The other day I did a workshop with students from Advanced Mathematical Economics III, which is more or less a pure maths course for economics students. It covers such things as mathematical logic, analysis and topology – all a bit intimidating for students who started out the degree with almost no mathematical background!
We had just spent an hour looking at relations: the definition as a subset of the set of pairs, the common usage as statements in natural language, various ways to visualise them, examples and non-examples of various properties relations might have, and proofs involving those properties. After all this point, one of the students said, with some surprise, “It really feels like it’s just playing around here.”
The student was right, of course, and I told him so! Pure maths is play. You have some ideas and you fiddle around with them to see how they fit together, how they’re similar or different, what other things are really the same things in disguise. Most of the time there’s no particular goal in mind, but even when there is, you often get distracted by something cool and end up somewhere totally different. This is exactly what play is. Watch any child in a sandpit and you’ll see the same behaviour.
It suddenly occurred to me that I have always seem maths as play, and have liked it the least when it looked like work. More than this, the maths courses I have had the most academic success in were the ones where I allowed myself to see it as play. These were the courses where I took the concepts I was learning and pulled them apart and put them together in new ways, where I tried to do things that may or may not be possible just to give it a go, and where I drew lots of pictures in vivid colour just because. And importantly, where I didn’t question what the point of any of it was but just ran headlong into whatever crazy idea the lecturer presented next to see what happened with it.
So perhaps the best way to approach a pure maths course is to see it as play! Surrender to “just playing around” and see where it goes. That’s the advice I gave to these students, and I’m hoping it helps them to have the freedom to learn.
Students make a lot of mistakes when doing their maths, but sometimes they will make two mistakes in such a way that their final answer is still correct. This happened last week with one student quite spectacularly, because his doubly wrong method of doing a particular problem always produces the correct answer.
Let me explain: the Maths 1A students are currently learning about vectors in Rn and one of their assignment questions gives them several lists of vectors and asks them to decide if they are linearly independent or linearly dependent.
What they are supposed to do (based on what they are shown in the lectures) is put the vectors into a matrix as columns, then do row operations until they can tell where the pivots will be. If every column has a pivot, then the vectors are linearly independent; if there is a column with no pivot, then the vectors are linearly dependent. Check out this example to see how it works (it has a little extra for later on).
The reason for this method goes right back to the definition of linear independence: the vectors v1, …, vr are linearly independent when the equation x1v1 + … + xrvr = 0, has only the trivial solution of x1 = 0, … , xr = 0. If there are any solutions where any of the xi‘s are not zero, then they are linearly dependent.
If you write out the vectors in full using their coordinates, then you find that using the first coordinates, you can make a linear equation involving the xi‘s, and similarly each successive coordinate produces a separate equation. So solving the vector equation is the same as solving n linear equations. And the students know how to do this: you put the coefficients of the equations into a matrix with one row per equation and do row operations. The final matrix represents a reduced set of equations, and if you can get to a stage where each row has a 1 in one spot and 0’s in the others then you’ve got x1 = 0, …, xr = 0. This would mean your original vectors had to be independent. If any one variable doesn’t have a pivot, then you can let it be any value at all and still find a solution. This is called a free variable. Since it can be anything, it could be something other than zero, and so your vectors will be linearly dependent.
So this is what the student was supposed to do. However, what they did do was put the vectors into a matrix as rows, do row operations, and then look to see if there was a row of zeros. If he got a row of zeros, then he said the vectors were linearly dependent, and if he didn’t, then he said the vectors were linearly independent.
His first mistake was to put the vectors in as rows. I am often repeating the mantra “vectors are columns, equations are rows” to students to remind them that for most situations, vectors really ought to be columns. We usually mutliply matrices on the left of a vector (Ax) which would only make sense if the vector was a column, and also this arrangement corresponds exactly to doing a specified linear combination of the columns of A. Finally, in this specific situation, the matching coordinates of each vector together make an equation, and equations are definitely rows, so this forces the vectors to line up their coordinates in rows. That is, they have to be columns (if we are to be solving equations anyway).
His second and much more fundamental mistake was to think that a row of zeros meant there had to be a free variable. (I know he was thinking this because he actually said it to me.) This is wrong because if a matrix does represent a set of equations, then whether you get rows of zeros at the end is actually not strongly related to whether there are free variables. Firstly, a row of zeros with a nonzero number in the answers position indicates no solution at all, independently of what the rest of the matrix is doing. Secondly, if there’s a pivot in every column, then there will be a unique solution regardless of how many rows of zeros there are at the bottom. Whether you get infinitely many solutions is all about the pivots in the columns, not about the zeros in the rows!
It’s not surprising that the student had this misconception because many high school students only ever see square matrices, and a row of zeros in a square matrix prevents there being a pivot in one of the columns and so does in fact indicate a free variable. But it doesn’t apply for non-square matrices, especially ones with more rows than columns.
I explained all of this to him and he was happy with what I said. But then he frowned and asked, “Then why did I still get the answers right?” Why indeed?
If my student had put his vectors in as columns and looked for zero rows, he would have gotten his answers wrong. This is because with the vectors as columns, the rows are equations and the aim is to solve the equations and look for free variables. Since zero rows do not always force free variables, he would be looking for the wrong thing and would have been wrong a lot of the time.
If my student had put his vectors in as rows and looked for columns without pivots, he would also have gotten his answers wrong. This is because pivots are supposed to refer to variables in equations, whereas this matrix wouldn’t actually represent equations at all, letalone ones related to the definition of linear independence.
However, he did neither of these combinations and instead put the vectors in as rows and looked for rows of zeros, and he got his answer right every time! The key to resolving this paradox is to figure out what the row operations represent when the rows of your matrix are vectors rather than equations.
Row operations basically perform linear combinations of rows. So when you do a sequence of row operations, you are actually doing a linear combination of the original rows. Therefore any row produced by this process is a linear combination of the original rows. So if you are able to produce the zero vector then it is actually possible to produce the zero vector by linear combinations! And it’s not the trivial one either. Hence the original vectors must have been linearly dependent. Amazing huh?
An important question arises: what if you wanted to know what the actual linear combination was? With the vectors-as-columns approach, you are literally solving to find the xi‘s, and so at the end if you pick a nonzero value for the free variables, you can find the rest of them and there you have it!
With the vectors-as-rows approach, the linear combinations you do are recorded in the row operations. We need a way to keep track of the linear combinations / row operations we have done. One way to do this is to reason that if the original vectors were the standard basis, then whatever final vector we got would tell us what linear combination we had done (for example, (1,3,-2) = 1(1,0,0) + 3(0,1,0) -2 (0,0,1)). So why don’t we do the same operations on the standard basis as we do on the original matrix?
What this means is that if you place an identity matrix next to your original matrix and do the same row operations on both, then whatever vector is next to the row of zeros when it happens will be the linear combination of the original rows that produced the zero vector! Check out the same example as before but by this new method.
Isn’t it amazing the things you can learn by being wrong first!
This comment was left on the original blog post:
Stephen Wade 20 May 2014: It’s funny how convention comes into play and that you could prove that either approach works. I had to check quickly, and I think this is right, that if you put vectors as rows in an m x n matrix A, then if dim ker A^t > 0 means you have linear dependence. Using rank theorem, dim ker A^t = m – dim col A = numbers of rows – number of rows with pivots = number of rows of zeros in the rref of A. So if the number of rows in rref of A > 0, you should have linear dependence. Cool 🙂
It is a well-known truth that assessment drives learning. Students will often not learn a particular topic or concept unless it is assessed by an assignment or exam. Fair enough – often students are not choosing to do a particular course for the sheer love of it, are they?
However, many lecturers take this truth just a little further and subscribe to the belief that assessment can actually teach. They put quite a bit of faith in what a simple assignment question can do for students: a lot of them believe that a well-chosen assignment question has the ability to teach students amazing truths about maths. They imagine the student doing the assignment question, struggling through it, and coming to an epiphany where suddenly everything makes sense. I have actually had lecturers in the past telling me about the great question they’ve written and how it will teach the students something cool. I think this is just a little unrealistic.
Through years of observation of students, it seems to me that actually, assignments don’t teach people, people teach people. Let me give you two examples from the last couple of days to illustrate.
The Maths 1B students are currently studying orthogonal projection and they have a MapleTA (computer-based) assignment due today. In it, there are some questions that give them a basis for a subspace and a vector and ask them to project the vector onto the subspace. The students need to do this and then type their answers into the computer. In the particular question I have in mind, there are two vectors to project onto the subspace, and the second one doesn’t change when you calculate the projection. What this means is that the second vector is already in the subspace, which is why projecting it into the subspace doesn’t do anything. I’m pretty sure that the writer of the question is hoping that students will notice this and wonder why it’s the same and remember that fact about vectors already in the subspace and feel the warm glow of learning.
But of course they don’t learn. Talking to students yesterday, they didn’t even notice the answer was the same as the input. They just noted their answer was correct and moved on. Luckily for these students, I was there to point it out and ask them why they thought that might happen and help them find the bit of their notes that discussed this concept.
My second example comes from last week’s written question. It asked the students to prove that each vector in a subspace can be written in terms of the basis in a unique way. This is quite a fundamental idea which is not covered explicitly in the lectures and it’s a pretty safe bet that the writer of that question was hoping that the students, through doing the question, learned this concept. And also, I reckon they also are hoping that the students learned how to prove that something is unique.
Only they didn’t of course. Almost every student who visited the MLC had dutifully written down the question, but the rest of the page was blank. They had no idea how to even start. Even those who had made a good start by writing down the definition of basis had no clue where to go from there. Since they had no clue how to start, they had no hope of finishing, and absolutely no hope of learning anything! Even those students who only needed a little prompting to solve the problem still had to ask about what was really going on.
See? The assignment question was certainly the fuel that was needed to learn those things, but it wasn’t the assignment question itself that did the teaching – it was me, or sometimes the students’ friends. It was the discussion with others that helped the students learn. They needed someone there to help them notice what was going on, and to help them turn it into a lesson.
I’m not saying you can’t choose good assignment questions that make it more likely for students to learn, I’m just saying that without also organising an opportunity to talk to someone as well, students will often not learn anything. Indeed, they often won’t do the question at all. So if you’re a teacher do remember: assignments don’t teach people, people teach peole.
This comment was left on the original blog post:
Sophie Karanicolas 21 November 2014: David, I have only just come across this amazing space that you are writing and creating in! I am really enjoying reading your blogpost on “People teach people”. I couldn’t agree more. Every teaching and learning initiative and every assessment requires people presence. The teacher as the coach and guide. I realised very soon in my career that students need to be well prepared and coached for any kind of assessment. They need to trial assignment and exam type questions and workshops these with teachers and peers to help them develop the critical thinking skills they need to undertake the challenges of assessment with confidence. This also enables them to engage with the content and reach deeper levels of understanding. I came across a student who was so disheartened because she couldn’t seem to understand a topic no ‘matter how many times [she] rewrote the notes’. (There are students out there who still believe that by rewriting notes they will somehow learn. Then they come across a question or scenario where they need to use their knowledge and they have that ‘blank page’ you referred to in your post.) Once this student started working with the teacher and her peers to answer these questions she immediately started to ‘get it’! We couldn’t wipe the smile off of her face and couldn’t find enough questions to give her! It opened the flood gates.. she had her ‘aha’ moment. So yes… people teach people!!!!
I love wrapping presents. I’d like to say it’s because of the warm glow I have inside from giving a gift to someone else – and that feeling is certainly there to an extent – but I’m sorry to say the main reason is because I like the process of wrapping presents itself.
I like putting the present on the paper and making a judgement of how much paper to cut; I like using the scissors like a knife to cut a clean edge; I like folding the edge of the paper so that it looks nice and clean when you fold it over the present; I particularly like the part where you do the fold-in-the-sides-then-fold-up bit on the sides; and most of all I like the part where it’s all finished and your present is neatly encased in a piece of paper just the right shape with all the bits folded in neatly.
Yes, I know I’m weird.
But I reckon I’m not that weird. My daughter at 10 years old, still likes reciting the alphabet, though she learned to do this 6 years ago. My other daughter at 5 years old, will write her name over and over and over and over, seemingly getting pleasure out of the simple act. A musician will sometimes play a song they know well, for the sheer pleasure it, and almost any person will go up to a piano and play chopsticks. Many people I know like the experience of making scrambled eggs, no matter how many times they have done it before.
It seems that all people derive some pleasure in doing things well that they know how to do well, even though they have done it before. There is something about the repetition that gives you a sense of pleasure. Perhaps your brain likes to have the electrical signals pass down the well-worn paths where it’s not so much effort. Perhaps the experience helps you remember the buzz when you learned it for the first time.
I think perhaps the second reason is pretty accurate because I see myself doing it all the time in my work as well: guessing eigenvalues, calculating integrals, adding fractions and drawing conics. I love them all. I jump at the chance to do them with students in the MLC because I love doing them, no matter how many times I’ve done them before. And every time I do them, I remember with pleasure the first time I figured out how to do them myself.
But whatever the reason, I do get pleasure from doing the integral of ex cosh x or (cos x)2 or 1/(x2 – 1) – integrals I have done a hundred times – and it coming out to the answer it ought to. It’s the same pleasure I get from wrapping a present.
Sometimes you just enjoy doing something you know how to do.
What do you think of when you hear the word “basic”? For example, when you see a topic in a maths textbook entitled “Basic Algebra”, what comes to mind?
In that context, most people interpret the word basic to mean “easy” or even “babyish”. They either feel put out that they are expected to go back to the beginning, or they feel embarassed that they need to.
Well it’s time to publically dispell this myth – basic does not mean easy or babyish, especially when it comes to maths!
In maths, the stuff you learn first is the stuff on which all subsequent learning is built. Something is basic in this context because it is the base we use to build other things on top. For example, on top of arithmetic we build algebra, and on top of algebra we build calculus, and on top of calculus we build differential geometry. The dizzy heights of fancy differential geometry are only possible because somewhere down below there is a strong base in arithmetic on which to build it. Without those basics, the rest of maths wouldn’t exist. That’s actually an exciting concept!
There’s even more to this concept of basic as foundational. If something is basic, that means it’s right at the bottom of the building. So there’s nothing below it to build upon. And if there’s nothing below it to build upon, then you just have to learn it as it is. People learning differential geometry have the advantage of being able to base their learning on calculus, but people learning the arithmetic of fractions have nothing further down to base it on. This makes the “basics” really tough to learn! So no-one needs to be embarassed to be struggling with the “basics” – all basics are hard to learn simply because they are basic.
This comment was left on the original blog post:
Nate W 17 October 2013: Such good insight David! Very encouraging.
There is a little trick someone played on me once as a child and I have been playing on the students in the Drop-In room this week. It goes like this:
Answer the following questions:
What would you find in a haunted house?
What do you call a meal of meat cooked in an oven?
What is the part of the country that is next to the sea?
When you have more than everyone else, what would you have?
What do you put in a toaster?
The answers to these questions are of course, a ghost, a roast, the coast, the most and… bread. You weren’t thinking toast were you? 😉
You may ask why I’m playing such a mean trick on my students, when normally I am adamant that we shouldn’t make our students feel stupid. Fair point, but I think it will help them feel less stupid in the end.
You see it all started when one of the students was doing “volume of revolution” problems. Every problem so far had required him to take a 2D shape and rotate it around the x-axis, thus creating a solid 3D shape. The next problem, however, required him to rotate around a different line outside the 2D shape, thus creating a 3D shape with a hole in it. It said on the page it had to be rotated around a different line, and yet he still rotated around the x-axis anyway. “Why did I do that?” he asked. And in response I played the above trick.
The point is that humans are good at following patterns, so good that we don’t even know we’re doing it. In general this is actually a good thing – it means you can set a table, sing music, do jigsaw puzzles, count, learn languages and and even learn maths. But sometimes it fails us, because things don’t always fit into a pattern. Just because the first four answers rhyme with GHOST, it doesn’t mean they all will; just because all questions so far require rotating around the x-axis, it doesn’t mean they all will.
So for students, the message is to keep your mind open. Don’t just follow the pattern, but think carefully about what the problem at hand requires you to do. For teachers, we should be careful to put in more than one type of example, so that students aren’t encouraged to form a pattern that isn’t there. In short, for all of us: Beware of the toast.
The CLPD head administrator Cathy told me a story the other day about an experience she had on the train: She was sitting opposite a pair of students, and one was helping the other prepare for a test. The first student was reading out words from a stack of cards and the second was trying to correctly say what they mean. After listening to this for a while, Cathy leaned over and asked what it was they were studying. The students said “pure maths”.
This completely surprised Cathy, because not one word they had said in all that time seemed to be related to maths in any way, and some of them she had never even heard before. Now Cathy has worked in many different areas in her life, many of which were in academic institutions, not to mention her own experience with maths in the past. So it was quite a shock to her that she had never heard these words associated with maths before. “It was like a completely different language,” she said.
My response to this statement was, “Quick Iggle Piggle! Catch Makka Pakka’s Og-Pog before it hits the Ninky Nonk!” And Cathy immediately knew what I was talking about because she, like me, has a young daughter, and therefore watches ABC2 rather a lot.
You, however, may not regognise or attach any meaning to any of the words in that sentence at all. The question is: do you feel like an idiot for not knowing what I’m talking about? Of course not – it’s just that you happen to have never seen the TV show “In the Night Garden”.
So why do so many people admit to feeling stupid for not knowing specialised maths words? If you happen never to have come across that particular area of maths in your life up till now, that doesn’t make you an idiot. It just means you’ve never come across that area of maths in your life up till now.
If you feel stupid when you hear someone using unfamiliar words, just think of a phrase from some other area if life or learning where you’re pretty sure the other person won’t know any of the words. (Such as, “Quick Iggle Piggle! Catch Makka Pakka’s Og-Pog before it hits the Ninky Nonk!”)
Do you know the story of the Shoemaker and the Elves? Well, I’ve known it since I was very young. It’s a Brothers Grimm, and it goes something like this:
A poor shoemaker is down on his luck and can’t make enough to feed himself and his wife. All he has left is enough leather for one pair of shoes and he works late into the night preparing the leather but falls asleep at the workbench. In the morning the shoes are all made with such fine and perfect workmanship that they are snaffled up quickly by the next person to pass the shop window.
The shoemaker of course buys some more leather and gets it ready and tries the trick again. And again all the leather is sewn into wonderfully well-made shoes. Soon he and his wife are very well off.
Eventually they decide to ask the question of how this is happening, and they hide themselves so they can see who is making the shoes. As it turns out, it’s a team of little elves, who are all completely naked.
The shoemaker and his wife feel sorry for the little elves who have helped them so much and decide to make clothes for them, which they leave out the next night. The elves are so delighted with their clothes that they declare they don’t need to work all night anymore and dance away into the night.
Now those of you who have read Harry Potter may recall a character called Dobby – a house elf, who had to remain in servitude until such time as his master presented him with clothes…
Just a moment! Doesn’t that sound familiar? Of course it does. It’s right out of the Shoemaker and the Elves!
For no reason that I can see, I suddenly came to this realisation this weekend. Jo Rowling rose again in my estimation as being a very clever woman. And I sank just a little in my estimation because I knew this story from when I was very young – why on earth did I not see this connection earlier?
Still, it’s not worth kicking myself over it – this sort of thing happens all the time with learning maths. Students say to me all the time: “I just realised these things were connected! I never knew I didn’t understand how this worked until I suddenly understood how it REALLY worked!”
It’s nice for the feeling to happen to me for a change
The same experience has happened to me several times in the Maths Drop-In Centre recently – with different students from different courses – and it was such a strong pattern I need to talk about it.
The students are doing some algebra involving negative powers on the tops of fractions. Something like this:
\[\frac{1-x^{-2}}{1+x^{-2}}\]
Now they remember this rule (probably from school) which says that a negative power belongs on the bottom of a fraction but as a positive power. And so they do one of these:
\[\frac{1-x^{2}}{1+x^{2}}\]
or
\[\frac{1+x^{2}}{1-x^{2}}\]
Both of these are, of course, TOTALLY WRONG. But the students have a hard time being convinced of this fact.
The problem is, that that rule only works if everything involved in your fraction is multiplication and division. It doesn’t interact with the plus and minus that are trapped there on top and bottom of the fraction. And why doesn’t it interact with the plus and minus? Because the rule is based on the definition of what a negative power means. This is what it means:
\[x^{-2}=\frac{1}{x^2}\]
What this means is that multiplying by a negative power is the same as dividing by the matching positive power. And this gets to the heart of the issue: adding a negative power is not at all anything to do with multiplying it, so the nice “switch to the bottom, make positive” rule just isn’t going to work, because you have to do the addition first.
The rules for negative powers are colliding with the rules for addition, and for fractions, with unpredictable results! If only the students had been encouraged more to work from the original definition rather than it being all about remembering a rule. Then maybe the results wouldn’t be quite so unpredictable! If only the students had attempted a few things like this in the past in a situation where someone could notice it and talk to them about it! Then maybe they would have found this glaring gap in their understanding of algebra!
PS: If you’re wondering how to go about simplifying that fraction, then you have to first deal with the negative power using its original definition – which means it will become a positive power on the bottom of its very own little fraction. Like this:
There is a saying that goes “practice makes perfect”, but I’ve had several people point out to me that a truer statement is “practice makes permanent”. If you do something over and over, it will stick — whether it is the right thing or not.
This principle was brought home to me very clearly by some stories that my instructor told during the First Aid training I had at the beginning of the year (provided by Red Cross).
The first story involved a person who went to the aid of a shopper who had collapsed in a shopping centre. After ascertaining that the person was not breathing, they proceeded to put a plastic bag into the patient’s mouth and then performed CPR. Needless to say, the patient died.
The second story involved a teacher coming to the aid of a student who had gone into anaphylaxis due to an allergic reaction. The teacher fished the life-saving epipen out of the student’s bag and quickly jabbed it into her own leg instead of the student. I never did find out what happened to the student in this situation.
The question is: what on earth possessed these people to do these strange things when clearly it wasn’t going to help their patients?
Answer: that’s how they had practised it during their training.
You see, during our training, we practise CPR on plastic dummies. Part of the mechanism is a plastic bag that goes inside the dummy’s throat and holds the air as we breathe into the dummy’s mouth. In the extreme stress of the moment, the poor first-aider from the first story simply reverted to doing what she had done in her training. The situation is similar for the second story — the teacher had practised putting the dummy epipen into her own leg and so did exactly that with the real-life epipen. (Our instructor had the presence of mind to make sure we used the dummy epipens on each other.)
Of course, these stories came home to me as a teacher because I see the principle in action all the time. I see so many students doing practice exams with all their notes handy, when they are not going to be able to do that in the real exam. I also see students being sloppy with their writing and saying they’ll do it properly for the exam, even though I know full well that they will write sloppily in the exam despite their best intentions.
But the students don’t just inflict this upon themselves — we teachers do it too. We show students a short cut to do something in a particular case, hoping to fix it by showing the “proper way” later, only to find that they keep trying to do the short cut even when it’s not appropriate. And we also find it so hard to avoid doing things for the students — thus forcing the students to practise letting someone else do it for them.
I’m hoping the message of the plastic bag CPR helps me remember to get the students to practise what I want to make permanent, rather than something else!
