Visualise, Verbalise, Verify

This year I've been learning more and more about the importance of Spatial Reasoning in Math. I've learned that Spatial Reasoning is multifaceted and that you can be very good in one area and not so good in another. For example, the one thing that I love to do more than anything else is hiking. Now when I first started hiking and reading maps, I relied solely on man-made features such as paths and cairns and trig points. I knew that the contours on a map represented height but couldn't look at these to see what the hills looked like. Then one day in the hills, something funny happened: I looked at the 2-D map

and in my mind's eye could clearly see the shape of the hills in 3-D:

As a hiker, this is very powerful and now I must say (blowing my own trumpet) that I am pretty awesome at reading maps.
Yet I still have to think very carefully about 'left' and 'right'!
So at a session run last week by Ontario's Ministry of Education, I came across some great advice to help improve Spatial Reasoning:

Visualise, Verbalise, Verify

For example, consider the views of an object as shown below:

Visualising gets a student exercising her mind's eye to try to build a mental image about what the structure might look like. Ideally this should be done individually.

Verbalising gets students describing what they have just visualised; it forces the students to reason, to use spatial and positional language, to communicate with words and gestures. Sometimes the students will agree, sometimes they won't (and this is more fun!)

Verifying is when we allow students to create the structure to check if they are right. In the past, I have jumped straight to this stage but now I realise the importance of the getting students to visualise and verbalise. What is nice about problems such as this is that once a student has proven that their solution is correct, we can ask: "Is there another solution?" For example:

Spatial Reasoning is malleable. I am convinced that if we can get our students to visualise, verbalise and verify, then their Spatial Reasoning will improve dramatically and this will have a knock-on effect on the Math understanding.

Name That Fraction

I came across a fraction misconception last week that I've never seen before (or to be more precise, I've always missed before). We asked students what fraction was shaded:

Most said 'one-sixth' apart from one student who was adamant it was a fourth. Initially, I thought she had miscounted the parts but upon further questioning it was apparent that this student also thought that the following were also fourths: can you see why?

 

 

Basically she is describing the fractions using ordinal or positional language. What was neat about this is that before I had a chance to challenge her thinking, her partner did it for me (referring to the first diagram above):

 

"Well if that's the case, you could count from the right side and it would be called a third and it can't be a fourth and a third!"

 



The original student was pretty adamant that she was correct though which got me wondering what experiences she might need to understand how we name fractions. Perhaps she never had an opportunity (or not enough opportunities) to split a shape into equal parts like this as shown in this previous post . After the lesson I realised that I could also have challenged her fractional thinking by bringing in spatial reasoning and asked her if each of the following are fourths:

Or by asking her to name each of these fractions (will she name them 'firsts', 'seconds', 'thirds' and 'fourths'?)

It got me thinking how important spatial reasoning is in helping students understand fractions.

*            *            *

There are some other great ideas for thinking about how we can teach fractions in the Ontario Ministry of Education's latest document Paying Attention to Fractions. One of the ideas coming out of this is to avoid solely teaching fraction as a unit but rather to incorporate it as much as possible throughout other strands and subjects all year long.

Or to put it another way, Fraction Immersion.

Face the Facts: Math Minutes Cause Mathphobia

There are some things that I did when I started teaching which I definitely would not do now. At the top of the list is the Mad Math Minute. This might go under various names but essentially the idea is to do as many questions as you can in one minute. These questions were pretty much always calculation questions or recalling number facts. As a kid, I enjoyed these because I did well: I loved the ego-feedback that I got ('Top of the Class again'!). However, I don't think I ever learned anything new from doing these; I might have gotten fractionally quicker at recalling facts but never to the extent that it made me a better mathematician or a better thinker. To that extent, Math Minutes didn't help me. 

Now I am older and (I hope) wiser and I have spoken with so many teachers who have told me that they started to become Math-phobic when they began doing these Math Minutes. They tell me it's not that they didn't know the answers, but that the pressure of answering the questions quickly caused their brain to freeze. This led to low scores which led to low confidence which led to more nerves which led to more low scores and so on. No wonder they ended up hating maths.

Now my anecdotal evidence is one thing but it is backed up by credible research. Jo Boaler's excellent research points out that whilst these timed tests might have been given with the best intentions, the effect is that they lead to the beginnings of Math Anxiety for a lot of students. She refers to research from Sian Bielock that shows how that the stress caused by these tests impedes students' working memory- the area of the brain where we hold our Math facts! This is backed up in the book Learning to Love Math by neurologist Judy Willis. High stress, low interest situations results in a reactive brain (fight, flight, fear) that prevents effective recall of facts.

Curiously, those who lead the Charge of the Rote Brigade will never consider this compelling evidence.

This is not to say that students shouldn't practice Math though. The more they practice the smoother the recall. However, practice doesn't make perfect: practice of the right kind makes perfect.
Good practice, for example, might involve a game situation such as The Product Game which you can see me playing with my daughter below.

Indeed there are many board games and card games which allow students to use and practise their number sense (Monopoly, Yahtzee, cribbage etc.) One which I would certainly recommend is the excellent City of Zombies in which you must use your math skills to prevent a zombie apocalypse. When I see students try games such as these, I see them more engaged, more willing to take risks, and learning more. The opposite of what I see in a Math Minute.

So, if you permit me to use some Yorkshire bluntness:

Stop pretending: Math Minutes help no-one.