You can clearly see the change in speed in the split part of the journey. It was the first time I ever felt I understood electric current halving when a wire in a circuit splits.
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Unsolicited email from a student. I really needed this today.
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After observing some others teaching yesterday and also reflecting on my own experience yesterday, I have this to say to anyone helping students one-on-one:
SHUT UP.
They need time to process the last thing you said. They have thoughts and you need to hear them.
JUST SHUT UP.
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I've been trying to be really vulnerable when I do maths here on Twitter:
* admitting things I don't know,
* doing my thinking live rather than waiting till I get the final answer
* leaving wrong thinking rather than deleting
* asking people to explain
It's been interesting.
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This is my most-retweeted and most-liked tweet ever. Not hard to see why...
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#MTBoS
. We need to talk seriously about how hard the job of teaching can be. Lest we give the impression that finding it tough is wrong.
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We need to stop awards for teaching celebrating martyrdom. You shouldn’t have to devote every waking moment to teaching, to sacrifice your family and your resources and your health, in order to be recognised as a great teacher. It’s not a healthy narrative and we keep telling it.
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Sends email to myself.
Email notification sounds.
Ooh! Who could that be?!
Oh.
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I think deep down, most of us maths educators subconsciously believe in the idea of a perfect explanation for each concept that students will be guaranteed to understand.
I'm here to remind you such a thing does not exist.
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In pure maths things are agreed to be true when they are proven. But a proof is a written form of communication. So in pure maths things are not agreed to be true until they are *communicated*. I kind of like that communication is at the heart of mathematical truth.
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I hate it when people assume that because I choose to explain things with pictures and everyday terminology, that I don’t know the technical details.
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In the MLC today.
Me: Three quarters minus two thirds is... [looking up]... a twelfth.
Student: What are you *looking* at?
Me: The clock.
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What *is* it with people just giving students formulas for everything?
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Completing the square is not a method for solving quadratic equations, it’s a method for rewriting quadratic expressions. Yes this is an important distinction.
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Reprinting handouts for the coming semester, I was reminded of my “Useful Facts About Angles” and “2D and 3D Trigonometry Facts” handouts that I am rather proud of.
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I think this is the biggest maths idea I've ever had.
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I love students who are brave enough to ask questions about why their answers are wrong. I learn so much from those students.
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When people ask for an explanation in maths, they often do not want a proof. They want something that makes them feel like they understand, and this is often a story or a picture or an analogy or a rationale, not a proof.
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If you draw lines from the corners of a square to the centre of each edge, you can make rhombuses of area 1/5, 1/4, 1/3
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Slapping a cute cartoon kid on your maths task does not make it engaging.
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A rant about school homework: if my boss regularly gave me a task in the last hour of the work day that was due by 9am the following morning, I would be discussing their unreasonable management practices with their superiors.
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A child is not being “lazy” if they choose to use the supports that are explicitly outlined in their IEP.😡
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@howie_hua
I could give them a common denominator:
3/4=15/20, 4/5=16/20.
But they’re just one 20th apart.
So I’ll multiply top and bottom by 2.
3/4=30/40, 4/5=32/40.
And now 31/40 is between them.
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I must remind myself:
Not doing as good a job as I hoped is not the same as doing a bad job.
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Competitive four-colouring at
#100factorial
. Two teams. Colour a region so that same colours don't touch by edge. Last valid move wins.
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A Desmos graph I've been playing with, to zoom in on a point on a function to see how the slope changes as you move along.
#MTBoS
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I am worried that reasoning is seen as an extension or enrichment as opposed to something that all the students should be doing.
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Rather fond of this method of completing the square, especially because you can use it to go direct from factored form to vertex form.
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@BeneFactumGames
Fascinating. You can actually do it without limits. You just need to cut it into thin enough slices so that you can move the triangle end of each from the back to the front.
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When you draw three skew lines in space, and then find all the lines that meet all three of them, these lines are called a REGULUS.
All the lines of a regulus together always form a hyperbolic quadric.
(Oliver is fascinated by my regulus.)
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Every time a door closes or a window opens, you change the topology of the building.
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Daughter K wanted to do an art activity together so I scrolled through
@anniek_p
’s
#mathartchallenge
. She was inspired by the isometric drawing and this spiral is what she produced.
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One aspect of society’s view of maths that I particularly hate is that if you’re good at maths it has to be the ONLY thing you’re good at. You’re especially not allowed to be artistic or literary if you’re good at maths.
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I’ve just realised 1/0 and 0/0 are undefined for OPPOSITE reasons.
1/0 is undefined because there is no number it can be.
0/0 is undefined because it could be any number.
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I reckon some students say “I’m bad at maths” so that you won’t try to help them, because their usual experience of being “helped” with maths is that it just makes them feel worse.
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How cool would it have been if instead of "the sample space", we had called it "the realm of possibility"?
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I've just designed a new set of data cards for use in stats workshops, especially with Health Science students. I'm very proud of them.
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BRACKETS HOLD THINGS TOGETHER
Brackets are not an operation
Brackets do not indicate multiplication
Brackets are not things you expand
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What I hope is an interesting task:
Draw a path following the lines of this grid that is exactly 1m long.
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@ddmeyer
It’s 17!
I know because I saw a word problem with the numbers 1 and 16 in it and I added them.
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Students told me today that one of their lecturers literally told the class in the first lecture that at least a third of them would fail. In what fucking world is that ok?! (Yes I said “fucking”.)
I’ve been angry all day and I can’t turn it off. Angry to the point of tears.
Because there are still people who believe their job as a university lecturer is to weed out those who don’t belong.
They never say it in those words, but that’s what they mean.
And so I’m angry.
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Over the past day this truth has become clear to me: A lot of little things combined to produce my daughter’s sense of hopelessness about her maths. It would be so easy for any teacher to unknowingly get a child to this stage.
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To all who are also not ok:
I want you to know it is alright not to be ok. It was always alright not to be ok. The year 2020 doesn't have to be the reason you are not ok.
I want you to know there is no time pressure for how long it takes to be ok.
I wish the best for you.
I am not ok.
I don’t have the energy to explain why, and I don’t have the energy to listen to advice.
But I just want you to know I am not ok.
Because I also don’t have the energy to pretend I am ok.
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I have just had the moment of clarity I have been praying for:
I am sure this is still the job I am meant to have, and the place I am meant to do it. There are more people I will help here.
So I choose to fight. I vow to stay, and to secure the resources needed for this work.
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I tweeted this.
I was accused of plagiarism because it was on Reddit.
When I went to look, Redditter had admitted to copying me so it was removed there.
I googled "222 feel" to see if anyone else had copied me, and learned (to my chagrin) it's an "angel number".
Life is weird.
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I have made a picture book "Maths Sheep Play Sheep" in honour of Children's Book Week, inspired by "Where is the Green Sheep?" by Mem Fox and Judy Horacek. I hope you like it.
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"Equilateral triangle" is way too posh. I wish we called them "regular triangles".
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I want to say a big thank you to those who follow me here. I have a lot to process in my line of work, and it’s nice to be able to process it “out loud” and know it’s not just flying into the void but at least a few people are seeing it, and a few of those get something from it.
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I wish people understood better that there is no such thing as a maths brain and even if there were, having one wouldn't make you a superior being.
I wish mathematicians understood this too.
What are the mathematical ideas you wish the general public understood better?
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Today is C and my 20th wedding anniversary. She is the strongest person I know and my best friend. I am so happy to have spent these last 20 years with her, and so happy to know it is still just the beginning of forever with her.
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Any old common denominator will do. It doesn't have to be the lowest.
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