So, in my previous post, inspired by MathIsFigureOutAble podcast, episode 81, I wrote about an important building block for single digit addition facts. As I mentioned previously, the episode is titled “part 1,” so I’m looking forward to listening to what Pam Harris and her co-host of the podcast have to say about single digit addition facts in their future episode(s?). But, I just wanted to write some important ideas about addition that I would want students to understand as they go beyond single digit addition facts. Some of the ideas are not strictly about addition as an operation, but they may have more to do with students’ understanding of numbers and also the base-10 numeration system (Hindu Arabic system). I tend to think the understanding of operations (including ways to actually carry out calculations) and understanding of our numeration system develops reflexively, not sequentially.

The first idea I would like students to understand is the idea of ten as a unit. There is big difference in understanding 36, for example, as 30 and 6 versus 36 as 3 tens and 6. Because English number words (number names) have an obvious pattern, it is not that difficult for children to see 36 (thirty-six) as 30 (thirty) and 6 (six). However, many young children who can say 36 is 30 and 6 do not necessarily understand that 30 is three tens. This understanding is not going to happen with the study of single digit addition facts. In fact, with English teen number names, many young children have difficulty understanding numbers 11 through 19 as 10 and some more. Thus, the idea of composing numbers 11 through 19 as 10 and some more is as critical as the idea of composing numbers through 10 as combination of 2 numbers for sums of 10 or less. I don’t think Kindergarteners will develop the understanding of ten as a unit when they study numbers through 100. Yes, they will learn the number names for decade numbers, twenty, thirty, forty, …, but for many of them, it is just a sequence of words with no numerical significance. You may want to try something like the following:

Arrange some counters/base-10 blocks like this:

Hide these blocks under a cover, then gradually reveal them and ask how many they see:

I have seen many children answer this series of questions by saying something like, “three,” “three, thirteen, and twenty-three” and “thirty-three, forty-three.” For those children, the number word sequence, “ten, twenty, thirty, forty,…” is just like the sequence “one, two, three, …”

One place where this understanding may be fostered is when students think about addition (and subtraction) of multiples of ten, like 30 + 40. At least, helping students moving toward that understanding should be a big goal when we teach addition of multiples of ten in G1. However, research by Steffe and his colleagues show that this understanding, ten as a unit, is a very challenging understanding, and some G2 students are still constructing this idea. Some people argue that because some East Asian languages, like Japanese and Chinese, has number names that more clearly align with the base-10 numeration system (i.e., numbers like 15, 37, 81, are named ‘ten five,’ ‘three tens seven,’ and ‘eight tens one’), those children understand this idea more easily. However, I have see Japanese first graders (G1 is the first year of schooling in Japan) struggles with the series of questions above, too.

Somewhat related to the idea above is the idea of you can only add two numbers that are referring to the same thing – the idea that you can’t add apples and oranges. Initially, this idea will be understood more in the context of problem situations – you literally cannot add the number of apples and the number of oranges. However, when students start thinking about addition like 30 + 40, and with the understanding of ten as a unit, they can see 30 + 40 as the situation where we have 3 tens and 4 tens, thus we can add 3 and 4, not 30 and 40, to find out how many tens we have. The Common Core State Standards, 1.NBT.C.4 explicitly addresses this idea. By the way, addition problems shouldn’t be written vertically until/unless students have this understanding. When they do, it makes sense to write two numbers vertically, aligning the place values, thus, two (or more) numerals in the same column are referring to the same units, thus can be added. I have seen some US textbooks where single digit addition problems are presented in the vertical format, and that makes no sense to me. Presenting the vertical notation so early seems to be an indication of the imposition of a particular algorithm.

Another important idea is that the idea that if you know the 100 single digit addition facts (i.e., 0 + 0 through 9 + 9), we can add any to numbers, no matter how large/small they may be. I know there are people out there who do not think algorithms (for the basic 4 arithmetic operations) aren’t important, and I agree that we should not be imposing algorithms onto students. But, developing/devising calculation methods that can be used with any numbers (starting with whole numbers) that are based on our numeration system is a critically important part of any student’s mathematics learning progression. I think we have had MANY issues with teaching of algorithms in the US, but I believe those issues are more about how they were taught (and sometimes what they were taught in the sense that one and only one algorithm – *the* algorithm – was emphasized). But, there is nothing wrong with algorithms themselves, and students need the experience of devising algorithms (see my previous post on algorithms here).

Again, I’m not saying we should be imposing any particular algorithm on students. Rather, teaching of algorithms should mean helping students’ own strategies into a written process. Of course, before we can do so, we have to help students develop strategies, particularly strategies that can be applied to any numbers. Making a particular strategy that applies only certain types of numbers seems to be rather ineffective/inefficient. I realize sometimes (often?) we see students thinking of numbers as if they are simply a collection of single digit numbers that are somehow glued together. Thus, when they have to calculate, for example, they don’t see, for example, 153 + 199, they don’t see 199 is just one away from 200. Furthermore, more often than not, when you are trying to mentally calculate, it is often helpful to think of numbers as a whole. For example, to add 37 + 46, it might be easier to think of 37 and 40 more will be 77 and 6 more will be 83. However, we have seen many students who try to picture these numbers written down vertically and try to apply “the standard algorithm” with them and get confused. So, it is important for students to maintain the meaning of numbers they are working with. Again, I contend that those students who jumped to algorithms blindly are most likely those on whom a particular algorithm was imposed, and they don’t understand why the algorithm works.

Finally, I want to stress that computational fluency is not about speed. Rather, fluency is more about students flexibly selecting and using computational methods based on the particular set of numbers to operate. So, we don’t want students to blindly apply an algorithm – even if it is devised by themselves – when they have to figure out 253 + 199, for example. But, the idea is for students to be able to flexibly select an appropriate (for them) method from repertoire of methods. To do so, algorithms should be a part of their repertoires. I have previously written about computational fluency, in which I referenced the three foundational ideas Susan Jo Russell mentioned. Having the knowledge of a large repertoire of number relationships is an important foundation, but if we just leave students with those 3 foundations – as important as they are – without taking the next step to develop algorithms seems to be a big disservice to students.

I’m sure some people may not agree with me, but I would like to hear about your perspective in that case.