Mastering Math: Exploring the Various Types of Word Problems in Math


I love teaching story problems in Math with any class I get. Watching my students' eyes light up when they figure out a solution on their own is so rewarding! The way that I teach problem solving today is much more structured and has been very successful, saving lots of time and tension. I use the Cognitively Guided Instruction (CGI) approach which, I recently discovered, has been around since the 1990s. Believe me, now that I have seen the extent to which this approach makes teaching easier, I am convinced that it’s definitely a game changer! If you haven't heard about this approach before, then let me give you an explanation with some examples to make it easy for you.

The Cognitively Guided Instruction (CGI) method was developed by researchers Thomas P. Carpenter, Elizabeth Fennema, Penelope Peterson, and others at the University of Wisconsin-Madison in the 1999. It's main focus is on how children think and use their existing knowledge to solve story problems in Math. It explains how their reasoning develops and gives teachers and educators an insight into how to guide their thinking. From concrete objects to abstract, this approach progresses from easy to hard; from joining problems (addition) to multiplication and division problems. 

A Teacher's Idea

Now let’s take a look at each type using examples to demonstrate how progressively what they should look like.

1. Joining Problems

In Cognitive Guided Instruction (CGI), "joining problems" are math problems where students are asked to combine groups of objects to find a total. For example, Maria had 5 apples. She bought 3 more. How many apples does Maria have now? These problems help students practice adding and understand the concept of putting things together to get a larger number. Students can model directly using concrete objects.

2. Separating Problems

"Separating problems" in CGI involves breaking down complex math problems into smaller, manageable parts. Instead of tackling a big problem all at once, students handle one piece at a time. This makes it easier for them to understand and solve the problem step by step. Always start with result unknown, then to change unknown and finally to the start unknown types of story problems. See picture examples below:

3. Part-Part-Whole Problems

"Part-Part-Whole Problems" are a type of math problem that involves breaking a whole into parts or combining parts to make a whole. For example, if you have 8 apples and you know that 5 of them are green, you can figure out that the other 3 are red. It's all about understanding how the parts add up to make the whole or how a whole can be split into parts. This helps students develop number sense and understand relationships between numbers.

4. Comparing Problems

In CGI, "comparing problems" are math problems where students compare two quantities to find the difference. These problems help students understand the concept of more and less, and how to determine the difference between two amounts. For example, "Maria has 8 apples and her friend has 5 apples. How many more apples does Maria have more than her friend?" students would compare the two amounts (8 and 5) to find the difference (3 apples). Comparing problems are like figuring out who has more or less of something and by how much.

5. Multiplying and Dividing Problems

In CGI, when it comes to the hardest type of story problems, “multiplying and dividing problems”, the teacher simply listens to the different ways students solve these problems and encourages them to explain their thinking.  The teacher also assists them in making connections between the different methods used in solving those problems. This way, they build a deeper understanding of how multiplication and division and they gain confidence in how they work independently. These problems require the use of standard algorithms or a set of steps that students apply.

Parts of a Word Problem

 In CGI we talk about the different parts of a word problem that are unknown or missing. I use the BME (Beginning, Middle, End) story model to anchor the parts: 

In a "result unknown" problem, we might know how many apples we started with and how many were added, but not the total

In a "start unknown" problem, we know how many we ended up with and how many were added, but not how many we began with

In a "change unknown" problem, we know the starting number and the ending number, but not how many were added or taken away

Each type of problem helps students think about numbers in different ways, making math more engaging and fun!


In CGI, the teacher acts more like a guide, asking questions and encouraging kids to explain their thinking. This way, children develop a deeper understanding of math concepts and feel more confident in their problem-solving abilities. I hope that this post gives you a clear understanding of how story problems should be taught developmentally so that your students would be successful independent learners.

A Teacher's Idea

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