I first learned about Problem Types with Cognitively Guided Instruction in the book Children’s Mathematics. The group of researchers studied children and found that when given open ended questions children move through a progression of understanding… when given no explicit instruction. 🤯

It blows teachers minds to think that children can develop understanding with no explicit instruction. However, when we think about how children learn it actually makes sense. 

You won’t want to miss this episode all about Problem Types in your math classroom! 

On This Episode on Problem Types & Why They Matter:

• 3 types of joining problems
• 3 types of separating problems
• comparing problems

Space to Explore Math Problems

I’m passionate about bringing high quality math instruction into every child’s classroom. That’s why I’m here. I believe that all children should have the opportunity to develop a mathematician mindset, to FEEL and BELIEVE they are mathematicians. 

In order to do that, they need the space to explore word problems and bring their experiences and knowledge to the problem to solve in any way that makes sense to them. I first learned about this with Cognitively Guided Instruction in the book Children’s Mathematics. The group of researchers studied children and found that when given open ended questions children move through a progression of understanding… when given no explicit instruction.

🤯 It blows teachers minds to think that children can develop understanding with no explicit instruction.

However, when we think about how children learn it actually makes sense. We don’t teach babies to walk, they explore, make mistakes, and learn. I’m not teaching my 5 year old how to play the piano… he’s exploring, matching tones, and learning. 

I didn’t teach my 3rd graders what it meant to pass out the markers fairly to all tables, but I gave them the markers and they figured it out. 

We provide opportunities in safe environments for students to explore math and solve in ways that make sense to them… that’s REAL WORLD learning. 

Over time, as Cognitively Guided Instruction supports, students’ strategies get more efficient and effective. You’ll see students passing out those markers 2 at a time to the tables instead of by 1s. More sophisticated might be coming up with a plan before passing them out by knowing that if there are 5 tables and each table gets.

What are the Problem Types:

However, one of the other things that came out of Carpenter and his colleagues’ work are problem types. There are 15 problem types that children are exposed to. Joining problems, separating problems, part part whole, comparing problems, and multiplication & division problems.

Notice that the problems are labeled– joining (the action), separating (the action), and differently part part whole.
There are 3 types of joining and 3 types of separating problems and the difference between each is where the unknown is. 

Let’s Look at a Start Unknown Joining Problem

Willard has bolts in a bowl while working on his car. He goes to his tool box and gets 14 more. Now there are 22 bolts in the bowl. How many bolts were in the bowl to start with?

How would you solve this problem?

Maybe you would start at 22 and draw 22 circles then cross off 14? 

More likely though, you’d start with what you have 14 bolts and then count up until you got the 22 bolts. 

What would the equation be that matches your work?
What would the equation be that matches the problem?

We ask students to write two equations because sometimes the way they solve is different than how the problem is written. This is an important aspect to notice and wonder about. Students can notice patterns and really conceptualize the start, change, and result when writing equations.

Separating Problems

Separate questions are those that are often thought of as subtraction. The action is something is leaving or being taken away. Again, 3 problem types based on the location of the unknown 

Then, part part whole. You can have an unknown part or an unknown whole. Which is more complex?

I have 21 shirts. 13 are blue and the rest are red. How many red shirts do I have? 

There are 13 blue shirts and 8 red shirts. How many shirts are there?

Compare Problems

Then compare problems, which lend themselves well to measurement.

Compare quantity unknown: Mom has 6 cups of coffee and Dad has 2 more cups than mom. How many cups does dad have?

Compare difference unknown: Joni has 78 beads and Ida has 92 beads. How many more does Ida have? 

Compare referent unknown: Romy has 89 legos in her tower which is 7 more than Louis. How many legos does Louis have in his tower? 

I find this is the group of problems that trips the MOST people up. These problems can seem overwhelming and make teachers think students must be taught HOW to solve these. However, if students are given time to explore and an authentic reason to solve then they have life experiences and knowledge of how to do this. These are actually very common in life. Running races, playing with cars, throwing balls, PLAY includes a lot of students comparing. Siblings compare all the time.

So, as teachers it is not about teaching them exactly the steps to follow to solve these… you know line up the drawings and count up or cross off. But instead, it is about asking the right questions and making sure students understand what is being asked. 

The best thing you can do with problem types as you’re launching more complex problem types is to get students asking questions and wondering about the problem. This isn’t something I always do in the launch but is one way to help students break down the problem and truly understand it. Get them asking questions. What do you wonder? They might say things like I know how many Romy has but how many does louis have? So does that mean Romy’s tower is bigger?

Wrap Up

Today we discussed the addition & subtraction problem types. We will have to come back and talk about multiplication & division (we have enough for a whole episode! Stay tuned)

We talked about how the problem type is determined by the location of the unknown. Some are more complex than others based on that location of the unknown.

Having students write an equation to match their work and the problem helps students draw connections and notice patterns in operations. These make for lively math discussions. 

Comparing problem types is often complex… try to find real world scenarios that your students can relate to. That will help them “get” what you mean by comparison. 
A launch strategy that can help students understand the problem is wondering and asking questions about the problem.

I hope this episode feels really actionable and helpful as you think about the types of problems you are giving your students. Many times our curriculums spend one day on each problem type… which isn’t enough. Students need lots of time to explore and refine their strategies and approaches to problems. Consider using a problem solving routine to allow students to interact with word problems daily.

I teach CGI and math discussions in Word Problem Workshop. If this is a way of teaching you’d like to bring into your class, be on the lookout for more information soon about WPW teacher training!

Links Mentioned in This Episode:

📘 Children’s Mathematics Cognitively Guided Instruction

👀 Look for more information coming soon about WPW Teacher Training!

Have Questions? 📱 My DMs on IG are always open @hellomonamath

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Have Questions? 📱 My DMs on IG are always open @hellomonamath