Well, I recently wrapped up a whole series of videos all about looking at the different strategies that kids use when solving addition and subtraction. I got a few messages during this video series about people asking, “Will kids actually come up with these strategies on their own?” 

I’m Christina Tondevold, The Recovering Traditionalist, and I hope you stick around because today we are looking at Invented Math Strategies: Do kids actually develop them on their own? In our quest to build our math minds so we can build the math minds of our students.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.)

The Development of Arithmetic Concepts and Skills (Studies in Mathematical Thinking and Learning Series) by Baroody & Dowker

Children’s Mathematics, Second Edition: Cognitively Guided Instruction by Carpenter, et. al

Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction by Fosnot & Dolk

I know this from personal experience but there’s also research to back it up, and I want to talk about both of those today. First of all, I want to share a story that I think I’ve shared a lot, so you may have heard this before. But both my husband and I grew up in the same small town. We had a lot of the same educational experiences. And yet when I was an adult, I was still solving math problems very procedurally. I thought doing mental math was doing the procedure in your head. 

It wasn’t until I went back and got my master’s degree that I was introduced to this different way to think about solving math problems. And I distinctly remember the conversation with my husband. He wasn’t my husband at the time, we were just dating. But I told him I learned about this way that kids think about numbers and solving addition and subtraction problems in particular, because that’s the one we had just talked about. And I started explaining to him the strategy and I was like so amazed by it and he looked at me like, kind of like, “Are you stupid?” Like, “Doesn’t everybody think that way?” And that’s what he said, “Doesn’t everyone think that way?” And I was like, “No! I have never thought that way about numbers.” 

I don’t remember the exact problem but it was something like 29 + 36. And this kid in the video that we had watched had taken one from the 36, given it to the 29, and said 30 and 35, and added it together. And I was like mind blown. I’m explaining this to him and he was just kind of like, “Why wouldn’t you solve the problem that way?” And I was like, “No, you stack it. It’s 29 and you put 36 underneath, and you add the ones and you carry it over,” like the whole procedure thing. 

That was one of those moments where I was like we basically had the same education, but he thinks differently about these problems. How was that? He wasn’t taught that way. He wasn’t taught that way to think about those problems, yet he did it naturally and why didn’t I? Why didn’t I? 

It was because I think back on it and I’m sure there’s lots of different reasons why but a big reason for me was that I was a rule follower. You tell me how to solve something and I can follow that rule and procedure. Whereas my husband is kind of a rule breaker. He looks at something that somebody tells him and he’s like, “Hm, I think there might be a better way to do that.” And that’s basically what he did his entire life around mathematics, is that the teacher would say, “Here’s how you do it,” and he’s like, “Why would I do it that way when I can do it this other way?” that was faster for him and made more sense. And I have been thankful for his way of thinking about math and also that I had my eyes opened to it because I’ve been able to expose my own personal children to thinking that way as well. We have never directly taught them a strategy. 

We’ve been very intentional about building their number sense and helping them understand place value, the value of understanding 10s, and the power of 10s, and grouping 10s. And those two things have been very foundational in helping my own personal kids build their strategies and how they look at numbers before they ever attempt to solve it.

Are you seeking evidence for why you can’t or…how you can???

Now there is definitely research that backs that up, but before I go into the research, I saw a quote just today actually on Instagram from a coach that I follow. She’s a personal wellness coach. Because there’s always things that we can do better and I love following people who help me grow in all different kinds of ways. In her world it was about why we decide not to do things to become healthier. 

But this quote applies even in education. And the quote was, “Are you seeking evidence for why you can’t or… how you can?” And a lot of my training with growth development and personal development, my own understanding, is this idea that our beliefs, whatever our belief is, it’s the whole growth mindset stuff too. If we believe that we can’t, we won’t. And the reason is that internally we have this, we want to prove that we are correct. 

So whatever our belief is, we go and seek things that justify that belief. So if you start off thinking, “Well, my kids can’t do that. “My students aren’t able to do that,” you will find evidence that supports that every single day. But if you start with that belief that your students can invent their own strategies, you will find evidence of that every single day. 

Now there is a ton of research out there about kids inventing their own strategies, but I’m going to reference three of them real quick for you because these are my top three favorite all-time things that I’ve ever read. 

Research from Baroody & Dowker

The first one is really kind of meaty and it’s one that unless you do this day in and day out, it wouldn’t be one that I would recommend you go buy. But it’s by Baroody and Dowker. It’s a whole series, all kinds of stuff in here, about how kids develop their understanding of arithmetic and how those concepts and skills get developed. But one important part, and this comes from page 111 for anybody who does have the book. 

This is a picture from the book right there. And they talk about how “conceptual knowledge will precede directly related procedural knowledge if a procedure is not demonstrated in everyday life or school, or frequent experience with a concept either in everyday life or school precedes instruction of a procedure.” 

So what that means basically is that if your students have not already been taught a procedure, that ups your chance of them inventing their own strategies. Because if they’ve already been taught a procedure, they think they need to follow that procedure. You’ve got a ton of rule followers like me in your classroom who just want to follow that procedure. Whether it came from a teacher or it came from a parent, if they have been introduced to a procedure, it’s less likely that you can help them build these invented strategies.

 The other point that they make is that it comes from frequent experience with the concept, either in everyday life or school. So we, I say this all the time, “it can’t be taught, it’s caught”. Kids need to have experiences that build their number sense, that build their place value, and then they will use those experiences when they are solving problems. We see it all the time, kids who know quickly what 7 + 7 is and then they also know 7 + 6, it’s related in there, because of their experiences with football, watching football, playing football. But if we have students who are not football fans, 7 + 7 is not something that they’ve experienced a whole lot, and thus 7 + 6 is something that they haven’t experienced as well. So those experiences play a really, really big role in what our students are going to be able to do in relation to invented strategies.

Research from Carpenter, et. al

Now one of my all-time favorite books is Children’s Mathematics by Thomas Carpenter et al. This was the training that I received that changed everything in me. That was one of those videos we were watching, was a CGI, cognitively-guided instruction video, and how kids were developing their own strategies without the teacher telling them how to do it. 

Now the part I want to tell you from this book is about how they reference that really how a student decides to solve multi-digit problems really does depend on how they are solving single digit problems. 

So if you are working with 5th graders and every experience that they have had leading up to 5th grade has been on memorization, following rules and procedures, then yes, it’s going to be more difficult to get them to invent their strategies. But if you are helping from very early on, helping kids invent their strategies on things like 5 + 6, 9 + 7, those strategies that they’re using on single digit addition, multiplication, subtraction, division type stuff is how they will end up solving multi-digit problems. It all layers together.  

This book is full of ideas and activities and the research that goes on into explaining how we help develop that for students. And it is one of those books that if you do not have this book, you need to get it. It is an absolute must-have for any elementary teacher.

Research from Fosnot & Dolk

Another 2nd must-have of mine is the book by Fosnot and Dolk. So there’s a whole series, but this one is on constructing number sense and addition and subtraction. And in this one, they really talk about the whole landscape of learning. Like there is so much to go into but one of the points that I wanted to bring up from this book was that if people haven’t already been taught an algorithm, they prefer to work from left to right, just like we read.

But our algorithms that we tend to teach kids start from the right and we work to the left. So when we don’t teach strategies to kids and we let them invent their own, their natural tendency is to start with the larger numbers, add those, and then work their way over. And she references research by Kamii, which is another great reference. 

There’s just tons of research out there to back up this claim that kids naturally will invent their own strategies if we allow them to. And I bet if you open up the belief in your own mind that your students can do it, you’ll start to see more and more kids who are actually doing it. 

So you have a choice. You can choose to think that kids can’t invent these algorithms and these strategies on their own, and so it’s our job to just teach them. We’ve got to teach them the rules and procedures. 

Or you can choose to believe that students will invent these strategies with guidance from us by allowing them to experience concepts around numbers, building their number sense, building their place value, and then help guiding them through their invention of these strategies. 

Now kids don’t just magically wake up and say, “Oh, I see how to do this.” Remember, it is tied to how they are thinking about single digit numbers, the number sense and experiences that they’ve had that help determine whether or not they will invent these strategies. So focus on building their conceptual understanding of numbers and the byproduct will be that students will invent these math strategies on their own. All right, I hope that this video helped you build your math mind so you can go build the math minds of your students. Have a great day.

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