Maths Mnemonics – Frameworks to make sense of maths

At a recent Maths faculty meeting, we had time to share teaching ideas and strategies.  It was fantastic that, even amongst very experienced teachers, there was this desire to learn especially from each other.

I really liked what @markmikulandra shared. He shared some mnemonics, a good learning strategy.

DAD – Draw A Diagram

This is a fantastic problem-solving strategy and it is easy to remember.

FSC – Factorise, Substitute, Calculate

This is really useful for equations. It is true that most equations are easier to ‘attack’ when you factorise first. There were some discussion on some students’ reluctance on this quasi-trial-and-error approach. That is, for these students they want to solve it on the word go. Anyway, I think this works because the fundamental skill involved in factorising is identifying the building blocks. Abstracting from this, factorising is an important problem-solving strategy not just in maths, ie if you can identify the factors that make up the problem, solving it is easier because you can focus on each factor (eliminate, mitigate or fix)….but, I digress.

N.ATEL – Number, Algebra, Trigonometry, Exponents, Logarithms

@markmikulandra introduced N.ATEL to his students (and us, his colleagues) as a metaphor for Avatar’s Na’vi.  So, N.ATEL is like an avatar to the world of mathematics. When these elements mix in higher mathematics, students sometimes struggle with their strategies as in using algebra in solving trigonometric problems (eg differentiating x sinx, x is algebraic and sin x is trigonometric). Problem is, most of junior maths are taught in strand-approach and it’s rare for students to see the elements together. Mix them together, as they do in real life and higher maths, and confusion often arises.  This mnemonic can help make sense. I think N.ATEL can be introduced in junior maths to provide a real-life context (the irony of an avatar doing this is quite amusing 🙂 ), a big picture.

In the spirit of sharing and mnemonics, I also shared my own – GGSC – Goal, Given, Solve, Check for approaching word problems (see older post).

All these mnemonics are learning strategies. They are frameworks to help make sense of mathematical ‘blobs’. These frameworks are forms of, even as they develop, mathematical thinking.

What’s the best way to teach Maths

I was very good in Maths back in high school – I was one of the best there. I wasn’t as good at university – just a tad above average; there are some truly clever people out there (but I digress). It’s not clear to me now whether I enjoyed Maths because I was good at it or that I was good at it because I enjoyed it. I did well, too, I think because I persevered with practice – lots of pencil-pushing, especially in Algebra – my favourite maths strand. The closest we got to technology was a scientific calculator!

As a teacher now, I cannot even imagine teaching the way I was taught….certainly not for every lesson. I’d like to say that it’s because I want to be a good teacher and engage my students, blah, blah blah. There is that, of course. But also, I get bored and a bored teacher is seriously bad news. Just as much as a teacher’s passion is infectious, so too is boredom.

Variety is key for me. Undoubtedly, access to technology makes this easier for me. Even if technology is not physically present in the classroom every lesson, quite often my ideas  have been ‘harvested’ from the web, and increasingly from my PLN via Twitter.  The amount of time I spend trawling the web for ideas is quite significant and that includes following the leads from my PLN. The more I explore, the more ideas open up and ultimately it’s the lack of time that stops me dead on my tracks.

Today, I had a review session with my year 8 class. A recent assessment revealed that many students really haven’t understood key concepts I have taught.  I found it interesting, however, that they got the Extension topic – the one I did using the Vitruvian Man (see previous post).  When I can, I do try to teach mathematical concepts within a given context – make it real so-to-speak – and usually, the class is engaged. I realised today that engagement in the classroom is not enough. Not for maths.  (Yes, I give and check homework). I suggested that perhaps I should teach by the book, i.e. have more time doing pencil-pushing work. There was a rather loud and unanimous, “No!”

When a student deemed below average in Maths can enjoy Maths, surely that’s a good thing, right?

Another problem is that it would seem that my students struggle with transferrability. That is, they struggle to apply what they learned in a different context. So, it’s not that they didn’t really learn but that they can’t apply it in an unfamiliar context. For example, we can talk about Percentage Composition in the context of polls but they struggle with using the same skill with just plain numbers (“out-of-context”).

For now, I’m not sure which way to go.

Do I teach the way I was taught or teach the way I was taught to teach, i.e. emphasis on the learning processes and the learners themselves?

All advice welcome. I don’t promise to follow but I promise to listen!

The classroom as a community

We are social beings. We like to connect. (Positive) Connections are affirmations.

I love my year 7 maths class. They waltzed into the classroom today singing the Perimeter song I taught them 2 days ago -they were impressed that I trawled the internet for ideas to make my lessons interesting….for them. They were engaged in the discussion about Area. They rose to the challenge of my prodding questions that required recall of  topics they learned months ago. They glowed from being complimented on their mathematical thinking (use of shape and number properties for reasoning). They revelled in my digressions (granted, most were part of my connectionist approach) – we revised Properties of Shapes, Properties of Numbers, Adding Decimals, Powers/Exponents and Algebra. They bantered with each other but quickly were on-task as required. They were enjoying maths. Most of all, they enjoyed my declaration that I loved the class.

It’s taken me a while but many of the things I wanted them to learn – aside from maths topics – came together today.

  • a classroom is a community; everyone has a voice worth listening to and everyone is a resource for the other
  • use what you’ve learned previously to help with your problems now
  • you don’t need to be an A student to enjoy maths
  • self-reflect and evaluate and then challenge yourself; easy is boring
  • it’s not just about the answers, it’s about the thinking and reasoning
  • notes are reference items, especially if they’re not from the textbook
  • learning requires risk – have a go

It was lively, even noisy at times, but it was a good learning environment.

I thought I’d highlight some of the things I did right today, if only to remind myself later that these are good teaching strategies to foster learning.

Teaching Data (or ‘tools’, in general)

At our faculty day today, we had a guest speaker – Mr Stuart Palmer from PLC Croydon.

He was amazing in his passion, achievements, creativity and generosity with resources.

One of the things which really struck me is pretty obvious, i.e. teach tools in the context of their functionality. This is a philosophy I uphold with technology and, being a beginning maths teacher, have struggled with to apply in maths. It’s fantastic to be walked through a unit of work that applies this.

His unit of work (on Data) which, in essence, looks like this:

  1. Introduce the whole toolkit, e.g types of graphs (data display), measures, etc
  2. Introduce the language/jargon; a match-making activity with example-writing is a fun way to reinforce learning
  3. Work on a guided investigation using an article straight of the  news, preferably one  appealing or relevant to teens; his example was on diabetes and pregnancy
  4. Do some self-directed investigation with resources within a click away and accessible as often as possible

This is just a fraction behind the wonderful piece of work with built-in differentiation. However, this summary captures the point I’m trying to make.

With maths, quite often teaching and learning revolve around the concept and mechanics and not on real application (generally the fun bit). At least, not often enough. So his challenge to us was to think about how we can teach the concepts and mechanics within the context of the ‘fun bit’.

As mentioned, I already have this skill with regards to technology – I believe this has made me successful in my previous IT profession. Now, the challenge is on to transfer this skill in the educational context, within the classroom.

So today, not only have I learned, I’ve also been challenged. Then again, isn’t that what makes learning fun in the first place?

Reflection as part of learning

Our maths faculty head teacher set a ‘soft’ challenge a while ago to have some subjective assessments.

I took this on board and have devised an assessment task that incorporated self-reflection which was another skill I’d like my students to practice; 2 birds in one stone. Reflection is essential in developing sound mathematical thinking, apart from the fact that it is a good life skill itself. At its core, it necessitates looking at something from a different perspective such as looking back, looking deeper, looking from a different angle. From a Jesuit perspective, it also involves Action. Solving mathematical problems often involve all these.

Here’s what I’ve learned so far:

1. Reflections (or learning journals) are just as good for assessing understanding.

What a number is and what it stands for can be dramatically changed just by putting a tiny, little decimal point in front of it. For example, the numeral ‘1’ is a whole number and stands on its own. But as soon as you make it ‘0.1’ it now stands for 1 over 10.

This shows an understanding of the decimal point as the anchor that defines place values.

2. Aim for conceptual understanding; maths is more than just mechanics and shortcuts

I did not really understand the purpose of it and therefore was confused as to why we were doing it. It made me realize that just because you can do something and get the answers right, it does not mean that you don’t struggle with it.

Do I stop to check for conceptual understanding when the students have demonstrated they can do the mechanics?

3. Passion rubs off – case in point, Algebra

Many of my students expressed struggling with Algebra and yet have found it interesting, even fun. Algebra is my favourite maths strand and know that many are put off by it – often from what they’ve heard adults or older students say. With extra effort as well as creativity – and surely joy – many of my students have admitted to struggling with, but finding Algebra interesting; some even said useful and fun! No amount of objective testing will ever show this.

4. Students can be Teachers

I see myself in the reflections of my students. I can see where they’ve struggled and what they’ve enjoyed. I am learning new ways of perceiving topics in a way that is impossible to capture in ordinary class discussions – it is just impossible to hear everyone out every lesson. I can see that some students do appreciate the variety in delivery – modelling, direct instruction, investigation, problem-solving, questioning.

5. There can be some objectivity in the subjective

This sounds more like a philosophical discourse and, a lifetime ago, it was indeed (Philosophy of Man at uni). For now, all I mean is that there can be guidelines to help in marking subjective answers. I used Depth, Breadth and Details.

While I initially dreaded the prospect of marking these assessments, I was surprised to find myself enjoying reading my students’ comments especially the like the ones quoted above. These are just year 7 students. I look forward to Parts B and C of this assessment task.