How many of your students wouldn’t know where to begin if they were given this problem to solve?
“Allison has three candy bars to share equally among her and four of her friends. What amount will each person get?”
One of the greatest challenges for students is solving word problems. Several factors relate to this challenge including the following:
- Students may be encouraged by teachers to look for and use key words which can be misleading because the same key word may mean different things in different situations.
- Students may lack the mathematics language needed to solve the problem.
- Students often have trouble determining the conditions or known information in a problem and therefore find it difficult to find an entry point that will ultimately lead them to the solution.
Encouraging students to use simple pictures and diagrams to conceptualize and solve a problem is one way of addressing this complex issue. How many of your students would be able to start the word problem above if they had the visual below?
Helping your students become proficient at developing visual representations on their own can provide them with a powerful tool to solve complex problems even outside the realm of mathematics. But which types of visuals can be effective when teaching the various concepts included in the standards?
The Common Core State Standards in Mathematics (CCSSM) Progressions documents, informed both by research on children's cognitive development and by the logical structure of mathematics, include many visual examples that can be used to communicate mathematical ideas and to provide sense-making aids for solving problems. A great feature of this resource is that it is already organized by common core standards’ domains and grade levels, therefore provides guidance on visuals that can be used effectively in teaching various concepts.
Embrace the practice of including a variety of visuals and graphics to aid students in solving problems and learning important mathematics content. Below is an example of a 6th grade percent problem aligned to 6.RP.A.3c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given part and the percent, illustrates the value of using visuals.
What visual representations could you provide to guide students to solve the following problem on their own?
"Cameron found a jacket marked 20% off. If the original price is $90, how much is the discount?"
Following are three steps, along with example hints and visuals that a student a student could work through to find the solution to the problem on their own.
Step 1: The first step includes clarifying information in the percent bar diagram intended to support the student in identifying the known information in the problem.
Subsequent hints guide the student through determining the correct response.
Step 2: Once the known information in the Challenge has been identified, the second step directs thinking from percent to the equivalent monetary value.
Additional hints such as the one below are included in the tutorial that guides the student to determine that each 10% section is equivalent to $9.
Step 3: The final step, directs the student back to the original Challenge in determining the monetary value of a 20% discount on the jacket.
Additional hints including graphics guide the student to the correct response of $18.
Visual may help students reveal conditions that are not obvious when just reading the problem. Drawing a picture or diagram has long been included as a sound problem solving strategy for a very good reason. It works! Additionally, pictures and diagrams can be useful in keeping track of various stages of a multi-step problem.
The inclusion of pictures and diagrams not only benefits and is valuable for the visual learner, but provides an opportunity for all students to use reasoning to solve challenging word problems conceptually. For special educations students, explicit systematic instruction that involves extensive use of visual representations appears to be crucial, Gerstung and Clarke (2007, p. 2).
The CCSSM emphasize the importance of both the content and practice standards. Encouraging students to utilize pictures and diagrams improves problem solving skills while at the same time affords the opportunity to conceptualize and learn the mathematics. Both of which are necessary knowledge and skills in becoming college and career ready.