R emember that Singapore math problem that went viral? The one about Cheryl’s birthday. Lots of people had something to say about it: It’s too difficult. How are parents supposed to help with math homework? That’s why the Common Core Math Standards are a terrible idea. Totally unrelated, but people used the viral Singapore math problem to support all kinds of ideas about math and math problems.
Truth is, the Singapore math problem is a special type of process problem, often found in math contests, (and not your typical classroom) in which the process of finding a solution to the problem is the goal, not computation like adding or multiplying. This process can be a diagram or a table or another problem solving strategy that helps students use reasoning and logic to come up with a solution. One of the most important Common Core Math Practice Standards encourages teachers to have students make sense of problems and persevere in solving them. Process problems, like the Singapore problem, are one of the best ways to do this. Process problems are one of the best ways to help students develop problem-solving skills and mathematical thinking that they can take to college and life.
But process problems are really difficult to construct and most math textbooks feature very few process problems. The most common type of math problem is the translation problem. Translation problems are the typical word or story problems that you see in textbooks and the type of problem that most teachers create as practice after they have taught students how to add or divide. For example, you teach the steps of subtraction with numbers and symbols (23 – 12= 11) then say that Nisha has 34 apples and gave 13 away to all of her friends. How many apples does Nisha have? This problem doesn’t really help develop problem-solving skills. Student extract numbers to plug into an operation they just learned. But there are 5 ways that translation problems can be crafted to help students develop problem-solving skills and improve math understanding.
1. Use word or story problems to teach computation. The idea here is to give students a problem, like the Nisha problem above, and have them figure it out, before you teach computation. That way students develop their own understanding of how to solve a problem instead of receiving this information from the teacher. The best way to do this is to adopt an inquiry-based learning or learner-centered approach. In this approach, you divide students into groups or partners and have them figure out how to solve the problem. Most students solve the Nisha problem through a drawing or diagram in which they draw 34 apples, cross out 13 and count the remainder. If students figure out how to solve this problem on their own, they are less likely to forget it. Also, there are students who will solve this problem using strategies that you, the teacher, never thought of. To discover these strategies, you ask students to communicate math understanding. More on this later.
2. Problems should contain missing, unimportant or inconsistent information. To really develop problem-solving skills, students need to be able to analyze a word problem to determine which information is important and which information is irrelevant. For example:
Ikenna just turned nine. He has eight guests at his birthday party. If Ikenna’s father wants to buy toys for all the children at the birthday party and each child gets two toys, how many gifts does Ikenna’s father need to buy?
The fact that Ikenna turned nine doesn’t seem important. The missing information is that Ikenna’s father wants to buy 2 toys for all the children at the party. Since Ikenna is one of the children at the party, the dad also needs to buy 2 toys for Ikenna. As such, Ikenna’s dad needs to buy toys for 9 children. Hmmm… but Ikenna also turned 9, like the number of children. Is that important now?…You get the idea. Students need to analyze the problem and plan the solution. What’s nice here is that the solution to this problem can involve various problem-solving strategies like drawing or diagrams and ‘acting it out.’ The problem also involves various operations, including addition (8+1), addition of sets (2+2+2+2+2+2+2+2+2), or if students are advanced enough, multiplication. Be sure to remind students of the importance of units as the answer is 18 toys, not 18 as most students tend to say.
3. Problems should be culturally relevant to students. Problems should be based on realistic ideas that feature familiar settings and ideas. If you are teaching in an urban area, you should craft problems that are relevant to the urban setting. One well-meaning teacher created a math problem that asked some urban students to arrange an equal number of horses among six paddocks. Many students didn’t know what a paddock was and couldn’t solve the problem.
The main way to resolve this issue is to know your students, their backgrounds, and have a sense of their worldview. Further, crafting problems that require students to know information that may be based on socio-economic class, ethnicity, race or language is a form of bias. It is bias against those students who don’t understand the context of the problem because of stuff beyond their control (class, ethnicity, etc). And bias for those students who do understand the context of the problem.
4. Students must communicate their mathematical thinking. Students need to explain how they solved a problem. This is another activity that needs a learner-centered approach to mathematics. When students are finished solving a problem, ask them to explain what they did either to you, their group or partner or to the whole class. It’s the ONLY way to know what’s going on inside student’s brains, where learning is taking place. Really listen to what students are saying and ask questions to clarify.
Sometimes it’s hard to listen when a student didn’t solve the problem correctly. But in listening, you may discover misunderstands that show gaps in your instruction and can use this information to adjust your teaching to promote understanding.
Student should also communicate how they arrived at an answer in writing. And we’re not just taking about procedure here (I divided both sides by 2 then found the square root of both sides), we’re also talking about rationale–why a student did what they did. For example: I found the square root of both sides to keep the mathematical statement true. And here you will also see if there are gaps in your teaching. And if there are, you use this information to adjust instruction to improve student understanding.
5. Students create math problems for other students (and the teacher) to solve. This is a great strategy, especially for more advanced students. This really gets to the nitty gritty of developing problem solving ability as it forces students to think about all the variables they need and the processes involved in solving them. Guide this activity with the steps in this post so you get meaningful results.
Now this stuff isn’t easy to do. Even I forget sometimes to include missing or unimportant information, especially when I’m creating problems on the fly to help students understand a concept. It’s easier if you plan problems ahead of time and becomes easier with practice and time.
I’m sure you can think of other ways to craft better translation problems. If so, I’d love to hear your ideas in the comments below. #heidiholder #redloheducation
