In a previous post I discussed how we need ways to think about functions that are not curves on an x-y-plane. Well I have a seasonally-appropriate one for you: the Advent Calendar.
The advent calendar I have in mind is the kind where there is a little cardboard door for each day in December up to Christmas, and behind each door is a little chocolate. (Yes I know it might just have a picture, but seriously, the chocolate ones are better, right?)
Most of the time when we’re not drawing graphs, we talk about a function as a sort of machine, which takes some sort of input and produces some sort of output. This is a very dynamic view of function which I like very much. I imagine putting an object (usually a number) in a funnel at the top and the machine churns and whirs and gurgles until a new object (usually another number) shoots out of a chute at the bottom.
But a function doesn’t necessarily have this time element. The set-theory definition of function is simply a correspondence between one set and another, so that every object in a domain has associated with it one object in a codomain. (The domain is the set of things we usually call “inputs” and the codomain is the set of things from which we choose the “outputs”.) In this sense the “output” is there all the time whether we calculate it or not.
This is where the advent calendar comes in. The chocolate is there whether you open the door or not. Opening the door to see for yourself what shape the chocolate is corresponds to what we do when we calculate the value of the function. But fundamentally all the numbers in the domain have a value for the function before you calculate it, just like all the unopened doors have a chocolate.
I find this particular picture works well for vector calculus, where every point in a plane or in space has a function value, which may be a number (in the case of a “scalar field”) or a vector (in the case of a “vector field”). In the vector field case, the vector you find doesn’t really interact with the ordinary points, and indeed the vector “output” at one point doesn’t really interact with the ones at the other points. It’s almost as if at every point there is a little room where the vector lives all by itself. All we need is a little door to let us into this little room…
Recently I was a guest at a planning meeting for a certain school and ended up in a session where we discussed how we can better support students in terms of their wellbeing. We were shown a news report highlighting the fact that the suicide rate in professionals of this particular discipline is four times higher than the general population. One of the major factors mentioned in the news report was that professionals in this discipline are very unlikely to seek support from anyone when they are struggling, having been trained too well to be self-sufficient while they were students.
Later, we discussed in small groups ways to support student wellbeing, especially with regard to helping them develop skills they can take forward into their professional lives, such as time-management. It was a heartwarming thing to see academics so concerned with the wellbeing of their students.
However, a small number of people expressed concern that giving students support might be “mollycoddling” them. They worried that students wouldn’t learn the coping skills needed to deal with demanding professional lives if they were given support. I do agree that it is important for students to learn those coping skills, but I am not sure that it is entirely healthy to respond to the fear of mollycoddling by not giving support.
Some lecturers I have met in other disciplines in the past seem to think that by refusing to give support they are doing the students good. They seem to think that if a student asks for support they are just lazy and expecting the lecturer to do it for them. For example, I talked to a student who asked a lecturer about one of the expectations for an assignment, but the lecturer refused to give the information and just said “it’s in the handbook”. Another student was confused by an assignment question and the lecturer said something like, “Well if you had done any preparation you would know what the definitions of those concepts were”.
But most students are not lazy most of the time. The first student in the previous paragraph had showed me the handbook itself and the information in it was contradictory, and I encouraged them to seek help from the lecturer. The second student had been in the MLC every day for the previous week going over all the resources at their disposal before they asked the lecturer. They were anything but lazy.
Admittedly there are a small number of students who actually are lazy, but you just can’t make the assumption that your student today is one of them. There is no telling whether a student is asking because they always hope someone will do it for them, or because they’ve tried everything they can themselves now they need your help.
My greatest fear is not that I might be mollycoddling a lazy student. No, my greatest fear is that I might be teaching someone that it is wrong to seek support. This was one of the identified causes of the high suicide rate originally: a lack of ability to seek support. It seems to me that for some students, receiving support today could mean the difference between life and death.
These comments were left on the original blog post:
Fiona Brammy 17 December 2014: David, what a read! I am a little shocked that we have staff who don’t feel that support is fundamental.
Peter Murdoch 17 December 2014: Sometimes I wonder if what appears as laziness is simply the fact that the right domino hasn’t been triggered by the right support so that comprehension falls into place. Surely all educational activities, in classes or not, ought to be designed to support students to find understanding and through this the confidence to support what matters in the world around them.
David Butler 17 December 2014: I like that way of saying it Peter. They might actually be working very hard, just not in a way that makes it possible for them to understand or complete the task they have been given.
When students learn about functions at school, we spend a lot of time forging the connection between functions and graphs. We plot individual points, and we find x-intercepts and y-intercepts. We use graphing software to investigate what the coefficients do to the graph, and discuss shifting along the x-axis and y-axis. We make reference to the graph to define derivatives and integrals. Some teachers help students to recognise from the formula of a function what general shape its graph ought to have, such as recognising that a quadratic function must have a parabola-shaped graph. (I wish this last point was much more strongly pushed, actually.)
However, there is a problem with all of this that students come to think that functions are graphs you can draw. Their idea of a function is a curve drawn on a piece of paper, or at least something that can be drawn as a curve. And this is can cause some serious issues later on.
The first problem comes when they investigate certain pathological functions where the graph is not drawable, but they are still perfectly good functions. For example, consider the Dirchlet function which is 1 when the input is rational and 0 when it’s irrational. It’s a perfectly good function but good luck trying to draw it!
This one’s not insurmountable – students can usually imagine a graph of two ghostly lines with the property that a vertical line which meets one of them in an actual point misses the other. They’re just extending their definition of what it is to “draw” when they draw a graph.
The real problem comes when we move on to functions where the inputs and/or the outputs aren’t ordinary numbers. The simplest case is a function like f(x,y) = xy. This takes a point in R2 and produces a number. Many students struggle to understand these functions because they don’t have a way to draw them. “A function is a curve”, says their experience, but where is the curve here? We get around this by extending their picture of what it means to “draw” a graph: We locate the point (x,y) on a plane and then the output we draw as a vertical height or depth. What this produces is not a curve but a surface.
This is a good start, but unfortunately at this point we also often tell them about level curves (or indifference curves if they’re in Economics). Many students at this time simply come to see a specific one of the level curves as the function itself, instead of all the level curves together as a description of the function, because their experience says that a function is a curve.
And now the real trouble starts: what about a function which takes a vector of 3 or 4 or 7 variables and outputs a number, like they meet in microeconomics or statistics? We don’t have enough dimensions to “draw” the graph then. And what about a function that takes a vector in 3D and produces a vector in 2D, like they meet when doing linear transformations in Maths 1B? And what about a function that takes a real number and produces a point in 3D, like they meet in geophysics? And what about a function that takes a complex number and produces a complex number, like they meet in Engineering Maths? What hope do those students have of understanding functions like that when their only understanding of function has an x-axis, a y-axis and a curve?
These functions are most emphatically not graphs, at least not in a way that you can draw. (I can hear pure mathematicians saying something about the definition of function being a subset of the cartesian product and hence essentially a graph, but you can’t draw it can you?) At the very least they are certainly not the curves students are familiar with!
I believe we need ways to represent functions that don’t involve drawing a curve on two axes, even for functions that can be drawn this way. When we introduce non-curve functions we place a huge burden on the students’ imagination, which can prevent them from understanding what’s going on. My idea is that if they can be familiar with multiple ways of imagining an ordinary number-to-number function, then new types of function will be a little less alien to them, because they will have ready-made ways to imagine them.
Today’s blog post is about my experience attempting to become better read in the area of education research, and I’m sorry to say I’m not going to be glowingly positive about it. As the title suggests, it just seems to get out of hand so quickly.
Let me explain.
The MLC’s job is to support all students in learning and using the maths they need or meet in their coursework. An important part of this job is to support the people teaching the coursework itself to do their teaching in ways that will most help students to learn.
While I have many good ideas, I wouldn’t be doing my job properly or in a scholarly way if I didn’t check out what people already say about teaching. Moreover, there’s nothing like an academic to not take good advice unless it is backed up by peer-reviewed research!
So I try to read education research literature about the courses and concepts the students I help are learning.
And there is the first cause of the can of worms: the students I help come from all sorts of different disciplines and even within the one discipline they are learning all sorts of different concepts. Every day there is at least one new concept that I have to wonder about how it could be taught better. And so I have an ever-increasing list of things to look up in the education literature.
Then, when I come to look up the education literature online, there are any number of papers which may or may not actually be about the concept I am interested in today. If they are related, then they usually introduce at least a few new terminologies or refer to other people’s work which I then need to look up. Alternatively, they aren’t related, but they are usually related to some other thing I am also interested in. So the list of papers I am interested in reading gets longer again.
And then the final problem is that education research is not nice and neat but never fully or adequately answers the question, and usually leaves you with more questions than answers. (As I have discussed before: Frayed Research.) So the can of worms is fully open now and they are wriggling all over the place.
I’m not sure how to deal with this problem. I may need to figure out a specific area of interest and just ignore everything else. (This is more-or-less what I did during my maths PhD.) It’ll be hard though, because I really am interested in a lot of different things, and I feel like I am letting people down by not looking into things as carefully as I would like.
For now I’ll try to wrestle with the worms as they come. You’ll see a new category of post called “Education Research Reading”, where I talk about a paper or few that I have read and what I think about it. It may not be systematic or thematic, but I hope you’ll come along for the ride.
(Don’t worry, though. You’ll still see the standard fare of object lessons, metaphors, teaching ideas and musings about the coolness of maths.)
These comments were left on the original blog post:
Sophie Karanicolas 5 December 2014: Dear David, don’t despair, there is some good stuff out there, they are just hard to find. We will find a good one for you to read! Have a great weekend.
David Butler 5 December 2014: Thanks Sophie — but that’s part of the point. In some areas there is too much good stuff out there! Better than no good stuff I suppose…
Maureen Coffey 16 December 2014: “… education research is not nice and neat but never fully or adequately answers the question …” Indeed, this is because pedagogy or more specifically didactics still lacks the underpinning of a scientific framework anyone can agree on. If nurtritionists wre split about the idea of whether intestines played a role and if chewing was truly necessary for digestion they’d probably be fired from faculty and rather be treated in mental homes. But if educators fail to accept research on how the brain functions and instead expound lofty theories they seem to still be admired … How would different subjects require different teaching if different foods do not require different “stomachs”?
It was the Uni of Adelaide Festival of Learning and Teaching last week, and as always there was a string of people telling us about the great things they’re doing with their teaching. As much as it can get a bit weary sitting through presentations all day, I really do love seeing that there are people excited about doing their best for student learning.
There are two of these people that I want to hold up high as a shining example, and they are Catherine Snelling and Sophie Karanicolas from the School of Dentistry. They have won all these awards for their good teaching, but this is not what I think is the exciting thing. The exciting thing is that they did it through good old fashioned giving it a go.
They thought it would be a good idea to have videos for their students to view, and they didn’t wait for a professional to shoot it for them or to have professional voice training, they just set up a camera and recorded themselves at the whiteboard. They thought it would be a good idea to have students talk to them online, and they didn’t bother to build a whiz-bang tool to do so, they just found out where their students already were on Facebook. Finally, they thought it would be good for the students to see detailed diagrams of dental anatomy, and they didn’t go out to buy fancy state-of-the-art teaching tools, but simply drew really good diagrams in colour on the board.
When you listen to these two people speak you can tell they love to help students learn and that they believe that anything is possible. It was impossible for me not to be inspired by their infectious can-do attitude.
In our bridging course (and indeed in Maths 1M and Maths 1A and several other courses) there is a section on differentiating logarithmic functions. One of the classic questions that we ask in such a section is to differentiate the log of some horrifying function, with the intention that the students use the log laws to simplify the original function first and then differentiate. There is something about this particular type of question has long bothered me and I only just figured out how to resolve my issue with it. I’m so excited I need to share it somewhere!
This is a guest post by MLC lecturer Nicholas Crouch
So it has to be said that I do like what Jamie Oliver does. I have always liked watching his shows and some of the messages that he aims to get across to the community are ones that I believe in. However when it comes to following his recipes and repeating what he does on television, well I figured it was like any other cooking show: the steps are there, but it required being a competent cook to begin with before you would get anything edible from doing what he said to do.
With this in mind when I saw his show on how to make pasta, I was keen to give it a try as I have always loved the idea of making my own pasta. In fact I had given it a shot in the past. My pasta took a very long time and really turned out rather ordinary (and for 3 hours of cooking there was about 5 mins of eating).
This time, however I listened carefully to what Jamie told me, created myself a mental list and believed that he was not going to lead me astray (and when I listened carefully, he did say pretty much don’t bother trying unless you have a pasta machine, which might be a lot of the problem with my first attempt). I followed his instructions and in very little time at all I had beautiful pasta which those who tried it all commented on in a very positive way. So why was this attempt so much more successful than the first?
Well in thinking about that question I felt there could be some lessons that all who aim to pass on their knowledge could learn from. Jamie’s shows could be considered like a lecture, where he can talk about a topic, but there is no opportunity to give feedback on what the student has done. What did he do that was worthy of note?
Firstly he was clear about what exactly to do. Most cooking shows are, but often get distracted by other less important details. He told me just what to do.
Next he sign-posted. “If this is the situation you find yourself in at this point, this is your problem, and this is the solution.” Or “Do this until these conditions are met”. For example, I think he mentioned that if your pasta is too dry you can add more water to make it softer (but not too much). However, do not over complicate it with all possible cases, just the important ones. He also gave some explanation as to why we would do something in that way.
Lastly he inspired me to believe that I could do exactly what he had done and achieve a similar result.
So what are the lessons about teaching?
Firstly, be clear, don’t overcomplicate something. You probably know a lot more than is required about your topic. It can be very difficult to cast your mind back to a time when you didn’t know that topic extremely well. However this is what you need to do to distil the essences of what you are trying to teach.
Secondly, signpost (OK I should know the proper pedagogical term for this but I don’t). By this I mean, talk about the reasons for making the decisions that you have, not just the steps that need to be taken! For a lot of what I do, this comes down to trying to state the connections between what is being asked and what I am doing. Even to the point of listing my options and then saying why I would chose this option.
As for inspiration, well that can be hard to give people advice on. I personally sound excited when talking about certain topics, and it is infectious. Other things that work are giving your topic some historical significance, or talk about how it is used.
The last part is keeping the goal in mind. In Jamie’s case, we want to eat what we are working on. In the case of teaching we want the student to have a greater understanding. So look back at what you have done. (I know about this term – the cognitive closure) But this also can be a time where we talk to students about how they could re-order the concepts for themselves so that they don’t just have the linear connections in their head. The way the material has been presented to them, we want them to do more than that with it. So perhaps connect the dots for them, even mind map it for them if you have to!
So next time you are in front of a class, sound excited and have pasta!
If you are the student in the class, create your mental list and have confidence in your instructions!
I had a long chat with one of the students the other day about rotation matrices. They had come up in the Engineering Physics course called Dynamics as a way of finding the components of vectors relative to rotated axes. He had some notes scrawled on a piece of paper from one of my MLC tutors, which regrettably were not actually correct for his situation. I know precisely why this happened: rotation matrices are used in both Dynamics and Maths 1B, but they are used in different ways (in fact, there are two different uses just within Maths 1B!). It’s high time I made an attempt to clear up this confusion, especially since three more students have asked me about this very issue in the last week!
In Maths 1B, you learn about Linear Transformations, which are a special kind of function that you enact upon vectors in some dimension to produce vectors in some dimension. It turns out that all linear transformations can be described by representing your vector as a column of coordinates and multiplying it by a matrix. Each linear transformation has its own matrix that works for all the vectors it acts upon. Rotations happen to be a type of linear transformation and in two dimensions there is a formula based on the angle you rotate that tells you what the matrix is. I’ve included just such a matrix in the picture here.
One reason this works is because multiplying a matrix by your standard basis vectors of (1,0)T and (0,1)T gives you the first and second columns of your matrix respectively. But multiplying by the matrix has the same effect as the rotation transformation, so to figure out what these columns actually are, all we have to do is rotate the points (1,0) and (0,1). If you do this, then because of trigonometry, you get the two points (cos θ, sin θ) and (-sin θ, cos θ), which are indeed the columns of the matrix.
Let’s just make sure we know what’s going on here before we move on: You have a point in the 2D plane, you take its coordinates as a column, you multiply this column by the matrix, and you produce a new set of coordinates, which is a new point. So your matrix in effect moves your point from one place to another. The point with coordinates (1,0) moves to the point with coordinates (cos θ, sin θ); the point with coordinates (0,1) moves to the point with coordinates (-sin θ, cos θ).
So now we have that a rotation matrix has cos θ on the main diagonal, sin θ in the bottom left corner and -sin θ in the top right corner. And it tells you where a point moves to under a rotation of θ anticlockwise. (It’s worth noting that it also works perfectly well on the components of vectors imagined as arrows.)
The problem is that over in Dynamics, a rotation matrix does not look quite like this! In particular, the minus sign is in the opposite corner. Why?
The answer is that in Dynamics the rotation matrix is not a description of a transformation of the points or arrows themselves, but a description of how their coordinates change when you transform the coordinate axes. The points themselves don’t move at all, it’s the coordinate axes that move and we just relabel the points with new coordinates.
The reason this works is again because of the standard basis vectors. The point (1,0) has its coordinates recalculated according to the new axes, and its coordinates turn out to be (cos θ, -sin θ); while the point (0,1) also has its coordinates recalculated and its coordinates turn out to be (sin θ, cos θ).
You may notice that this is precisely what the coordinates would have been if you did rotate the points themselves, but in the opposite direction to the original rotation matrix. This makes sense. If you turn your head to match the new coordinate axes, then this is precisely what has happened. Basically, if you rotate the coordinate axes one way, the points “move” the other way relative to the axes.
And this would be the end of the story, except that in Maths 1B you also rotate coordinate axes, and yet the rotation matrix is somehow still not the same as the one in Dynamics! Why?
The reason is that in Maths 1B we rotate axes in the context of equations of curves, and this is quite a different situation from when you rotate axes in the context of the points themselves.
Imagine I have an equation which describes a curve. A point is part of the curve if its coordinates satisfy the equation, and it’s not part of the curve if its coordinates don’t satisfy the equation. But what if I relabel all the points with new coordinates according to a new set of axes? I want an equation for my curve so that a point is on the curve if its new coordinates satisfy the new equation. How do I achieve that? Well I do already have an equation, it’s just in terms of the old coordinates. So if I have a point in the new coordinates, to tell if it’s in the curve, I just need to figure out what the old coordinates are and sub them into the old equation. It ought to be possible to make one equation that encompasses both of these actions – the transferring to the old coordinate system and the subbing into the old equation.
Did you notice what happened there? In order to create an equation that described the same curve relative to the new axes, I had to begin with the new coordinates and transform them into the old coordinates. Let me repeat: I had to go from new to old. The coordinate transformation matrix in Dynamics goes from old to new. To go in the opposite direction I have to have the minus in the opposite corner.
So that’s why the matrices are different. In Dynamics you are moving the axes but not the points, and finding new coordinates for the points. In Maths 1B you are moving the points, not the axes, so the rotation appears to be in the other direction. Or alternatively in Maths 1B you are moving the axes, but you already know the new coordinates and you want the old ones, so you actually are doing the calculation in the opposite direction.
We had students the other day from Maths for Information Technology and their task was to form the contrapositive of a several statements. Given a particular statement of the form “If A, then B”, the contrapositive is “If not B, then not A”, so mathematically the problem is not actually very difficult. However grammatically the problem is much harder than it looks.
Consider this statement: “If it is raining, then there are clouds.” If we compare this to my generic example above, we see that A is “it is raining” and B is “there are clouds”, so by my own rule, the contrapositive ought to be this: “If not there are clouds, then not it is raining.” This is obviously not a grammatical English sentence! A correct version is, “If there are not clouds, then it is not raining.” By giving people a generic rule, we are getting them into trouble with their grammar. This may seem like a small thing, but there are plenty of students for whom English is not their first language, and even students whose first language is English often don’t have a very good command of the rules of grammar!
There’s no easy way around this in the way we present these generic rules, except to make them aware that they need to think about the grammar of the sentence that they write when they do this, in particular, where the word “not” has to go.
But it gets worse! Consider the statement “If f: R → R is differentiable, then f is continuous”. According to my above rule, A is “f:R → R is differentiable” and B is “f is continuous”, so taking into account the grammatical placement of the word not, we get the contrapositive is “If f is not continuous, then f:R → R is not differentiable.” The students we worked with did this exact thing, and they could tell there was something odd about it, but they couldn’t quite figure out what it was.
The problem is that the part saying “:R → R” is not technically part of the if-then construction. It could have been stated in a completely different sentence like this: “Let f: R → R. If f is differentiable, then f is continous.” Then that lead-in sentence isn’t an if-then construction, so it isn’t part of the contrapositive.
And here’s where the grammar gets particularly tricky. The fact that this little bit of the sentence can be pulled out into a sentence of its own means that grammatically it is called a “relative clause”. A relative clause gives more information about a noun in a sentence, without interfering with the verb. You see it in sentences like “My brother, who is in Canada at the moment, says hi.” I could have said: “My brother is in Canada at the moment. He says hi.” Of course it wouldn’t have had quite the same impact as the first sentence, which is why we say it the way we do. Another example is “If Catherine, your wife, is a kindy teacher, then she is clearly awesome.” The relative clause here is “your wife”, which is telling more information about who Catherine is before you go on to say stuff about her. This sentence is closer to the maths sentence above, but it has one very important difference. In the maths sentence I mentioned f twice; in this English sentence I didn’t mention Catherine twice. Instead, I used the pronoun “she”. We could have done the same in the maths sentence too: “If f:R → R is differentiable, then it is continuous” would become “If it is not continuous, then f:R → R is not differentiable.” It would have been much more obvious what was wrong with this sentence – we haven’t told the reader what f is, or indeed even mentioned f at all until the end. This makes it obvious that we ought to move the relative clause to the first part of the sentence when we form the contrapositive.
Other than the strange tendency of mathematicians to not use pronouns, there is something else that prevents us from seeing “:R → R” as a relative clause: the flexibility of the notation itself. Maths notations that include a verb in their meaning can be read aloud in multiple ways depending on context. Many students do not actually realise this, mainly because they never read their maths aloud. For example, consider this bit of maths: “Let x = 12. Then x = 4×3 = 2×2×3.” This is read aloud as “Let x be equal to 12. Then x is equal to 4 times 3, which is equal to 2 times 2 times 3”. That “=” was read loud as “be equal to”, “is equal to” and “which is equal to”. In our contrapositive example, consider these three sentences: “Let f: R → R.”, “Suppose f: R → R.” and “If f:R → R is differentiable…”. The first is read aloud as “Let f be a function sending R to R”, the second is read aloud as “Suppose f is a function sending R to R”, and the third is read aloud as “If f, which is a function sending R to R, is differentiable…”. But they all look the same!
The flexibility of our maths notation makes for easy writing, but sometimes it makes for difficult grammar, especially when it masks those pesky relative clauses!
This comment was left on the original blog post:
John Baez 8 September 2014: Very interesting analysis. A digression: you write “Other than the strange tendency of mathematicians to not use pronouns….” This reminds me of something the computer scientist Tom Payne told me: “mathematicians are people with an extraordinary ability to keep track of many pronouns”. His point was that variables in mathematics serve as pronouns. Instead of saying “he”, “she”, and “it”, which breaks down when you have more than one he, she, or it, we introduce new pronouns (variables) as needed. Ordinary people lose track of all these pronouns.
On the train a while ago I overhead some people talking about Heston (the celebrity chef). Apparently he had been doing a series on giant food. It involves him trying to figure out the physics and logistics of trying to produce food on a giant scale – for example, a three-metre tall soft-serve ice-cream cone.
After describing all the care and effort Heston took to produce this giant ice-cream, the first person declared, “He’s very clever.” Her friend’s response was, “He has too much time on his hands.”
Clearly this person could easily do a better job than Heston if she wanted to, but she chooses not to because she has so many more important things to do with her time. Apparently Heston’s not clever, he’s just idle.
I had a strong desire to lean over and ask her if JK Rowling had too much time on her hands, or if Stephen Spielberg had too much time on his hands, or Michaelangelo had too much time on his hands. If you think about it, what they did was more or less for their own enjoyment too and wasn’t “important” either.
Of course, it wasn’t Heston I was really indignant about. The statement brought up several unpleasant memories when people have said this to my face when they have seen me making models of fractals, crocheting hyperbolic coral, drawing digits of pi on the pavement or solving puzzles, or even just doing maths in my own time. They seemed to feel that they needed to make those things seem trivial.
Perhaps they felt cheated that they don’t spend more time doing things they actually enjoy. Perhaps they felt like I was making them look stupid and they needed to make me feel bad for it. Or perhaps they are just grumps who are unable to share in others’ fun.
Now that my indignation has faded a little, I feel sorry for them. I remember what it was like to be in a situation where I felt it was somehow wrong to choose to do things I enjoyed, and it wasn’t a pleasant place to be. It can colour your view of the world and frankly it does make it difficult to enjoy other people’s fun.
Still, it’s no real excuse for making people feel bad about things they have spent a lot of time acheiving. Sure, they may have a lot of time on their hands, but at least they are using it well!
