Assembling the heterogeneous elements for (digital) learning

Getting back into maths – it is not linear

As mentioned last week, I’m slowly commencing the process of becoming a high school teacher in information technology and mathematics. As part of that process I’m trying to get back into maths – its almost 25 years since I did my undergrad math courses – and build up a network of folk and ideas.

So, via the well known @coursa, I find @katierosenkranz (a pre-service teacher with a math minor) and her mention of a “prezi” – Math is not linear. Not only does the content of this presentation resonate, the number of additional connections being added to my baby math network is rapidly increasing. Including @pvnotp author of the presentation. Though I do admit a need to connect with some more Australian folk.

Math is not linear

The presentation essentially argues that the simple linear presentation of math topics within schools has some problems. Problems that can be addressed and learning improved by drawing on an understanding of the connections between the different math topics.

In part this gets at the whole idea of teaching as a logical, sequential process where the expert builds the foundations for student learning. It’s the assumption that guides most teaching of computer skills to students. You have to have them in a lab with an expert watching what they are doing helping correct mistakes and misunderstandings. Compare that with the approach taken by the Hole in the wall project.

This idea is also connected with the TEDxNY presentation from Dan Meyer

The topic areas

Alison’s prezi includes a slide that contains a list of math topic areas. I’m simply going to treat the following as an initial bit of revision of mine. Actually, in many cases, I’m not sure I ever could have described what these topic areas were about and how they connected to the real world. Even though I was good at maths, I think I was better at playing the high school math game, rather than gaining deep understanding. This is itself one of the problems I’m hoping to avoid in my teaching.

If I’m going to break this linearity, I’m going to have to develop the knowledge. It won’t be complete and it is basic, but it’s a start. The topics are:

  • Algebra;
    So, part of pure maths along with geometry, analysis, topology, combinatorics and number theory. And what’s generally taught in high school is elementary algebra. Ahh, personal connection, Rob’s work in ring theory is abstract algebra.
  • Geometry;
    AKA earth measuring. Hadn’t made the connection with the name. Am sure there could be some interesting connections made with kids through that name. One of the oldest forms of math.
  • Set theory;
    The study of collections of objects, of sets. What’s interesting is how the Wikipedia description of when various topics in set theory are introduced in formal schooling seemingly illustrates Alison’s point. Some basic stuff in primary school, and then the more advanced stuff in undergrad. It’s also somewhat symptomatic that I can understand what it is from the Wikipedia description, but I’d be hard pressed to explain practical relevance to a 13 year old.
  • Number theory;
    Okay, either the Wikipedia article introductions are getting worse or I’m tiring, I’m finding it harder to explain/understand number theory. Of course, that’s probably because I’m trying to think of practical examples, and to some extent that’s not what number theory is about. Though there are some important practical applications of number theory findings (e.g. cryptography), practicality is not the immediate aim of a number theorist. It’s on the properties of numbers etc. that they are focused.
  • Calculus;
    Interesting that wikipedia uses a “study of” description for calculus (study of change), geometry (study of shape) and algebra (study of operations and their application to solving equations) on the calculus page, but not on the others.

    An example of the perils of top-down de-composition? The loss of the whole that arises from dividing a description of mathematics on Wikipedia into separate pages written by different authors?

    Calculus focuses on limits, functions, derivatives, integrals and infinite series. 🙂 They like their terminology, don’t they. Perhaps that is a big part of the problem. Mathematics really is a language and we’re teaching the language by dividing it up and teaching it separately. e.g. we’ll teach you French by dividing words up into topic areas such as politics, economics, the arts etc. No wonder learners find it hard.

    Interesting, this Wikipedia page attempts to connect to the practical applications.

  • Statistics;
    Ohh, it’s the “science of the collection, organisation and interpretation of data”. Closely related to probability theory. Or it’s a branch of math dealing with the collection and interpretation of data..differences of opinion.

    Statistics starts with a population of objects. Objects which have properties, information about these properties are gathered. Rather than study the information about all objects, a sample is taken. Statistics analysis then either describes the data gathered or draws inferences from that data.

  • Probability theory;
    Concerned with the analysis of random phenomena. The mathematical foundation for statistics.
  • Combinatorics.
    The study of finite or countable discrete structures. An interesting comment in connection with Alison’s posts “Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry”. i.e. it connects with many of the topic areas.

Interestingly, this list does not include arithmetic, which is what most people think math is.

While, the above process has been useful in refreshing some basic ideas. It’s also revealed just how much work I do have to do. Some of which will come back quickly, e.g. observing some of the mathematical equations in the above I found myself remembering aspects of them from 25+ years ago.

I also found it interesting, but not that surprising, that the Wikipedia pages aren’t that easy to learn from. I’m wondering if there aren’t online resources that do a better job of explaining these for non-mathematicians. Which in itself raises an interesting question, in a few years will I claim to be a mathematician or simply a maths teacher? What implications will my answer to that question have about my teaching?


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  1. I’ve just gone through Dan Meyer’s stuff yesterday as part of a linkhole from the Global Education Conference as well as Wolfram’s talk about more or less the same thing from a different angle.

    There was a session from the Math 2.0 interest group that provided many links and people to connect to …

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