Do We Truly Understand Place Value?

Number Talks are a wonderful way to see where our students are with their mathematical thinking. As a part of a daily routine a Number Talk promotes number sense and mathematical reasoning. In this post I take a closer look at what a Number Talk can reveal about our students’ understanding of mathematics and how they might be used to promote a fresh perspective.

James Tanton shattered my understanding of the vertical algorithm. More than that, he helped me to see how poorly I understood place value and that many of my students function with the same misunderstanding. What made the experience more humbling was that it took him less than two minutes to do this. Imagine a simple addition scenario involving two three digit numbers, something like 236 + 543 = How do you solve this? The mathematically inclined will know that there are many ways to achieve an answer. Undoubtedly the mathematics teachers reading this will be well armed with strategies involving rounding, or partitioning that make the addition more manageable. Most people with years of experience in the traditional mathematics classroom deploy the vertical algorithm. It probably looks something like this:

The vertical algorithm worked left to right.

The vertical algorithm worked left to right.

The average person knows that to solve the equation you work right to left. If you ask a student to verbalise the process you hear something like, “first you add the 6 and the 3 to get 9, then you add the 3 and the 4 to get 7 and the 2 and the 5 to get 7, the answer is seven hundred and seventy three”. The fun begins when you demonstrate how to solve this but reverse the order. Instead of working right to left, work left to right, just like you do when you are reading. “First you add the 2 and the 5 to get 7, then you add the 3 and the 4 to get 7 and then you finish by adding the 6 and the 3 to get 9, the answer is seven hundred and seventy nine”.

Do this with a class of students and by this point they will be howling. “You did it wrong!”, “That’s not how you do it” or my favourite “You have to start with the 6”. Claiming that the answer you got is the same as the answer they got doesn’t help. Some will point out that it only works because you picked small numbers. Some throw words at you like “trading”. Many will resort to the highest form of classroom reasoning and argue “But that’s not how you do it”.

Another example of the vertical algorithm worked left to right. The answer might leave some unhappy.

Another example of the vertical algorithm worked left to right. The answer might leave some unhappy.

So you offer to change the numbers. Make them larger, be sure that when the digits in each place value are added they surpass the magic number of ten. Try something like this:

Again explain to the students how you solve this beginning with the seven in the top left corner. If you want to really mess with their heads, start with the four but be prepared for claims that you always have to start with the top row. “First you add 7 and 4 to get 11. Then you add 6 and 9 to get 15. Then you add 8 and 5 to get 13. The answer is 11 hundreds and 15 tens and 13 ones or what might be playfully expressed as eleven hundred, fifteenty and thirteen”. In the interests of conventional counting it can and should be seen that we can unpack this number into a simpler form. Our fifteen ones allow us to add one to our collection of tens. We now have 16 of those and we can easily move ten of these into our collection of hundreds. We end up with 12 in our hundreds column, 6 in our tens column and 3 in our ones column and can call our answer one thousand, two hundred and sixty three.

What does this reveal? Our students have learned to follow the vertical algorithm but they may not truly understand how or why it works. The fact that we can work it backwards, or middle out, or upside down should not come as a surprise. We should see that in our numbers we have collections of ones, and tens and hundreds etc. and that we can combine these and have totals of any value. I proved this thinking to a student by offering them $10 notes. They didn’t mind the idea of having eleven such notes, or twelve or more, even though the idea of putting 15 in the tens column just minutes earlier seemed like the work of the devil.

What does this have to do with Number Talks? I have taught many classes who can perform page after page of vertical algorithms without error. There are any number of text books which provide just this sort of practice. I can dress up the question by wrapping it in a seemingly real world problem, something like “John has 768 watermelons, he buys 495 watermelons at the market. How many watermelons does John have?” (The only sensible answer here is too many) Regardless of whether the students get the answers right or wrong, a page full of vertical algorithms tells me very little about their understanding of the fundamental aspects of place value that it exploits. But, a short number talk will.

In a number talk I am inviting and requiring students to explain their thinking. Mathematical reasoning becomes more important than correct answers. Ask students to solve an addition like 68 + 95 in a number talk and you will know which students understand place value. While participating in the Number Talk students share numerous approaches to each question. They share and hear a range of strategies. Provide students with a whiteboard so they might make their thinking visible and you open new possibilities. Include the option of an extended Number Talk using concrete materials and you allow for diverse representations of mathematical thinking. In each instance the students are revealing how they understand number and each response offers new insights to the teacher for future learning. Number Talks by design close the gap between student performance and teacher action to address and remediate misunderstandings.

The particular misunderstandings revealed in our reversal of the vertical algorithm are beautifully addressed by Tanton’s use of “Exploding Dots”. The basic premise is simple. You can add dots into a place value box until it reaches a set value. In Base Ten that value is 10. Once you have more than ten dots in a box they explode and one dot appears, as if by magic, in the box one place to the left. If you model the above addition problem with dots the process becomes very visual and it is much easier to understand why you can start with any column. The process is not done justice when explained in words, it is one of those things you have to try for yourself. The website Exploding Dots is a great place to start. The diagrams show the three stages in the process.


Above the question 768 + 495 is modelled in dots. There are orange dots to represent seven hundred and sixty eight and green dots to represent four hundred and ninety five. Clearly some of our boxes have more than ten dots, so we get some explosions as below.


Finally we get to an arrangement that is mathematically stable and we can easily read off an answer that everyone is likely to be happy with.


Modelling the addition question we posed above with dots might not be the norm. It might take longer and require more space, but it does ensure that students understand what is going on. Exploding Dots can be used for so much more than addition. As an introduction to place value, perhaps beginning with binary counting, Exploding Dots provides a strong foundation from which mathematical understanding can be built. If you are keen to correct some misunderstandings amidst your students, definitely explore the world of exploding dots. It can be a great addition to you Number Talk routine.

By Nigel Coutts