Want to listen to an expert talk about the most essential addition strategies our young children should be learning for future success?
Dr Elida Laski
This is episode 10 and today I’m speaking with Dr Elida Laski who is an Assistant Professor of Applied Developmental Psychology at the Lynch School of Education at Boston College in the US.
Elida’s primary research focus is mathematics in early childhood and she has published many papers and won many awards for her work, including being highlighted in the Wall Street Journal, Science Daily and CBS Radio. For our purposes, however, her most outstanding qualification is that she was, originally, a Kindergarten teacher, so she knows what she’s talking about!
If you want to check out her research further, pop over to The Thinking and Learning Lab.
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Now to the interview.
Elida Laski, welcome to The Early Childhood Research Podcast. Thank you so much for joining me today.
Thank you so much for having me, Elizabeth. I think what you’re doing is really important.
What is decomposition?
Today we’re talking about one of your areas of research, and that is the early use of decomposition for addition and its relation to base-10 knowledge. When I hear the word ‘decomposition’ the first thing that pops into my head is the picture of an apple core slowly rotting away in my garden. What does ‘decomposition’ mean when we’re thinking in terms of mathematics and young children?
Well, I think your image of an apple core rotting away captures the essence of decomposition, it’s breaking down. And that’s how we refer to it with the mathematics of young children. But decomposition is just the breaking down of a problem that might seem difficult or more complex into simpler problems that are easier for children to handle.
So, I could give you some examples of the kinds of problems children might use decomposition on and how they would break them down.
Types of decomposition
There are a number of different kinds of decomposition.
- One of them that is more familiar to early childhood teachers is the breaking down of problems into known facts. For example, if the child was given the problem 6+8 they might automatically know that 6+6=12 and then 2 more would be 14.
- Another kind of decomposition is when you have 2 double digit numbers and you break down the process of adding those together into first adding the tens and then adding the units. For 38+23 you would say 30+20 is 50, 8+3 is 11, 50+11 is 61. It allows them to perform mental math without having to worry about the algorithm of carrying the ones.
- And then the final kind of decomposition that’s really quite common in the East Asian countries, and one that has been shown in my research to be really important for helping children do mental math, is known as going ‘through 10.’ That one would be, for example, if you had 24+9 the goal is to break down the 9 to get to the next possible decade. So you’d say, ‘24+6 is 30’ and then just add whatever’s left over, so you’d have 33.
So the kids really have to have a good understanding of how those numbers all go together to be able to do this.
Yes, they need to understand the difference between the tens digits and the ones digits, and how those are composed to make a bigger number which is the idea of the base-10 structure. They also have to have quite a bit of fluency with how to break down 9, for example, into various ways. Sometimes 5+4, sometimes 6+3 and so forth.
So it just pops into their head when they see the problem.
Right, so they know if they need to borrow 6 to make 24 become 30, they don’t have a lot of extra demands to figure out what’s left over from the 9.
Base-10 number structure
So, it’s understanding that base-10 number structure that gives the children the ability to do these calculations?
That’s right. Base-10 number structure has 2 basic elements to it:
- Generally understanding that the number system is organized around decades. That lets children see the decades as an important jumping off point for that ‘through 10’ strategy, for example.
- Understanding that the place notation of numbers reflects the recursive nature of the decade. So that a 3 might actually represent 30.
If we talk about children between the ages of 3 and 5, what kind of activities can we do with them to encourage familiarity with base-10?
Count to at least 30
One thing that research suggests is important is counting much further than early childhood teachers tend to count to. In order for kids to understand the recursive nature of the number system, the important of a decade, they have to count well beyond 10. It’s not until 30, for example, that we start seeing that it just starts over with 1 to 9 and then the decade starts over.
So if they’re only practicing 1 to 10 or 1 to 20, they’re not actually being given the opportunity to see how that happens.
Label 2-digit numbers carefully
Exactly. Another thing that can be really important with younger children is when you start labeling double digit numerals to be quite careful to do so in a way that doesn’t lead to certain misconceptions.
For example, sometimes a teacher or a parent might say, ‘that’s 12, it’s a 1 and a 2.’ But, in fact, it’s not a 1 and a 2, it’s a one-ten and two units.
One of the very common mistakes that young children make when they’re identifying double digit numbers is to think of it in a way where they think 76 is a 7 and a 6. That shows a lack of base-10 structure because they don’t understand the 70 as a double-digit numeral.
And I think some of the ways we label the room for children leads to these kinds of misconceptions even though we don’t intend to.
Board games based on 10
I did a post on one of your previous papers and it was to do with board games and the way that it’s preferable to do 1 to 10 and then above that, 11 to 20 and so on. That’s what you’re really talking about, isn’t it? The visual representation so the children can see the 10, the 20, the 30, they’re building on each other.
Exactly. With a board game like Chutes and Ladders, sometimes known as Snakes and Ladders, it snakes around. What happens when it’s organized in that way, we think, is that it doesn’t highlight for children the relationship between numbers and the important role of decades and the recursive nature of decades.
Because half the time they’re going backwards.
Exactly! And 14 is up above 6 rather than above 4 so they’re not seeing the relationship. We think one of the reasons the research board game worked is because of the way the numbers were structured. They could see that the further they got along, the more decades they passed.
Addition strategies: 1. Count All
Let’s backtrack a bit because when we’re talking about young children learning to add, being able to decompose numbers is not going to be our top priority. Firstly, we want them to be comfortable with basic numbers and counting. When children are shown, say, 2 small groups of glue sticks, they’ll start pointing and counting from 1 and keep going until they’ve counted the last glue stick. In your paper you call this the ‘count-all’ strategy. How effective is this strategy, and just how young can children be, and still do it effectively?
This is the very first strategy that children tend to use when confronted with addition or arithmetic. Children as young as 2 and a half or 3 can execute the strategy depending on the size of the sets they’re being asked to add. A 2 and a half-year old might add 2+1 but it’s not until they’re 4 that they can accurately execute it for 6+4.
The problem with this strategy is two-fold.
- It can become very cumbersome and labour intensive as the size of the addends or the sets becomes larger.
- It can be error-prone. So you’re adding, for example, 6+14 you’re much more likely to make some accidental counting error.
And then it’s very frustrating because they have to back to the beginning and start all over again.
Or they end up with the wrong amount but it’s just one off and they can’t figure out where they went wrong.
Addition strategies: 2. Count On
So, then they go to the next step up which is the ‘count on’ strategy. Instead of counting the first number in an equation, they’ll start with that number and then add the second number by counting upwards from there. At what ages can kids use this strategy effectively?
It depends on their socio-economic level. In recent research we’ve found there’s huge differences in children from upper class families relative to those from lower class families. With some children being able to use this strategy quite well by the age for 4 and then some 6-year olds still struggling with it.
Encouraging counting on
There’s actually a transition in this strategy where the first thing you have to do is just be able to hold one of the addends in your mind. So if the problem is 2+5 you have to be able to hold 2, and then start counting from there, 3, 4, 5, 6, 7. So not only do you have to hold one of the addends in your mind, but you also need to be able to enter the count stream from somewhere other than 1. Most children just practicing rote counting always start with 1.
Going back to your classroom, one of the things we could do with preschoolers is invite them sometimes to start at 3, or start counting at 6. That would help prepare them for using the ‘count on’ strategy.
Counting on efficiently
The other thing is that children take a little bit of time to use the most efficient version of the ‘count on’ strategy, which is counting on from the largest addend first. If it’s 2+5 you don’t actually want to start at 2. It’s more efficient and less error-prone to start at 5 and then only count 2 more.
And to be able to do that children need to be able to identify which is the larger number. They also need to conceptually understand that it doesn’t matter in addition what order you go in, as long as you’re combining them.
So there are a couple of pieces that children have to understand before they can start executing the count on strategy well. But it’s also much more efficient and so it’s worth trying to move children toward it.
Addition strategies: 3. Retrieval
After this, things get trickier and the next step up is the ‘retrieval’ strategy, which relies on children remembering learned facts. Just how important is it that kids can remember specific facts, and what are the most essential facts that we should be giving our kids plenty of opportunity to learn?
Retrieval is incredibly important.
There’s a very large body of research that shows that the ability to quickly and accurately retrieve arithmetic facts not only matters in early arithmetic, but plays a really important role later on in say, 3rd and 4th grade when children start doing more complex computations.
What happens is that we can only hold so much in our mind at once, so we’re adding 2 double-digit or 2 triple-digit numbers. If we’re having to solve every column and think really hard about that we lose track of where we are in our overall computation.
The decomposition we were talking about before is a much more efficient mental strategy when you’re faced with doing arithmetic in your head.
We need to know the sums within 10
So the most important arithmetic facts to ask children to start learning are the sums within 10. Because that’s usually the way they start breaking down problems to be able to execute a decomposition strategy.
You’re talking about number bonds? So 2 and 8?
So if the teacher says the number 6 then all the kids can shout ‘4,’ is that what you’re talking about? Number bonds to 10?
Correct. We call them here in America fact families. Basically all the different ways you could decompose 10, all the addition problems that would lead to a sum of 10.
We need to know how smaller numbers can be broken down
You’re also saying for the other numbers, 9 for example, if the kids know they can do 5+4 or 3+3+3 that that’s going to help them a lot as well.
It’s going to help them a lot to be able to think quickly to use these mental strategies but also to be more accurate with more complex arithmetic because they’re thinking about less which means they’re attending to more.
When I think about kids decomposing numbers when they’re young I keep thinking ‘how do they hold it in their head?’ That seems so hard for them.
Exactly. It impacts what we call in psychology, working memory. We’re very limited in how much we can hold in our head. But if those facts are SO automatic it won’t take up much working memory space.
There’s 2 ways to get to the retrieval strategy.
- One way is to just drill kids, which is not usually very comfortable with teachers, and we don’t want that to be the centre of math learning in early childhood.
- But another way is just lots and lots of practice using these other strategies like count all and count on. The more times you’ve solved problems using these strategies the more likely you are to start associating an answer with a particular problem.
So, just over and over in fun ways, I assume.
What about mastery of addition strategies?
If we have children who haven’t mastered counting all or counting on, should we expose them to decomposing numbers as a strategy?
It depends on how we define ‘mastered.’ In addition strategy research there are 2 ways of thinking about that.
- Have they shown an ability to use it at all? Even though it may not be their preferred strategy.
- Has it become now their predominant strategy?
Some children will use ‘count all’ with easier problems, but with harder ones they’ll use ‘count on.’ What the research suggests is that as long as they have started to show some evidence of being able to use ‘count on,’ it’s better to push them toward ‘count on’ above and beyond ‘count all.’
It’s better to just tell kids, ‘you know what? This is a faster way of doing it! You should do it.’
I think the same is true for decomposition. Once the child, even if they’re not using ‘count on’ all the time and don’t seem to have mastered it, if the child shows some evidence of knowing how to break down a 3 or knowing some of the twins like 5+5 is 10. If you give them 5+6 why not ask them, ‘you already know that 5+5=10 so 6 is only 1 more.’
Encourage kids to use new addition strategies
So the research suggests that it’s important to invite them to use it and sometimes encouraging them to use the most efficient strategy as opposed to always giving them the freedom to choose.
When we’re teaching children maths should we be teaching them the strategies that get them to the right answer the most often? Or should we be asking them to chill out and not worry so much about the answer? A lot of children get stressed over wanting to be correct, so they don’t want to try a new strategy? Should we be encouraging them to ‘have a go?’
I think so! To one extent math is about being exact and it’s important that they learn to arrive at correct answers. But there was a really interesting study done by Rob Siegler, where he looked at the kinds of strategies children of different confidence thresholds, he called them, used. There are some children who are risk takers and they’ll always try a really efficient strategy but they’ll never get to the right answer.
And there’s some children who have low thresholds for risk and they will always use ‘count all,’ the most cumbersome strategy, but they’ll always get it right.
What’s important in early childhood is trying to help children have a balance there. To choose the strategy that’s most likely to lead to a pretty good answer, if not the right answer, but also to do it in a way that’s efficient. Sometimes that means asking them to work outside their comfort zone, to take a little risk, but starting to get familiar with that strategy.
Every child is different
For a lot of kids in their learning, it is their personalities that we’re dealing with as well, isn’t it? People think we’re just teaching kids facts, but children are all so different and they have different beliefs about themselves and they have different worries and concerns so as a teacher we need to approach each child differently sometimes to psych them into using a different strategy.
Absolutely. In my work with children there are some that will react very well to just being told, ‘look, this strategy’s faster, do it this way now,’ and other children who don’t seem to be willing to take that direction. What has been shown to work with them is what we call challenge problems.
It’s counterintuitive, but if you have a child, for example, who’s confronted with 5+4 and 6+3 who continues to use ‘count all,’ part of it might be because they don’t see any benefit to counting on because counting all is pretty effective. Then, if you offer them a problem that’s 6+18, all of a sudden it seems to make that light switch go off so they realise that counting all is NOT going to be a good solution, or ‘it’s going to take me a really long time.’ Then they understand the benefit of some of these strategies we’re trying to teach them.
In Australia, and most of the rest of the world, our children use the metric system, which means they’re dealing with the base-10 consistently whether they’re talking about numbers, or if they’re talking about weight, distance, cooking measurements and so forth. Do you feel that the continued use of US customary units such as feet and inches, pints and quarts puts your kids at a disadvantage? Why or why not?
Goodness! That’s a whole political can of worms! I would say, at least not in early childhood, because they’re mostly not working. I think it could potentially put them at disadvantage in that opportunities to practice these skills and to see other contexts in which the base-10 system is at play. But we don’t see in the research, when we look across national studies, any significant differences between the US and, say, Germany, countries where there’s only a difference in the metric system.
So, the kids get it in the end? It just seems to give them more to learn.
It’s making them more cognitively flexible, right?
Of course, just like being bilingual!
Top 3 suggestions for early childhood teachers
What are your top 3 suggestions to early childhood teachers related to teaching math effectively to young children?
Don’t underestimate children
My very first suggestion would be not to underestimate young children. I think they’re capable of much more than we thought even 10-15 years ago. I don’t mean to push them in a way that’s uncomfortable, but not to underestimate them. For example, even just 10 years ago in the US the standards for Preschool and Kindergarten were only to count within 10-15. And we know that children are capable of thinking about, and working with, much larger numbers than that.
Take them past the concrete
My other suggestion would be to not hold religiously to the idea of progression from concrete to abstract. Too often we think we have to start with concrete materials, and introduce concepts with concrete, and the research suggests that children can simultaneously be thinking about concrete and abstract depending on the size of the number. Sometimes we persist in the use of concrete for too long and it actually becomes a hindrance for children.
We’re holding them back, is that what you’re saying?
Yes, either because they aren’t developing these mental strategies because they don’t need to bother. They have constructs in front of them so there’s no impetus. Or they’re just not challenging themselves to start forming a mental representation of the number which we know is really important.
I’m not saying that using manipulatives and concrete materials in Preschools isn’t important. In fact, I believe it is. But my suggestion would be to think carefully about when you can also push children towards mental, or help them transition from concrete to abstract, faster than you might be accustomed to.
It goes back to what you were saying before, sometimes we’re underestimating what our children can do.
That’s right. And my third suggestion would be to not be afraid to just tell kids things. It’s not always necessary for them to discover everything on their own.
I agree with that.
Like I was saying before, ‘it doesn’t make any sense that 6+7 would be 8. That doesn’t make any sense, just think about it!’ Or, ‘you’re doing it a really slow way, here’s a faster way.’ Sometimes children need these little influxes of information in order to stop spinning their wheels and to move forward.
Sometimes if you’re waiting for children to discover it, some children will discover it a lot earlier than others, and it puts the others at a disadvantage, because they’re just thinking about the cake they’re going to have for afternoon tea, they’re not really thinking about the mathematics! So just give them some explicit teaching and let them go on their way.
That’s right. And there’s a whole lot of research with older children, too, that shows that, we were talking about working memory a while back. In fact, sometimes the discovery process actually interferes with learning because thinking about all the different things you could possibly try or attempt puts a lot of load on working memory. So your resources are spent on a lot of irrelevant ideas. The children’s resources, I mean, not the teacher’s.
It’s all about balance
There are a lot of different philosophies of teaching, aren’t there, and so depending on how we expect our children to learn it really can change what children do learn, how quickly they progress, and how much fun they have while doing it.
I used to be involved with literacy before I started doing math research and I can parallel what’s happening in math now, with what happened in literacy 20 or 30 years ago. It used to be whole language or phonics, and now we know it’s actually a little bit of both.
I think the same is true in math. It’s not skill and drill, but it’s also not just conceptual ‘discover it.’ What children need is a balance of both.
That sounds like a great place to end this interview. Thank you so much for being on, it’s been great to listen to you.
I just think the way you translated that patterns study, I was so impressed. And I think what you’re doing is so important. I was a Kindergarten teacher and I can remember never having any resources and struggling. I think you have a real knack of taking research and translating it into lay language.
I wish I had that knack because then my research would get to more teachers! Thank you so much.
You’re very welcome!
The Cult of Pedagogy Podcast
I mentioned earlier that The Early Childhood Research Podcast has joined The Education Podcast Network. While this podcast is the only one specifically aimed at early childhood (at the moment), there are plenty that cross over that you might like to check out. I was listening to episode 33 of The Cult of Pedagogy Podcast the other day that I really enjoyed, called ‘5 Powerful Ways to Save Time as a Teacher.’ If you want some time-saving ideas you might like to check that out!
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