Reasoning in Maths

My year 8 class started a unit on Reasoning in Geometry today. Still going along the theme of teaching big-picture style as I did with Equations, I introduced this topic with a lesson inspired by @ddmeyer’s “How we talk about shapes” and @edteck’s (Peter Pappas) “Basic Communications exercise”.

The premise is this….

Reasoning requires knowledge of  facts and language (read, write, talk – words and symbols).

Anyway, rather than just state my premise, I went through the activities outlined in the linked blog posts above. The activities certainly made it clear to the students why we name shapes (in the way we do) and how literacy in words and symbols facilitate communication.

This places a big emphasis on literacy in Mathematics but that’s because I think Maths is a language in itself, a man-made tool to help people make sense of – and manipulate – our world. To further emphasise literacy, I also defined Geometry, listed terms we will use as well as discussed properties of shapes and why it makes sense to call a triangle a triangle, etc.

The very nature of the activities engaged the students and a quarter into the lesson, some even said “I’m really enjoying this lesson” or later yet, “I feel I’m really learning” (they do usually learn something but awareness/articulation of learning takes it up a notch). Throughout the lesson, they  also saw how their learning – over time – is valuable to build on; such is the spiral nature of maths education.

To make sure the lesson flowed as planned, I created a powerpoint.  It’s good to go for anyone who wish to use it but you really must read first the posts by Dan Meyer and Peter Pappas to see where I’m coming from.

Am I wrong to emphasise  literacy in maths?

Problems vs Exercises

Problems vs. Exercises

Inspired by a blogpost by David Cox (@dcox21), I decided to use the same problem and added an extra fraction. Anyone following me for some time would see that I value mathematical thinking and, as David said, this is a good one to show problem-solving skills. I think it’s also a good one to revise, reinforce and connect mathematical skills (or tools as David calls them).

Simplify:

algebraic fractions

I asked the class what their first thoughts were upon seeing this problem on the board:

  1. “It looks complicated”
  2. “It’s hard”
  3. “It looks like a big problem with lots of little problems”

They were all correct, of course, and I told them so (affirmation is good). I then told them that, in fact, they already have all the skills to solve this problem and they looked at me to as if I’ve gone mad. I suggested that, because it did look complicated, we look at the little problems that make up the big problem to make it easier (note the use of their responses; affirmation is good – oops, said that already).

I then asked what the problem looked like, that is, what’s familiar about the problem.

1. Dividing fractions
2. Adding fractions
3. Algebra – use of pronumerals/letters

Upon revising the above, we added 2 more:

4. Multiplying fractions (division of fractions as multiply by inverse/reciprocal)
5. Order of Operations (fraction bar as a grouping symbol)

And so we set off to solve the problem and they asked for more to practice on, an Exercise as David puts it. This was the desired and expected effect. I used a similar approach when I introduced Decimals to my year 7 class last year (note to self: must share/blog this resource).

I should have also pointed out that this sort of thinking/questioning/problem-solving approach can be applied in real life. Sometimes problems we face in life can seem hard and complex yet often, with chunking (or breaking down into smaller bits), we find that we have the skills/tools to solve them.

Percentage Composition

I wanted to make this lesson a bit more interesting as well as incorporate opportunities for connections with other maths topics.

I knew my year 8 class students loved music so I devised a very simple poll on Rob Thomas (one of the students was going to his concert): like him, don’t like him, don’t know him. This was a good opportunity to revise Frequency Distribution.

Just by fluke, our 3 fractions turned out percentages of recurring decimals (another connection). In that sense, it’s really not a very good first example! Anyway, someone asked if the percentages will total a hundred (I love it when they think!). So, we added and of course it didn’t. We got a result of 99.8%.  This opened up a discussion on the consequences of rounding/truncating decimals.

I made a statement that the most accurate way to represent not whole (fractional) values is via fractions. Nods all around until I asked, “‘Do you believe me?” The consensus was yes because I was the teacher. I was quick to point out that they should not always believe everything teachers say – they have to think if it’s reasonable; teachers are humans, after all, and can make mistakes. Besides, I really wanted to challenge my students to develop their thinking and reasoning.

So, I went back to the Frequency Distribution Table and 3 fractions. Adding the fractions naturally gave a total of 1 and nearly everyone said that’s 100%, as expected. Joy!

Were it a more able class (we stream our Maths classes), I could have pursued more connectionist opportunities but I did sense I’ve pushed enough today. I know I’ll refer back to this lesson when we actually do Data later in the school year. The class was engaged because they knew the meaning behind the numbers.

By the way, the statement is a fact I had truly learned in my previous career in IT. Particularly when I worked for a bank calculating interests, calculations in programs (software rather than curricular) were designed to stick to fractions as much as possible to minimise rounding errors.

In summary (more for my future reference),

Learning strategies: connectionism, motivation and engagement via meaningful and relevant examples, question/reason validity, compare/contrast

Fractions, Decimals and Percentages – provide different ways to represent fractional values, of which fractions are the most accurate

Love2Learn – some strategies

I’m a teacher but, unlike most teachers I know, I am not passionate about teaching. Rather, I’m passionate about learning.

This blog will be an attempt to record my learning…as a teacher, as a “student”, as a human being. The challenge is to document – via this blog – one item I’ve learned a day. Obviously, some days are packed, like today, when I attended a professional development course. The fun will be in dissecting days when learning isn’t so obvious.

Learn To Learn

This was the title of the course I attended today. There was some theory:

  • learning is an assessable product but it is also a (teach-able) process
  • learning is a factor of skill and will (cognitive and affective) as well as reflection (metacognitive)
  • learning is a cycle of planning, acting and reflecting/evaluating
  • learning necessitates a questioning mind (in my opinion, questioning is a manifestation of opening one’s mind, a necessary premise for learning it self)
  • learning is personal/individual as well as social/interactive

I really enjoyed the strategies and examples, some experiential, that were shared. The list is massive so  I’ve picked just a few here, possibly the ones I’m most likely to adopt.

  1. Metacognitive thinking, aka thinking about thinking: KWHL, self-assessment (Journal, learning log)
  2. Critical Thinking : PCQ/PMI, analogies (similarities and differences)
  3. Creative Thinking: SCAMPER
  4. Questioning: Progressive Questioning, Socratic
  5. Remembering: RAP, 30-word/short summary, analogies (making connections)

KWHL

K: What do I know now? (Foundation)

W: What do I have to learn?  (Goal)

H: How can I learn more (Plan and Activities)

L: What have I learned (Evaluation)

Journals – Potential scaffolds:

  1. What I recall and What I wonder
  2. What did I learn and what did I not learn (questions, confusions)

PCQ

P: Pros, positives, plusses, advantages

C: Cons, negatives, minues, disadvantages

Q: Questions, consequences, what-ifs, interesting

SCAMPER

Substitute, Combine, Adapt, Modify/Magnify/Minify, Put to other uses, Eliminate, Reverse/Rearrange

This has obvious application in design-oriented task but may also be used to challenge assumptions and the status quo.

Progressive Questioning

Have the open-mind of a 5-yr old who constantly asks What, Why and How. Of course, the purpose is not to ‘annoy’ but to get a deeper and/or broader understanding.

Socratic Questioning

The list is massive but the crux is asking questions to clarify and probe. Apart from the overt given, questioning should encompass assumptions, reasons, evidence, perspectives, implications.

RAP

R: Read and highlight key words/phrases

A: Ask key questions

P: Put into own words