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?
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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.
KP® Ten-Frame Tiles, Professional Learning, Teaching and Learning
A while back, I was talking with a fourth-grade teacher at my school. She questioned, “What did the third-grade teachers do last year? I just started my fraction unit this morning, and none of the kids knows anything about fractions!” Rather than explaining what I already knew (since I was the one who taught the third-grade fraction unit the previous yea!), I simply asked her to bring her class to my math lab after lunch.
When they arrived, I had several manipulatives out on the tables – representations they had used in third grade. Before diving in, though, I simply asked the students to sing a song with me. I started the tune, which was about numerators and denominators, and almost everyone joined in. The teacher, now standing in the back of the classroom, threw her hands up and simply smiled.
You see, far too often, teachers begin mathematics units where they think they should start rather than tapping into where others left off. In this case, my colleague simply began the fraction unit as her textbook directed rather than talking with the previous year’s teachers to find out where they had left off. Had she simply inquired, she would have known which manipulatives, which visuals, which vocabulary, and which instructional strategies to use in order to tap into students’ prior knowledge.
Whether you’re a teacher, a coach, or a site leader, facilitating conversations among colleagues within and across grade levels is critical to connected learning. As discussed in our previous blog post, Math “Rules” That Expire, taking time for such conversations is critical. In the case of common visuals, here are three ways to ensuring students use visuals to “see” how inter-connected mathematics truly is…
- Connections within grade levels: Invite teachers within a grade level to map out the math concepts to be taught throughout the school year. Under each concept, have them list the manipulatives, diagrams, and other visuals to be used. Finally, ask them to look for opportunities to use common visuals across the year. The more overlaps that exist, the greater the opportunity for children to see mathematics as a system rather than isolated skills.
- Connections across grade levels: Once each grade level has compiled a year-long list of visuals and mapped out the connections, ask them to engage in cross-grade conversation about how visuals might be used to connect the mathematics from one year to the next. The more connections they make, the better students will be able to tap into prior knowledge and use that prior knowledge to learn new ideas.
- Connections using common visuals: The list below includes some of the common visuals that help students see mathematical connections…
- Manipulatives: pattern blocks, base-ten manipulatives such as KP Ten-Frame Tiles, unifix cubes, two-color counters, Cuisenaire rods, fraction bars
- Diagrams: number lines, bar models/tape diagrams, number bonds, ten frames, place-value charts, arrays
- Other visuals: hundred charts, multiplication charts (may be used for skip counting, multiplication facts, arrays, equivalent fractions, etc.), spreadsheets, Excel graphs
We owe it to our students to make these connections explicit. And we owe it to our teachers to ensure they have time and space to discuss and internalize these ideas.
Please share your thoughts on the visuals most likely to help students make connections. Do you have experiences with any of the listed visuals? Do you have other favorites you would like to recommend?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
One more thing: A great example of common representations is KP Ten-Frame Tiles. They are so superior to base-ten blocks! Check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks.
KP® Ten-Frame Tiles, Math, Professional Learning, Teaching and Learning
“Multiplication makes bigger and division makes smaller.” “You can’t subtract a larger number from a smaller number.” “Improper fractions should always be written as mixed numbers.” “You always divide the larger number by the smaller number.”
We’ve all heard these statements…and I’ll bet we’ve all said them at one time or another. And every time we say these, we further perpetuate misconceptions that are difficult, if not impossible, to un-teach in future years.
Far too often, teachers inadvertently teach mathematics in an overgeneralized and imprecise way. We teach tricks that promote nothing more than memorization. The result is that our students misunderstand the very ideas we are trying to illuminate. How many teachers have had to contradict their colleagues from previous grade levels because what was taught was so painfully limited in scope?
So…how can we stop this practice in its tracks and work together to teach mathematics correctly and coherently? Here are a few ideas to get you started.
- Read Up! First, I’d like to introduce you to two recently-published articles that drive this point home: 13 Rules That Expire and 12 Math Rules That Expire in the Middle Grades. These two articles simply and eloquently outline overgeneralized, commonly-accepted, inadequate strategies we need to abolish from our vernacular. Seriously – check them out – you’ll likely find one or two or more to work on.
- Talk to Your Colleagues: Professional discourse provides foundational and at-your-fingertips opportunities for growth. After perusing one or both of the articles above, talk to your colleagues. Last month, I had a group of 40 teachers and leaders read and discuss these articles, and their conversations were robust.
- Create School-Wide Agreements: Take some time during grade-level and staff-wide meetings to discuss these ideas. Then come up with a list of five to ten school-wide agreements, identifying the concepts, vocabulary, procedures, etc. that everyone, or no one, will use on campus. This will make for a lively conversation!
I’m in the process of working through these steps in my own school district right now. Won’t you join me on this journey to abolish rules that expire by engaging in professional discourse and by creating school-wide agreements?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Preview for Next Week: When selecting visuals and manipulatives to use within and across grade levels, choose wisely. Focus on ones that allow for connections throughout the grades. A great example is KP Ten-Frame Tiles. They are so superior to base-ten blocks! More on this topic next week…for now, though, check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks from the get-go.
Professional Learning, Teaching and Learning
“Many hands make light work,” claimed John Heywood more than 600 years ago. And we still find this to be true today. Especially in the world of education, when we join forces to serve a collective goal, our work becomes magnified and exponentially impactful.
By way of example, please humor me as I tell my own story. This past year, my school district adopted new mathematics resources. Because we had varied perspectives for what mathematics teaching and learning should look like, I brought together diverse teams on each campus to work and learn together. Each team had one representative from each grade band, K-2, 3-5, and 6-8, as well as an instructional coach, a SPED teacher, and an administrator.
Initially, during phase one, each team had the responsibility to represent their grade bands, taking inventory of what was working, what materials were needed, and what PD was requested. This one-way communication allowed them to take ownership of the needs of their staff.
However, there was a second intention, phase two, if you will, behind the formation of these teams: equipping the members as teacher leaders, both individually and corporately. Here’s the plan I implemented for evolving the teams from phase one to phase two…
- Monthly Meetings: During both phases one and two, I met with each team about every four to six weeks. During phase one, I gave the team assignments to talk with everyone they represented about their needs: materials, professional development, strategies and methods, etc. During phase two, I am now giving them things to take back to their teams: number talk strategies, games, manipulatives strategies, word problem methods, etc. This helps their colleagues begin to see them as the experts.
- Summer Symposium: The best way I’ve found to transition from phase one to phase two is to provide a common experience for the teams. This year, I offered a Summer Math Team Symposium where the teams came together to examine mathematics content across the grade-levels, pedagogical strategies, and leadership techniques. They created vision statements for what mathematics teaching and learning will look like on their campuses. And they planned for how to share important ideas with their colleagues.
- Teacher Leadership: An amazing (and deliberate) by-product of this effort is the empowerment of teachers to see themselves as leaders on their campuses. It’s thrilling to hear teachers confess how their discomfort with mathematics has been overshadowed by the summer experience and how excited they are to support their colleagues. In just the past month, I’ve seen the teams blossom as they presented mini math sessions for their colleagues, offered vision-casting seminars for their staffs, and tutored their colleagues in areas of discomfort. This effort is resulting in teachers taking on new challenges and supporting their colleagues unlike anything I’ve witnessed before.
So…are you interested in creating or taking part in such an effort…or at least in hearing more about how you might do something like this within your own sphere of influence? You don’t have to be a site or district leader to get this started. Sometimes the best efforts begin within the teacher ranks. We would love to hear your thoughts on building camaraderie amongst the math teachers on your staff.
And for those of you following Peggy’s journey with KP Ten-Frame Tiles, please click here for her latest entry. You may find great ideas in here to take back to your colleagues to inspire new conversations about the teaching of mathematics.
As always, we would love to hear your thoughts regarding these ideas or your own journey. Please continue the conversation in the comments section below.
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 Phoenix this Wednesday morning, August 29, please join us for our FREE leadership event taking place in the downtown area. You can learn more by clicking here.
