Kids Love Problem Solving

Kids love problem solving. I really believe this. They like to figure things out. If given the right environment, students like to challenge their thinking and struggle with concepts.

I was reminded of this last weekend when I checked my “Shared With Me” file on Google Drive. I had a number of files from students titled “Math Puzzles.” A few of my students created puzzles to challenge their classmates.

Everyday we start our math lesson with a math puzzle and some mental math problems to build number sense and improve our problem solving. My students love completing these tasks, so some of them they made their own puzzles.

This got me wondering if students really don’t like math, or do they just not like how math is typically presented. I had six students who spent their own time creating puzzles. I didn’t tell them they had to do it or assign it for work. They enjoyed the challenge and wanted to show their creativity. It doesn’t get much better than that – creativity, engagement, and problem solving. Now I have to keep that excitement going.

I know… So I… To find… Therefore…

Teaching problem solving is a tough task. Students have to read a word problem, understand what to do, complete the computation, give an answer, and then determine if the answer is reasonable. So how do teachers best support students in that endeavor?

There is no magic bullet to help students become better problem solvers. They need to be exposed to different problem types. They need to see, hear, and learn how other mathematicians around them are solving problems. Some students also need a structure to help them solve problems and explain their thinking.

One strategy I’ve use is called “I know… So I… To find… Therefore…” This helps students work through a problem and explain their thinking. Here’s a breakdown of each of those parts.

• I know… What important information was given in the problem?
• So I… What did you do to solve the problem?
• To find… What answer did you get?
• Therefore… What is the answer with the correct label?

Let’s look at the following problem to walk through the process.

David started his coin collection with 14 coins. He added 3 coins to his collection at the end of each month for 5 months. How many coins are in David’s collection after 5 months?

Students would answer the question with: I know David has 14 coins in his collection. He adds 3 coins each month, and this happens for 5 months. So I added 14 + 3 + 3 + 3 + 3 + 3 because he gets 3 coins after each of the 5 months to find 29. Therefore, David has 29 coins in collection after 5 months. (Students could also say they added 14 + (3 x 5) instead of using repeated addition.)

This takes some modeling at the beginning of the year, and I always supply a graphic organizer with the “I know.. So I… To find… Therefore…” components already listed. I created a copy of the graphic organizer on Google Docs.

Students pick up on the process pretty quickly. Using the graphic organizer slows their thinking, which improves focus during the problem solving process. It also helps improve a student’s explanation by giving them a structure to follow.

Last week, my students were working on problem involving fractions. The problem was: A quarter is 1/4 dollar. Noah has 20 quarters. How much money does he have? Explain.

I gave my students graphic organizers. (We’re not quite ready to take the training wheels off yet.) Here are two examples of how an average student completed the graphic organizer.

I have adjusted this approach over the years. I’m sure I’ll continue to make slight changes in the future to try and improve the supports I’m providing for my students. What strategies and supports do you use with your students to help them be better problem solvers?

I read an article a few years ago that really changed the way I talk to my students in math class. The article is called 13 Rules That Expire. It was published inNCTM’s August 2014 issue of Teaching Children Mathematics.

The gist of the article is that there are math “rules” used in classrooms. These are often presented as rules that always work, but in the case of these 13 rules, there is an expiration date. The article goes into greater detail about each of the rules, including the expiration date (or expiration grade). My goal is not to thoroughly discuss all 13 rules in this post, but wanted to at least list them.

1. When you multiply a number by a ten, just add a zero to the end of the number.
2. Use keywords to solve word problems.
3. You cannot take a bigger number from a small number.
4. Addition and multiplication make numbers bigger.
5. Subtraction and division make numbers smaller.
6. You always divide the larger number by the smaller number.
7. Two negatives make a positive.
8. Multiply everything inside the parentheses by the number outside the parentheses.
9. Improper fractions should always be written as a mixed number.
10. The number you say first in counting is always less than the number that comes next.
11. The longer the number, the larger the number.
12. Please Excuse My Dear Aunt Sally.
13. The equal sign means Find the answer or Write the answer.

There is one rule I wanted to discuss a little further. Before reading this article, I was a big fan of teaching students to solve problems with the help of keywords. This often included making an anchor chart with the keyword on one side and the meaning on the other side (i.e. in all means add, how many more means subtract).

One common keyword strategy to teach students is that the word altogether means you should add. While that does work most of the time, it doesn’t always work. Take for instance the following problem:

`Jimmy read 11 books in fourth grade, and Brian read 8 books. Jimmy, Brian, and Elyse, read 37 books altogether. How many books did Elyse read? `

If students attack word problem by simply looking for keywords to help them solve problems, they will most likely see altogether and add 11 + 8 + 37. If students are thinking about the context of problem, and not just the keywords, they will see that they have to subtract 19 (11 + 8) from 37. This requires students to understand what is happening in the problem and not simply use the keyword as their only problem solving tool.

Teaching problem solving to our students is not an easy endeavor. It takes many tools in the mathematician’s toolbox. It takes constant repetition and exposure to many problem types. Teaching problem solving is not easy, but it is one of the most important skills we teach in any subject area.

My students have been working hard this year to improve as problem solvers. We consistently take time to talk about what makes effective problems solvers and practice the skill of problem solving. Last week I talked about the Three Reads strategy we use in our classroom.

Another strategy I like is giving students the answer and having them create the problem. Recently I gave my students three prompts:

1. Write a division problem where the quotient would be 6r3.
2. Write a division problem where the answer would be 5 1/3.
3. Write a division problem where the quotient would be 7r3 but a mathematician would add one to the quotient to report the answer as 8.

Students had the option to brainstorm with a partner before writing. Each student had to create their own problem. My goal was to get students to think about the structure of math problems – narrative and expository text combined.

One of the first things I noticed was students struggled to create the complex narrative structure that exists in most fourth grade word problems. This made me wonder if one of the obstacles for young mathematicians is they struggle with the narrative component of a word problem.

Most of the problems students initially wrote for #2 were similar to this: There are are 16 cookies for 3 kids. How many cookies does each person get? These problems lacked the character names and any extraneous information that often appears in rigorous problems. They also lacked the need for multi-step problem solving.

The conversations with students after they wrote their problems was wonderful. I had some of the students go into their math books and look at similar division word problems. This helped them better understand the structure. Other students practiced writing some problems with me. In both cases, we talked about the “story” at the beginning of problems with characters and a scenario which creates the necessity to solve a math equation. One student actually said, “Ahhh!” The lightbulb went off.

This exercise made me realize the value of students looking at math problems to analyze the structure of how a problem is put together instead of trying to solve it. All of the problems we revisited had already been completed, so the student could focus on how the problem was written.

It’s another tool in the problem solving toolbox which I hope will continue to grow for me and my students.

Problem Solving

I was walking into work one day and a colleague literally came running across the parking lot. She was frustrated and asked, “What in the world is going on with these math word problems?” I looked at her waiting for more detail and trying to not drop my smoothie. “These aren’t math problems,” she said. “They’re reading problems.” She had no idea how correct she was.

Word problems are as much reading problems as they are math problems. One of the challenges with younger mathematicians is getting them to slow down and read problems multiple times to understand the complex structure.

The higher-level thinking problems students are asked to solve go far beyond basic computation (5 x 5 = 25). Students have to read a complex problem, understand the context, know based on that context that they have to multiply 5 x 5, then multiply 5 x 5  to get 25, and finally write an answer of 25 with the correct label. Talk about challenging!

Math word problems are especially challenging for readers because the structure is unlike most of what we teach in reading. The authors of Routines for Reasoning state, “Reading in math – especially reading a math word problem – is different from reading in other subject areas… word problems combine both narrative and expository text… Therefore, word problems must be read several times with a different focus each time…”

In reading, we generally teach narrative OR expository text. In math, students often encounter both types swirled into one problem.

There is another challenge with the structure of math word problems. Students learn in reading that the main idea is generally at the beginning of a paragraph or section of text. Think of the main idea of a math problem as the question being asked. The main idea – the question – is at the end of the paragraph or section of text.

So how do we help our young mathematicians become effective problem solvers? Routines for Reasoning shares a strategy called the Three Reads. This approach requires mathematicians to read a word problem multiple times and sets a purpose for each read.

2. Read 1: Understanding the Context – Focuses on the general idea of what the problem is about.
3. Read 2: Interpreting the Question – Determine the question or questions being asked in the problem.
4. Read 3: Identifying Important Information – Look for the important information or words in the problem.

Let’s say students are solving this 4th grade released problem from the Pennsylvania state assessment, known as the PSSA:

David started his coin collection with 14 coins. He added 3 coins to his collection at the end of each month for 5 months. How many coins were in David’s collection at the end of the 5 months?