Game Play, Problem Solving, Professional Learning, Teaching and Learning
Years ago, I walked into a Weight Watchers meeting, shy, self-conscious, and unsure of what to expect. After signing my agreement and going through what would become my weekly ritual, I sat down to read through the stack of literature they gave to me. I read through many ideas, insights, and testimonials. Yet it was one very small side note that caught my eye. There, in an offset lavender box, were familiar words that resonated with me. “The Four Steps for Solving Your Weight Loss Problem.” Can you guess what those four steps were?
- Understand your weight-loss problem.
- Create a plan to address the problem.
- Solve the problem by following your plan, adjusting as needed.
- Reflect on your progress and check for new ways to address it.
Yes – these are the four steps I shared with you in last week’s blog post, So You Took Keywords Away – Now What? These four steps, immortalized by George Polya in How to Solve It, are the same steps for solving mathematical problems. It turns out that these steps find utility in all of life for beyond math class! And, interestingly enough, we were encouraged to move back and forth among the steps, reflecting upon and revising our plans from one week to the next.
Fast forward to today…I had some great conversations with several of you last week after sharing the problem solving template I use (click here to download again). One such conversation centered on the notion that Polya’s problem solving steps are not isolated, sequential steps. Rather, they are stages in a process that is both fluid and flexible.
Here are a few thoughts on ways to encourage and demonstrate flexible, fluid problem solving with your students:
- UNDERSTAND the Problem: This step has two parts. And you’ll likely revisit this step several times throughout the process as a way to monitor and adjust your progress.
- Understand the Story: Read the problem for understanding. Cover up the question at first, and relate to the context. Who is the problem about? What is happening? Where is it going? What story is being told?
- Understand the Question: Read the question to discover where the story goes next. What is being asked? Given this direction, what information in the story is useful? What are the quantitative relationships that emerge based on what the question is asking?
- PLAN a Course of Action: As with the Understand stage you’ll likely revisit your plan several times throughout the solution process, revising it as needed. Therefore, although you may be tempted to create a definitive plan, it’s likely that you will benefit more by coming up with something loose that you can revise as you go. The more flexible you are with your plan, the more likely you will be able to engage with the problem realistically, especially if you get stuck or need to back-track a bit. And remember that good plans includes both thinking strategies and a representations (see last week’s post for more on this).
- SOLVE by Implementing Your Plan: You’ll want to be flexible and fluid during this step of the process! Dive right in, knowing that you’ll be revisiting the Understand and Plan stages a few times to ensure that you’re on the right track. It’s okay to revise your plan once you gain new insights and notice new nuances within the problem itself.
- CHECK Your Solution and Reflect on Its Accuracy: This stage is so much more than using inverse operations to check for arithmetic accuracy. Did you really understand the address the original question? Does the solution match the original context? Does it answer the question? Did the plan work? Did both the thinking strategies and the representations help get to an accurate solution? Was the solution process messy or elegant? Might there have been a different plan that may have been more efficient or revealing? Does the final solution make sense? Does your solution need tweaking? Do you need to go back refine your process a bit more?
As you can see, problem solving is not at all linear, with siloed steps that happen in isolation. But, rather, problem solving is both fluid and flexible, allowing for movement in all directions throughout the process. Just as with real-life problems such as weight loss, job searches, or financial decision-making, moving back-and-forth among these steps allows you dive in, refine your understanding, revise your process. And even when you get to the check/reflection stage, you’re not necessarily finished.
Let us know what you think – we would love to hear your stories! Do you have any real-life examples of when this problem-solving process could be helpful? Have you had the chance to see students engage with problem solving that revealed flexibility and fluidity?
—————
PS – Beyond Math Blocks Teacher Institute is just around the corner – October 25-26 in Phoenix, AZ. My colleagues and I are super-excited about what we have in store for you. Have you signed up yet???? Click here to learn more.
PPS – Check out Peggy’s latest Beyond Base-Ten Blocks: A Search for a Better Solution journal entry and learn more about how base ten blocks may be problematic as manipulatives representing number and operations in our elementary classrooms.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Math, Problem Solving, Professional Learning
Problem Solving Song
sung to Row, Row, Row Your Boat
“Problems come along in many different ways…
Follow these steps and you’ll find solutions all your days.
Understand the problem then choose a strategy,
Solve, then check your answer so it’s all that it should be.”
This song became the focal point of almost every demonstration lesson I presented as a coach. Every student on my campus, from kindergarteners through sixth graders, would belt it out. Sometimes we would whisper sing, sometimes caveman sing, and always with gusto. Then we would chant, “Understand, Plan, Solve, Check” three times.
This song captures Goerge Polya’s now-famous problem-solving process he wrote about in his 1945 classic, How to Solve It. This book has since become the centerpiece of school problem-solving strategies. Although some sub-divide the steps into 5-8 parts, almost every problem solving process out there is based on Polya’s “U-P-S-Check.”
A group of teachers I worked with a couple of weeks ago became enamored of the Problem-Solving Mat we were using to help children embrace the process and wanted something tangible for showing student work. The mat, linked below, walks students through the problem solving process and provides guidance and space to show work. This seems like a great follow-up for last week’s blog post, Don’t Fall Into the Keyword Trap!
Problem Solving Mat-U-P-S-Check
I’ve used this mat with students for years. Here are some simple guidelines:
1) Understand: Type the word problem in this space ahead of time. You may consider leaving the question off so students can interact with the situation first (see Target the Question in my previous post). Reveal the question only after students have spent time understanding the context, identifying the quantities and relationships, and anticipating possible questions.
2) Plan: Keep in mind that students often confuse representations and strategies when they identify their plans. Help them understand that strategies are the ideas they bring to bear on the problem, while representations are the means they use to show their thinking. For example, a student may use a skip-counting strategy to think about a solution and then use a number line to represent that thinking. In this case, a solid plan would include both the thinking strategy (skip counting) and the representation (number line).
Also note that the planning phase of this process may be not be sequential. Students may identify the plan up-front, or they may start working and “discover” their just-right plan during the process or even after they’re done. You need not insist that they create the plan and stick to it prior to doing anything.
3) Solve: Young children often use manipulatives to help them think through their solution. The Solve section on the mat is large enough for them to use manipulatives directly on the mat and then translate their representations into drawings. Students may also use this section to show their thinking by using numbers, pictures/diagrams, and/or words to convey their thoughts. You may want older students to sub-divide this section vertically so they can record the steps they used on the left and provide justification on the right.
4) Check: Finally, in the bottom row, students record the “simple” answer, typically a number with a unit label. They re-read the original question and answer the question in a complete sentence. Although there are many other ways to “check” a solution, this way reinforces the notion that the solution should make sense as an answer to the question posed at the beginning.
The format on this Problem-Solving Mat has been used with literally hundreds of students with great results. It’s even a focal point for problem solving in The Amazing Ten Frame Series by KP Mathematics. Try it out…and let us know what you think in the comments section below.
Happy Problem Solving!!!!
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
PS – Remember that the Beyond Math Blocks Teacher Institute is coming up in Phoenix, AZ, on October 25-26. My colleagues and I will be sharing this process and many others as teachers reimagine mathematics teaching and learning in grades K-5. We’d love to have you join us! Check it out by clicking here.
PPS – One more thing: Don’t miss Peggy’s latest journal entry on the amazing properties of KP Ten-Frame Tiles this week. It’s about place value. You may be surprised to see that there’s such a concept as the “infinite ten-frame.” Click here to view.
Math, Problem Solving, Professional Learning, Teaching and Learning
I’ve done it. You’ve done it. We’ve likely all done it at some point…used a “trick” to help us find the answer to a math problem. And, like magic, we have no idea why we got the right answer. It was just there!
Much like teaching algorithms without conceptual understanding, teaching keywords as a primary strategy for solving word problems is not only problematic, it’s harmful. It undoes logic, intuition, and reasoning. Rather than trying to really understand the situation, we pull out the numbers, find a word or two to indicate the operation, and voila, we have a solution.
If the keyword strategy for solving word problems was really that simple, we wouldn’t even have to read the word problem. We would only have to pick out numbers and words and then operate. However, just like the rest of life, it’s just not that simple. Key words are inconsistent (e.g., “more” can mean to add, subtract, or multiply, depending on the context) and, often, nonexistent (sometimes they aren’t even present in a word problems). Sure, they work in contrived situations when “experts” write them to conform to a formula. But trying to apply an over-simplistic strategy such as key words will let us down in real life far more often than it will help us.
So, if using a keyword strategy won’t work, how else might we go about solving word problems?
Here are three strategies I’ve found to be highly successful. If you find that you, or the teachers you work with, are falling into the keyword trap, please share these strategies – they really work!
- Bet Line: Present a word problem, revealing only one line at a time. First, present the title of the word problem and ask, “What do you bet is going to happen in this math story?” After listening to a few predictions, reveal the first line. Then ask, “What do you bet is going to happen next?” After listening to a few more predictions, reveal the next line. Continue asking “What do you bet is going to happen next?” after you reveal each line until it’s time to reveal the question. Ask, “What do you bet the question is going to ask?” This has been, by far, the most productive process I’ve used to help children really understand the structure and flow of word problems. Want to learn more? Check out this article from NCTM.
- Target the Question: Post one reading passage each week. The passage should include several quantitative bits of information that can be used to answer a question. Then, each day, post a different question that can be answered using some or all of the quantitative information in the reading passage. This helps students see that you don’t know what to do with a word problem until you know the question. And different questions will require that you “do” different things each time. For a commercial version, check out the Lone Star Learning website.
- Write Me a Story: Asking students to write their own word problems is a powerful way to help them grapple with the structure and syntax of story problems. Begin by providing an equation such as 24 x 6 = 144. Then, leave it up to the students to find a context that makes sense for the number set and operation you gave them. They need to be sure to include a statement of context, a quantitative scenario, and a question that makes sense. You can differentiate for students by giving them different numbers and operations. You can also make this more difficult by asking students to include additional information that won’t be used to answer their questions. You might also ask students to write multiple word problems that use the same equation.
So, there you go — three ways to move beyond a keyword approach for solving word problems. Next week, I’ll share an amazing template that provides students with space for working out their word problem solutions…so stay tuned!
Please join the conversation. What are your favorite ways to help students grapple with word problems? And please let us know how it goes if you use one of the strategies mentioned above!
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
PS – If you’ll be in the Phoenix, AZ area at the end of October, please consider joining us for the Beyond Math Blocks Teacher Institute. This unique opportunity is designed to support K-5 teachers and coaches as they reimagine mathematics teaching and learning. Learn more by clicking here.
