by Connie Rivera
I am an avid numeracy blog reader. As I read about the experiences and ideas of others, I find I’m challenged to think deeply about decisions I am making while teaching in my own classroom. The blogs I read inspire me with new ideas on a regular basis.
Lately, I’ve noticed ideas from different sources that are all ways to change how problems are presented. All of these teaching strategies produce a slightly different way of investigating a problem, even if it’s a basic word problem. Because each strategy comes at the problem from a different angle, students learn something more profound than if they had all the information laid out at once (which doesn’t happen in real life!). Instead, students are forced to look deeper into the problem or to develop new problem solving skills. These skills are useful for solving problems that appear in other academic disciplines as well as in real life, when learners need to research possible options and choose the best course of action.
The ideas I’ve come across have me connecting to so many other things that I know about teaching adult numeracy. When I read about them, I think about Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. All of these ideas increase opportunities for my students to make sense of problems before solving them. I am also reminded of Universal Design for Learning and how considering multiple means of representing the problems make the learning more accessible for my multi-level classes, which include English Language Learners.
All improvements start with making just one change to our instruction. Try some of the example problems in the linked resources below to get a feel for what’s possible. Can you think of a problem you already use in class that you can adapt using one or more of these strategies?
Strategy 1: Makeovers
Text book questions provide so much information that they are no longer a “problem” (in the sense of Mathematical Practice 1) for students. Makeovers consist of blocking out most of the information, usually given in a textbook, until students make sense of what’s there and ask for the information they need.
Strategy 2: Numberless Word Problems and Graphs
When all the information is provided to you, it’s easy to glance at the numbers, and perhaps key words, and jump to conclusions. Numberless problems force you to make sense of the problem before solving it. With this strategy, ask students for a solution pathway based on the information without the distracting numbers. Only after they’ve made sense of it do you plug the numbers in and the students solve the problem.
Strategy 3: Notice and Wonder
Notice and Wonder is an approach to problem solving that allows you to make observations about a situation, and then ask, research, and answer a mathematical question about which you are curious. I find this approach especially useful with my English Language Learners.
Strategy 4: Problem Solving Scenarios
These scenarios from NCTM’s The Math Forum and Dan Meyer lend themselves to using a Notice and Wonder approach to ask a question. Present a problem scenario without a question, then let students ask the questions.
Strategy 5: Reversing the Question
Beginning with the end is an idea for showing students a visual or calculations and asking, “What questions gave these answers?” or, “What are the questions these calculations find out?”
Strategy 6: “Open Middle” problems
Open Middle problems appear simple and procedural but actually involve deeper thinking. They have:
- A “closed beginning” – start with the same initial problem.
- A “closed end” – end with the same answer.
- An “open middle” – multiple ways to approach and solve the problem.
You can create this style problem yourself, but among the great collection of procedural problems, you will find some in word problem style such as the one in the link.
Connie Rivera teaches numeracy skills to adults of various skill levels, including court-involved youth and English Language Learners. She is also a math consultant, providing math strategies and support to programs implementing the College and Career Readiness (CCR) Standards for Adult Education. As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is President of the Adult Numeracy Network and a LINCS national trainer for math and numeracy.