Whether you’re working with your student at home or in a traditional classroom, it’s important to understand the progression of how a young child’s mind starts to comprehend word problems.
We start with real objects in the physical world, providing students with language to explore and manipulate objects as they begin to understand the relationship between numbers and words and words and numbers. Then, we move into quantitative pictures, counting things and creating lots of conversation through purposeful questioning.
As students master the first two developmental stages, we can move them into the next stage which is acting out story problems using CPA. This has nothing to do with the person who does your taxes, but everything to do with getting kids to express their knowledge of mathematics concretely, pictorially, and abstractly.
Story mats are an excellent way to help children solve problems with different types of real-world scenarios. We are so excited to bring you these Word Problem Story Mats – 10 story mats, 10 sample problems for each = 100 problems you can use with your littles for only $3.99! There are three great video tutorials that will show you the mats in action and help you learn how to use them most effectively.
These mats are designed to guide students through exploring their thinking by providing a visual image and allowing students to act out the problem. The visual picture might be a barn, or a house, or a dinner table – something that is familiar to the child that they can use to put counters on to represent a quantity.
Concrete Representation
Sometimes it’s difficult for our littles to use an inanimate object, like a counter or an M&M, to represent something that isn’t real. As we’ve talked about before, many of our children are really skipping over the imagination stage in their childhood. Because kids are so plugged in these days, very few children have imaginary friends anymore, and they might even have difficulty imagining how to put on a puppet show or play something like kitchen where pretend something is real. I vividly remember sitting in a play kitchen in a preschool classroom and pretending to pass out pizza to everybody that was sitting at the little table. A little preschool boy said to me, “You didn’t put anything on my plate. There isn’t any pizza.” I said, “Well, yes I did! I just put it right there! It’s the pretend pizza!” He was skeptical, and he represents many of our kids today that, unless it’s a real pizza or a rubber pizza that looks like real pizza, they have a hard time using their imaginations to represent real objects.
Depending on their developmental stage, it might take littles some time to see concrete tools as something that can mathematically represent something entirely different than what it actually is. When you start to use objects like counters or beans, or maybe even cereal if you’re at home, to represent things, our littles sometimes have a hard time transitioning to that. For example, if we’re using counters to represent eyes for an owl, we may have to explain that owl eyes don’t really look like that, we’re just representing how many there are. Or if we’re using one-inch squares to represent people sitting on a playground bench, we may have to help students envision a person in each square. Realistically, if we’re looking at a beach or a garden, we know we can’t really turn our counters into fish or flowers.
So, concrete representation is the first stage in this next level. We need to help kids start to understand that an object can be related to something. We can provide different scenarios – a tree, a bucket, a sandbox, a lake, a road, etc. – and have them pretend that an inanimate object represents something quantitatively from the story problem. For example, if we’re working with our “In the Tree” story mat, we might talk about birds or owls in the tree. Students need to be able to hear the story problem read aloud, and then use the counters to represent the birds or owls, even though they look nothing like birds or owls, as they “act it out” on the mat.
As students become more and more familiar with concretely acting out the story problem that they’ve heard read aloud, they are going to get more and more fluid within their understanding of bringing math into the real world.
Pictorial Representation
Much like with concrete representation, we have to retrain our students to think about pictorial representations differently. In math, and on our story mats, we have what we call a Quick Draw. In traditional writing and liberal arts, students are often told to use all the white space, and to accurately represent what they’re seeing – correct colors for hair, legs at the bottom, etc. Students end up with more color and description in this type of drawing. In math, less is more, especially in a quick draw.
If I’m counting ants on a playground (one of our story mats), I’m not going to draw individual ants out in my Quick Draw, because that’s not very quick. I’d have to draw their legs and then their antennas, etc. We don’t have time for that!
After we teach kids to transition from a real object to an inanimate object that represents something, we have to transition them to being able to draw a bird as an X or a circle, instead of a bird with a beak and eyes and feathers. I call it ringing and labeling – using an X or putting in a dot or whatever is going to represent pictorially what we’re talking about in the picture. Obviously, depending on the age of students that you’re working with, they may start by doing more detailed drawings. This is certainly okay because kids will naturally transition into a quick draw as they realize they can represent the object more quickly with simpler shapes. For example, if there were six ants crawling on the playset and two of them crawled away. How many ants are left? We could draw six circles, cross out two, and find out our answer.
Abstract Thinking
Once students can represent an object concretely with an unrelated object, and draw a picture of their thinking, we can add in abstract thinking. This can be a number bond or a number sentence – something using numbers to show what they’ve done concretely and pictorially.
This is not the right level for all students right away, and we don’t want to build to this level too early. In my opinion, it’s more important for students to understand the concept behind the digit, or what the subtraction sign represents, than to just be able to write it. If kids can’t make the connection between words and numbers or number and words, or are pushed through the standards too quickly, it can be detrimental to their math foundation. That’s why quantitative pictures and story mats are such great tools as kids can interact with these concepts in a playful, real-world way.
Word Problem Story Mats
Working through story mats hit all three aspects of this phase of development as students build problem solving skills. We have a set of Word Problem Story Mats, a Math4Littles publication, and some great tutorial videos that will help as you apply CPA to real life situations. Whether you’re a teacher or a parent, you can use these with your littles! Check out the previews in the videos below and get all 10 mats for $3.99!
For each story mat, there are 10 sample problems. The first three are focused on part-whole addition. Numbers four and five focus on subtraction, six and seven are missing addend problems, and the last three are deeper thinking challenge problems. These problems ask questions a little differently to encourage kids to think outside the box and use higher-order thinking skills in order to find a solution. For example, I saw six eyes. How many frogs were in the pond?
You can easily differentiate the level of any of the problems simply by changing the quantity based on the skill level of the students you’re working with. If you’re working on quantities to 10, you can adjust the numbers to make sure the total is 10 or less. If you’re working with younger students, you might want to keep the total at 5 or less.
You might also notice that the problems are pretty generic. We use lots of personal pronouns and general nouns. Insert the student’s name into the story problem, or the names of family members or teachers that are familiar to make the story problems more personal.
How to “Teach” Story Problems
Our goal, as teachers and parents, is to help kids become deeper thinkers. To do this, we have to remember that we aren’t the givers of all the information! Yes, we know the answers and how to solve the problem, but we need to let them figure it out for themselves because we know that they will remember it better if they do. Our natural instinct, if we notice kids beginning to act out a problem incorrectly, is to stop and fix it so they don’t end up “wrong.” But we don’t need to autocorrect students as they are solving problems!
Think about how we would handle this in reading, maybe in a guided reading group. If a student comes to a word they don’t know or says a word incorrectly, they’ll typically appeal and wait for you to give the word. As good teachers, we don’t do that though! We want them to engage in a little productive struggle and use the strategies they’ve learned to help them decode the word on their own. We might walk them through a few strategies – let’s look at the picture, let’s reread the sentence and look for context clues – but ultimately, you want them to decode the word. We want them to do the work because, if they don’t, the next time they come to that word, they’ll be right back asking for help again.
Math works the same way. Obviously, we don’t want to tell them the answer. We want students to have a little bit of productive struggle as they begin to solve the problem. But we also want to be careful about our questioning. We can technically not give away the answer, but still walk students through exactly how to solve the problem just by asking leading questions, which practically give away all the information they need. This is what we don’t want: “There were six frog eyes. Ok, everyone get out six counters. Now, let’s put them in groups of two because you know that frogs have two eyes. How many frogs were there? Ok, let’s count our groups of two to get the answer.” No learning took place here!
Instead, we want you to ask questions that are open-ended: Can you tell me more about what you’re thinking? I noticed that you have out counters…how many counters do you have? Why did you pick that many counters? The more we can let kids explain their own thinking, the better!
Additionally, when you begin questioning a student, most of them think they’re wrong because “you wouldn’t question me if I was right!”. So we encourage you to question students as they’re using these story mats, even if what they’re doing is correct. At the very least, it will have them going back and rethinking themselves Did I do it wrong? We want kids to be able to defend their answers and the way they found it.
Because we are encouraging deeper thinking, there isn’t one right answer in how to solve the word problems we’ve included in our story mats. If a student’s work is different than you expected, that doesn’t mean it is incorrect! When students are working, I encourage you to ask those open-ended questions to get them to do more of the talking and the deeper thinking. This is how we will get kids to get better at problem solving as they get older.
What’s next?
After students can easily move through all three levels of CPA with a story mat, it’s time to give them a blank story mat where they can work similar problems without the scaffolding of a picture. Developmentally, students at this stage can envision that scenario without the picture because they’ve worked through all the levels and built their foundation for math. They’ve done it with real objects, they’ve done it with quantitative pictures, and now they’re doing it on a story mat. We have one you can download for free – either in portrait or landscape.
You certainly don’t have to use the story mat from our website, but we do recommend having a designated mat or space to work the math problems. These mats are designated space where students can act out the story in a concrete way, and it helps eliminate some confusion as they switch back and forth between pencil and paper. It could be a dry erase board, it could be purple construction paper, but it just needs to be something to hold manipulatives!
This works well in a group setting, maybe with kindergartners, to have students come and sit in a circle with you with their blank work mat and a bowl of counters. Read the story problem out loud to them, and once they hear the problem, they can use their manipulatives to act it out the way they’re hearing it. This is a great way to know that the child is at that top point of being able to start to move into the early stages of using a visual model to solve story problems.
Next week, we’ll talk about a way to scaffold students in kindergarten and first grade to move through the progression into more traditional story problem format that will help them create success in applying mathematics.
