by Sara Donaldson, Ed.D., Wheaton College, Norton, MA
In their series of articles about rules that expire, Karp, Bush, and Dougherty (2014, 2015, 2017) discuss the negative impact many provided hints and short-cuts have on students’ future mathematics success and confidence. Many of these “rules” do not hold true as students move into more complex topics (e.g.; just adding a zero to the end of a number when multiplying by ten no longer works when you are working with decimals). And even when the “rules” do hold true, students’ reliance on them does not support their understanding of the patterns and structures that make math work and that help them develop the sense making and reasoning needed for long term success.
Rule number two of their original 13 Rules That Expire (Karp, Bush, & Dougherty, 2014) article is “Use key words to solve word problems” (p. 21). The authors explain that although key words can be helpful, when students are encouraged to scan problems for key words and numbers instead of first making sense of the overall problem situation, the everyday and multiple meanings of key words can lead to wrong answers and an inability to determine whether an answer makes sense (e.g.; left might indicate subtraction, but it could also just be identifying handedness). One strategy for helping students develop the ability to make sense of problems, instead of relying on short cuts, is to have them practice sorting problems based on the type of operation they would use to solve the problem. This task shifts the focus from solving problems to making sense of problems (as finding the answer is not part of the work) and allows students to determine the characteristics of problems that require adding versus subtracting on their own, thus helping to promote understanding and confidence.
Here is how I approach this type of lesson:
Preparation: Gather 10-15 one step word problems with approximately half of them for each operation (e.g.; half addition and half subtraction). You can make up problems, adapt problems from your curricular materials, or simply use problems from your math text or workbook. I try to choose problems with familiar contexts and easy numbers so students can focus on the structure of the problem without being overwhelmed by other details. Depending upon your grade level the problems could involve addition and subtraction of single-digit whole numbers or multiplication and division of fractions, making it easily adaptable for different parts of the curriculum.
Once you’ve chosen your problems, print them out and cut them apart. Each group or pair of students will need a complete set of problems. I put them into envelopes for easy distribution.
Step 1: Display two problems (one for each operation). Using a think-pair-share structure, have students determine what operation would be used to solve each problem. Then lead a short discussion around how students made their decisions. If students bring up “key words” as their strategy, push them to talk about what the word means in the problem and what the word indicates is happening in the problem situation.
Step 2: Explain to students that today they will be working in pairs/small groups to sort problems into two sets based on the operation needed to solve the problem. Emphasize that they will not be solving the problems, just sorting them. I usually encourage students to see each problem as a mini-story and to use what they picture happening in the story to help them determine the operation, just like they work to create a movie in their head to help them understand texts when they are reading.
Step 3: Group students and distribute the problem sets. I like to have students work with at least one other student on this task so they need to talk through decisions, however you could also start by having students complete the sorting independently before moving to step 4 where they will compare their sort with another student.
Step 4: Once students have sorted their problems have them work with another pair/group to compare their sorts. For any problems where they disagree on the operation, ask students to discuss their thinking and work to reach consensus. If a group is unable to come to agreement on a problem, ask them to put it aside so we can discuss it as a whole class during the debrief (Step 5).
Step 5: After groups have had time to talk through their sorts, bring the class together to debrief their thinking, talk through any problems which caused disagreement, and come up with some guidelines for determining the underlying characteristics of problems requiring each operation. With the problems displayed for all to see, have students make “We noticed that…” statements that generalize the patterns found for problems using each operation. For example, “We noticed that for subtraction we were finding the difference between two groups but for addition we were putting groups together.” Recording these generalization statements on anchor charts, along with representative problems, will serve as a good future resource for students that will promote their sense making and reasoning ability, as well as their independent problem solving ability.
Some groups of students are very adept at generalizing patterns, while others are not. Being prepared with questions such as, “Is that always true?” and “How is that different than in the (opposite operation) problems?” can be helpful. Additionally, pulling out a few problems that have similar structures and asking students, “What do you notice is the same about these problems?” is also a helpful scaffold.
Extending the lesson. In addition to sorting problems based on inverse operations, students can also sort problems that use the same operation, but which have different underlying structures. Connecting this type of sort to solution strategies helps students develop fluency as they come to recognize entry points for different types of problems and thus are better able to pull forward prior experiences as they tackle unfamiliar problems. The “Mathematics Glossary” of the Common Core State Standards for Mathematics identifies common addition and subtraction problem situations (Table 1) and multiplication and division problem situations (Table 2). Although students are not expected to be able to name these different problem structures, becoming familiar with them and recognizing that each type requires a slightly different approach will empower students and allow them to carry these strategies forward to similarly structured problems using more complex numbers in future years, as this knowledge and understanding will never “expire”, unlike the rules upon which many students currently rely.