WitCH 102: A Mind is a Terrible Thing to Map

Some years back, I enrolled in the teaching Masters at the University of Melbourne. I lasted three days.

I can’t remember much specific of the nonsense I was presented, but I do remember clearly a tutorial-workshop in which we students were asked to construct a mind map of something or other. My fellow students went happily to work but I had never heard of such a thing. So I asked, and the kindly tutors explained what a mind map was. My reaction, possibly vocalised, was “What’s the point?”

These memories came back recently, when a correspondent alerted me to a “mind map” blog post. The post, by a prominent member of the CLT gang, was about the teaching of quadratics to Methods students. The focus of the post was a mind map that the writer-teacher had drawn for their students, the diagram reproduced above. My correspondent was deeply unimpressed, and so am I.

There are obvious specifics of the diagram to dislike, but I had to think of why my unimpression was so fundamental. Maybe the issue is obvious, or just in my imagination. But I’m slow. Having discussed it with Simon the Likeable, I think I now understand.

39 Replies to “WitCH 102: A Mind is a Terrible Thing to Map”

  1. Just to be clear… is there one major piece of crap here to be found, or is it a case of lots of crap, spread not-so-thinly on the page?

    For example:

    1. The heading is misleading and pointless.
    2. The key is equally so.
    3. For a map, it doesn’t seem to give much direction.
    4. Are the green highlights meant to all be part of something? I can’t see it.
    5. Is “polynomial form” even a thing? I would call this “standard form”, but whatever.

    1. 0-1-2-3. I’m honestly interested in people’s reactions and thoughts. As I wrote, there are specifics I don’t like, but I don’t like the the entire thing more than the sum of my specific dislikes. Also, to be fair, you should probably look at the entire post. It didn’t do much for me, but presumably it was intended to.

      4. I’m not sure what you mean. There is text in the green rectangles.

      5. It struck me as a weird label, but yeah, not the main point.

      1. I thought (wrongly, perhaps) that if words were highlighted using the same colour it was to group them together somehow as related ideas.

        1. Oh, I see. Well, the bottom greens are all “things we wanna know”, and the top green is just green for the hell of it, I guess.

  2. I like that the teacher didn’t want to present quadratics as a series of disconnected processes. However, presenting them as a series of connected processes isn’t much of an improvement. Quadratics were so much more manageable once I understood them as singular objects, with various properties.

    The ways of determining these properties for a specific quadratic (turning points, intercepts etc.) almost always follows fairly easily from its definition. If we know the y-intercept is the point (0, f(0)), we can find the y-intercept of any quadratic. If we know what an ‘axis of symmetry’ is, and we have a good idea of the shape of a parabola, it follows that this axis is a vertical line which passes through the turning point and is halfway between the x-intercepts. If we know just a tiny bit of algebra, then we know that to get to the standard form of a quadratic we need to expand, and if we have half a brain we know that to get to the factorised form we need to factorise.

    My issue with the mind map is that nothing is allowed to be implicit; everything that could possibly be done to get one piece of information from another needs to be pointed out as its own distinct process to highlight ‘connections’. But this act of separating out each process based on (sometimes minute) differences between them really just makes the thing feel disconnected.

  3. I am not a fan of “mind maps”.

    On the other hand, I once worked with a colleague in education, and he was keen on drawing diagrams. I have found it useful to represent an idea or process with a diagram. When I read, I tend to take copious notes and include diagrams to help me understand my notes.

    Perhaps some people might describe some of my diagrams as “mind maps”. Maybe I would too if I knew what the term meant.

    One difficulty with such maps is that one is constrained to 2-dimensional representations. Many ideas are not suitable for this. I recall going to a lecture on algebraic topology where a complicated theorem was true only in dimension > 2. The lecturer used “proof by diagram”.

    1. I should add that there is a big mistake in the mind map above – and any mind map in education. The student must be at the centre.

  4. Mind maps, I remember these trite things that our teachers made us fill in middle school. Not on paper of course, but on some app on our iPads. I don’t remember why I didn’t complain; I certainly complained about other things. Probably because it was easy enough to write some random garbage then get on to what is truly important, playing video games.

    The most interesting part of the mind map exercise is that people share their own ideas. Whether, the mind map is crucial to this, is not obvious (and probably false.) I don’t see the problem with using a simple document (whether electronic or on a whiteboard) that the classroom works on together. Of course, this really only has a point in the humanities. I don’t think it’s useful in the hard sciences and mathematics, where teachers should just directly teach students the concepts, definitions, methods, and later proofs, instead of trying to force students to “discover” it themselves, or “create” stuff themselves.

    The teacher agrees, creating his own mind map. His reasons are as such: “Students can take exorbitant amounts of time to construct them, they can construct them incorrectly (relating knowledge in ways that is inaccurate), and even when they do have the info organised in their minds accurately, sometimes it can end up a pretty big mess on the page.” The real reason, should, of course be, that a teacher is meant to teach a particular thing, which the teacher should teach, and not force the kids to “discover” on their own.

    The teacher didn’t follow his own last point; his mind map is messy and I think it is intentionally so. I don’t know if it’s more the result of general clumsiness or just trying to make it look more cool or complicated than it is on purpose, but the map is very hard to read, with tons of crossings between the arrows, and the meaning of the text isn’t exactly clear. The graph is planar, so it doesn’t have to be like that. It’s fairly trivial to get rid of all the line crossings, even if you want to retain the “levels” of the graph.

    “‘How can I ensure that students have all these ideas connected in their minds in the same way that an accomplished mathematician would?'”

    I don’t even need to say anything about this.

    “A great model is presented to, or constructed by, students, but isn’t revisited.”

    I don’t know if this is just me, but this sound extremely egotistical. It seems more like he’s focused not on teacher quadratic equations, but rather on teaching the specific “great” mind map that he created.

    There’s also a bunch of allusion to mumbo jumbo like “relating” to what was previously learnt, and the “spaced repetition algorithm.” Let me just say this, won’t “relating” to previous mathematics and “generating” new long term memories be easily done with just following the exercises in a good, school mathematics textbook? To do advanced stuff in mathematics you have to know the intermediate stuff, and to do the intermediate stuff you need to know the basic stuff.

    This is wholly unrelated but I’m surprised that quadratic equations are being taught in Math Methods. For some reason I thought they were taught earlier.

    1. They are. Well, they were. In yr 9 and yr 10. But ACARA removed them. More or less (mainly more). Too difficult, for extension only (but you have to look hard to see the extension mention).

    2. Thanks, Rambler. Your “accomplished mathematician” quote is indeed telling.

      I don’t know that the diagram was intentionally messy, and it’s a little difficult to tell from the blog post how the diagram and its discussion was intended to fit in with the actual teaching.

      Also, your surprise that this quadratics stuff is being taught in this manner to (one dearly hopes Year 11) Methods students is very much related. It is a core part of what really annoyed me. If this kind of Map of the Territory was being presented to Year 9 or Year 10 kids, who are getting their head around the various new approaches, I’d not mind so much. It’d still be a bad and badly distorted map, but it’d be somewhat understandable. But to present this to senior mathematics students as This Is Quadratics is simply appalling. Yes, it is not nearly all the fault of the teacher-blogger, but they are still presenting it, to us, as the “accomplished mathematician” view, and it very much ain’t.

      1. I assume it was shown to an “accomplished mathematician” who gave it their imprimatur. That is certainly Lovells suggestion.

          1. “How can I ensure that students have all these ideas connected in their minds in the same way that an accomplished mathematician would?” The answer given is the lovelly mind map. How can that be the answer unless an accomplished mathematician confirmed it? Thats how I read it.

            1. That’s really funny. You really don’t know these guys at all, do you?

              The obvious interpretation is that the author regards themself, at least for this purpose, as an accomplished mathematician.

  5. Huh. So concept maps are called mind maps these days. Whatevs. To parasing Julie Andrews, these are a few of my unfavourite things:
    It implies you can always factorise a quadratic over reals. It implies the y-intercept can only be calculated from the polynomial form. Despite all the detail, it says nothing about the discriminant (why not have discriminant geq zero on the arrow from polynomial to factorised. And I think the discriminant is better placed on the quadratic formula arrow). “b tells us where the parabola sits” (wtf) It makes the simple look complicated and makes my eyes glaze over. It smells like try-hard humblebragging (which I detest). It is very misleading. Plus what red sed.

    I cant (and dont want to) imagine what the original looked like. Marty, since you mention it, some trivia courtesy of ChatGPT:
    “A mind is a terrible thing to waste” is a phrase that has been used as an anti-racist slogan. It was first used by the United Negro College Fund in 1972, which used it in its advertising campaign to raise money for scholarships for Black students. The phrase was used to draw attention to the importance of education and the need to help young people reach their full potential. The phrase suggests that it is a tragedy when someone’s mind is not used to its full potential, and that this is something that should be prevented. The slogan has been used to promote education and support for marginalized communities to ensure that everyone has the opportunity to reach their full potential. It also implies that a mind is a valuable resource that should not be wasted, and that everyone should be encouraged to use their minds to the fullest.

    1. Thanks, Anonymous. I agree with your specific criticisms, although I think there are more, and worse.

      I had in mind the UNCF motto when I came up with title of the post. I thought the original motto was great, and meant no disrespect by playing on it for my title.

      1. No disrespect taken. There may be some readers who are not be familiar with the original phrase or its origins. The motto is great and your title is clever and apt, the mind is being wasted with this mind map (I think mind crap is a better name for it).

        1. Yeah, I know you didn’t think it was disrespectful, was just confirming. And yeah, I was aware most readers would not make the connection, but I always like playing this way, even if only for myself.

  6. Some university academics seem to love mind maps. I find that they add clutter and confusion to even the most simple concepts. Worse than pointless, they are destructive.

  7. Two problems with mind maps are that they are overused, and that are associated with Tony Buzan (who wrote about them and introduced the term “mind map” in the 1970s) who has been aptly described as the “emperor of self-promotion.”

    This particular diagram looks like an unholy bastard child of a mind map and a flow diagram, taking the worst aspects of both. I can’t make sense of it, and I suspect that if it was given to students struggling with quadratics, that they would be even more confused and depressed. The idea of any diagram is to simplify a description, or an idea, and this diagram seems to do the opposite.

    1. Thanks, Alsadair. It wasn’t clear to me that the quadratic thing was a mind map, partly because I have no clear sense of what a mind map is supposed to be. And of course what to call the thing is the least of the problems here. But given the writer called it a mind map, I figured to stick with that and take a quick drive-by slap at Unimelb’s Masters.

  8. I very much hope any student reading that diagram ( I refuse the term mind map; mind map= pretentious pap) doesn’t just look at it and tell themselves they know everything about quadratics. The experience of looking at the diagram is passive. The convoluted series of connections is almost entirely external to the reader with almost no learning or understanding from reading it. Even now as I type I have forgotten what is in the damn thing. But anyone would want to be able to make all of those connections in their own mind i.e. an actual bloody mind map. It takes effort, practice and grit to do that
    The only purpose of the diagram for a student that might be of any benefit (that I could see) is to externalise their current state of understanding for others to discuss. But after the student has done the work, not before. Showing a student the quadratic diagram beforehand would be counterproductive and limiting of a students’ potential

  9. Have to admit to using the phrase
    ” polynomial form”. In my language “canonical form” actually means tp form.
    Too many errors on the map, but the idea..is it really so bad?
    Firstly, does it work for students?
    (Linda Hunt to Arnie -kinder cop: it was not by the book, but it worked!)
    I am using some aspects like the ones on the map. I like to ask: how do you get from polynomial form ( yes, what s wrong with that? You have to be creative) to tp form?
    ( correct answer: complete the square)
    What info does factorised form convey, without any working?
    And so on.
    May be silly for strong students, but i usually have many more average students.

  10. In calling this THE quadratic map, is the creator implying that this is everything a student/reader needs to know about quadratics? It feels like that might be the crux of the message, in which case the crap is pretty self-evident!

    Sure, all the little bits that don’t do much except distract from the more important ideas and seem to be some sordid attempt to say that a not-entirely-simple idea such as quadratics can be reduced to a set of “remember these formulas and you will be fine” statements… I’ve seen so many diagrams like this in textbooks though that perhaps I’ve grown immune.

    1. I hadn’t picked put on the definite article. But yes, the suggestion that the map is covering the whole terrain is a central annoyance for me.

  11. A week is a long time in WiTCH land, so here are my criticisms of the diagram, roughly in order of seriousness:

    1. There is nowhere a definition of what a QUADRATIC actually is. Not as a function, an equation… nothing.
    2. The loose way PARABOLA and QUADRATIC are used interchangeably is annoying. At no point is there any suggestion as to why a quadratic equation produces a parabola graph.
    3. Real solutions and x-intercepts – again, there seems to be an implication that a reader knows how these are related, except… every parabola has an axis of symmetry. Not all parabolas have x-intercepts.
    4. There is no clear “starting point” from which to read this map (sure, I get that real maps have no starting point either, except maybe a YOU ARE HERE sticker) and so the arrows don’t really make much sense.
    5. The “key” in the top left corner is next to useless.

    What did I miss?

    1. Thanks, RF. Given it’s intended to be a summary (“map”), I’m not sure everything needs to be, or should be, indicated. But it’s muddy in the way you suggest.

Leave a Reply

Your email address will not be published. Required fields are marked *

The maximum upload file size: 128 MB. You can upload: image, audio, video, document, spreadsheet, interactive, text, archive, code, other. Links to YouTube, Facebook, Twitter and other services inserted in the comment text will be automatically embedded. Drop file here