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.

Creativity in Problem-Solving

Problem-solving is arguably at the core of mathematical teaching and learning. That is, various concepts and skills are taught/learned in the hope of solving more and more complex problems. Logic is very much valued.

I have always sensed that mathematics (re: Logic) and creativity are contradictory.

I just read an interesting article on observations on Real Science : sometimes (often even) logic stumps scientists and creativity paves the way to solve problems. The article talks about diversity-induced far view talk reliant on metaphors and analogies. I learned today that the idea of far view, or distancing oneself via abstraction, is a strategy for creative thinking (read more on Scientific American). The way I see it, it’s a logical way of thinking outside the square or taking on a different perspective.

My point is, logic does not necessarily clash with creativity. Used in tandem, the likelihood of finding solutions is increased.

The challenge now is translating this into practical applications in the classroom! I already use metaphors and analogies in teaching (re: Analogies and Algebra post). I must encourage students to think of their own. Also, I need to harness more the potential brought by the innate diversity of  every class – this means, at least, more opportunities to work in group to solve problems…futuristic perhaps?

Tangential Learning

I’ve always been fascinated how learning is facilitated through fun, play and games. We have all had personal experiences attesting to this.

Professionally, I first heard about this by reading Marc Prensky’s work. Since then, I’ve always sought ideas on injecting fun activities into my classroom.

More recently, I watched the following video and first heard about Tangential Learning.

This video defines Tangential learning as

what you learn when you’re exposed to things in the context you’re already engaged in…’s the idea that some portion of your audience will self-educate if you expose them to concepts in the context that they already find interesting….

I’m not really a video-gamer, much less an author of one, but I lived by these principles and have indeed found that students were most engaged in their learning in such cases. Here are some examples:

1. Decimals

Throughout this topic, the students had to work out how many jelly beans will fit across their desks.  I worked through how they would have solved this problem as they learned more maths from primary school. It started with counting (teacher lines up jelly beans, student lines up jelly beans) to estimating to measuring and calculating. I should add that I had to throw the jelly beans lined up as they ‘carried my germs’. The students were aghast at such waste but they soon figured out that by measuring just one – and learning operations with decimals – they can earn their jelly beans. And so they did!

2. Algebra

I introduced this with a personal experience of wanting to learn how to play Rihanna’s Unfaithful on the piano. Not having played for years, this was a real challenge particularly because the music sheets were several pages long! As if learning Rihanna wasn’t cool enough, the students were particularly impressed that I used YouTube, i.e. this videoshowing me how to do it. The main point was that the video highlighted that the song only really had 3 repeating sections. Learn the 3 and you learn the song. So, not only did I get to learn to play the song, I memorised it really quickly. Patterns are the crux of Algebra. This has set the tone for the students to see that Algebra can be used in something so seemingly far removed from mathematics. Some of them also felt affirmed that teachers use YouTube to learn.

3. Reading Tide Charts and Timetables

The class was told that they were to plan a group’s day out to Manly beach using a wiki.  This YouTube video on wikis was really helpful to enthuse the class. The activity included collecting shells on the beach, watching a movie, catching public transport and be home in time for a particular TV show. I gave them all the web links they needed and they really did not mind reading tide charts at all; they knew it had to be low tide when they go shell-collecting. Some even appreciated learning about Sydney’s Trip Planner. All that besides, they also learned about the challenges of planning and collaborating.

Tangential Learning works and I like it. The real challenge is finding that context that students will be engaged in.

The importance of Affirmation

One of the most memorable questions I’ve ever encountered in my life was in a Philosophy class at uni…

How do you know you exist?”

This question encompassed differentiating real and dream worlds – How do I know I’m not dreaming? At the core of every possible answer explored then was affirmation by self (Déscartes sor sum ergo sum- I think therefore I am – comes to mind) and by others.

For me, affirmation is declaring what is true (perceived or real). It can be a mere full-stop or a full-blown exclamation mark.

As a teacher, I see that need for affirmation in my students’ faces. As a mother, I see it on my kids’ faces, too. It is certainly an essential part of learning as it helps guide learners towards the learning goal. It is an essential teaching and learning strategy.

I recently shared a learning activity with some colleagues. Often I feel insecure about sharing resources because I have little teaching experience. This time, I believed I really had something worthwhile (worth a post in its own right) – self-affirmation. Imagine my joy when an experienced teacher told me how good it was and how her students enjoyed it, too. That was affirmation by others.

Affirmations are motivating – I know because I’ve just experienced it.

Maybe it’s something I’ve always known. Blogging about this now helps me cement it further and add to my ever-growing teaching and learning strategies.

I need affirmations as a teacher; I am a learner after all. This is humbling and empowering at the same time.

I make no apologies for being a beginning teacher – I try my best. I know I have much to learn about teaching, learning, relationships, education/school culture (vs. corporate culture), teen psychology, technology, motivating and being motivated, etc. etc. etc. Every little bit helps.

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.