# 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?