How do you know?

Being able to verbalise the steps taken to arrive at an answer or a conclusion is a valuable skill. It is particularly important with mathematical thinking, even at beginning stages. Sometimes it feels as if the answer appears without the need for thought. Of course we aim for this with automaticity of number facts and times tables. However, I am referring to problems that may involve two or more different calculations.

Unless one understands the processes involved in solving one problem, it can be difficult to apply the same strategy in other situations. Sometimes it may be difficult for a teacher who finds the answer effortlessly to understand a learner’s confusion leading to mis-steps and miscalculations.

I generally collect my six-year-old grandson (G1) from school once or twice a week.  Now that he’s a big year one boy he thinks he doesn’t need to hold my hand to cross the road. However, there is a bit of traffic around his school and I prefer him to do so, telling him it’s for my safety too.

One day I, erroneously, told him he needed to hold an adult’s hand until he was ten. (Ten is closer to the age for riding a bike unsupervised.) While this post is about mathematical thinking rather than traffic safety, if you wish to do so, you may find some information about traffic safety with young children here.

Always on the lookout for teachable moments that encourage thinking, I then asked if he knew how many more years it would be until he was ten. It was a bit of a redundant question because, of course he did. He is an able mathematician, just like his dad who constantly extends his ability to compute and think mathematically.

When G1 didn’t answer immediately, I assumed that he was either ignoring my question as it was too easy for him and not worth answering; or that he was thinking of the implications of my initial statement about holding an adult’s hand until he was ten.  Then again, perhaps teacher-type questions from GM are sometimes better ignored, particularly after a day in school. I was happy to accept his silence as we continued across the road and didn’t press him for an answer.

Once safely on the footpath he said, “It’s four.”

“Know how I know?” he continued, pre-empting my question (he knows his grandmother well).

He held up both hands. “Because 5 and 5 are 10.” (He put down 4 fingers on one hand.) “And that’s 6. And 4 more makes 10.”

Although I hadn’t asked how he knew (on this occasion), I was pleased he was able to tell me, even though, in reality, he “knew” without having to work it out. On previous occasions when I had asked him how he worked it out, or how he knew, he hadn’t always provided an explanation. He may have shrugged, said “I don’t know” or simply ignored my question. Being able to explain his thinking demonstrates his growing mathematical knowledge and metacognition.

The ability to think through and verbalise steps is important to understanding. How many of you talk yourself through steps of a procedure you are following? I certainly do at times. While knowing that six plus four is ten is an automatic response for most of us. It wasn’t always so and we needed a strategy to help us understand the concept and recall the “fact”.

Opportunities for mathematical conversations occur frequently in everyday situations but are often overlooked. Recently I described some ways the children and I discussed different ways of combining five of us on an outing to show that 3 + 2 = 5.

Recently when two grandparents and two grandchildren were travelling together in the car four-year old G2, without any prompting from me, began describing how we could be combined to show that 2 + 2 = 4, for example

2 adults in the front, 2 children in the back, that makes 4. There were many different ways that we could be combined; and just as many that would group three together and leave one out, for example

1 driver and 3 passengers.

We had other mathematical discussions during that car trip. I had cut an apple for each child into a different number of pieces. Each was to guess how many pieces there were. G1 went first. I’d introduced him to prime number just once previously when I’d cut his apple into seventeen pieces so I didn’t expect him to be proficient with them. He certainly wouldn’t have been introduced to them at school at this stage.

These are the clues I gave him, the guesses he made and my responses supporting his growing understanding. I thought he did very well. He requests a guessing game each time I cut apple for him now. It’s sometimes a challenge for me to think of new clues.

I had cut 15 pieces for G2. G1 helped her work it out when I told her that she needed to count all of her fingers and the toes on one foot.

For an additional challenge I asked G1 if he knew how many fingers and toes there were in the car all together. He thought for a moment before giving the answer. Then proceeded to tell me how he knew as I started to ask. He explained that he had counted in twos because each of us had twenty, and counting twenties was just like counting twos but they’re tens. A quicker and more effective way that adding on twenty each time which I may have suggested he do.

Asking how do you know or how did you work it out helps children think about their own thinking. Listening to their responses helps adults understand where they are in developing mathematical concepts. Asking questions about their thinking can challenge and extend them further, but it is important to not expect too much and to support their developing understanding.

What maths did you engage in today? Did you even realise or was it automatic?

14 thoughts on “How do you know?”

1. Annecdotist

This is a great way of helping children to develop their critical thinking, which I know is one of your passions. But quite a complex skill to know how you’ve got to the answer and then be able to articulate it. It connects with a non-fiction book I’m reading at the moment which looks at how children develop their thinking through internalising conversations with themselves – and how this continues as adults, as I often catch myself doing when talking to myself! A good lesson in this post for us all at any age.

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1. Norah Post author

Thanks Anne. I appreciate your comment. Would you mind telling me the title of the book you are reading. It sounds like one for me. I can’t quite remember where, but I think it might have been on his blog – Geoff and I had a recent discussion about talking to ourselves. Sometimes that talking helps to make things clearer, even if it does draw inquisitive looks from others. 🙂

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2. Bec

A lovely post, Nor! I always enjoy hearing about the adorable conversations you have with the little ones. And how great that your habit of asking for explanations has made it habit for them to think about ‘why’!

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3. Sarah Brentyn

Love that he explained right away. I’ve been asking “how” since mine were very young. Math was, this past school year, my “how” focus. Great. You got the answer correct. How did you get there? It’s a struggle. Especially with my Aspie. *shrugs* “I don’t know.” We’ll get there. 🙂

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1. Norah Post author

I know. Being able to see the steps taken, and then verbalise them, requires an additional skill. It is possibly more difficult for those with Aspie, but you would know better than I on that one. Thanks for sharing. 🙂

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4. Sacha Black

Lovely post. I have been thinking about this a little recently. As I want to do a post on how I deconstruct novels. But it has sort of become automatic. I ahem gone past the conscious stage and so now trying to know how I do it is actually quite hard!

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5. roughwighting

What a good grandmother you are! I tend to quiz on things less mathematical and more theoretical/philosophical, which some of my grandkids love; others would prefer a ‘numbers’ game. Raising my kids, I was so non-mathematically inclined. Yet my son tested above the charts in math and now is a financier of some kind (darn if I understand what he does). On the other hand, my 6-year-old grandson announced to me that he had been alive 1,000 years ago, and just 6 years ago decided to return to this life. Now, THAT’s the kind of conversation I enjoy! 🙂

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1. Norah Post author

Hi Pam. Thanks for joining in the conversation, and for your encouraging words. What a good grandmother you are! I like to have those theoretical and philosophical discussions too, thinking about who we are and why things are as they are. You grandson’s comment about having lived 1000 years ago reminds me of a book by Wayne Dyer called “Memories of Heaven”, a collection of similar things children have said. Have you heard of it? I haven’t read it, but have read about it. Here is a link if you are interested: http://www.drwaynedyer.com/memories-of-heaven

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2. Steven

Lived 1000 years ago… well if you’re in to this sort of thing, it kind of reminds me of this video, “11 Creepiest Things Kids Have Ever Said”:

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