Problem Solving and the Common Core

Theory Behind Order, Problem Types, and Problem Contexts

“Implicit learning must reflect the desired explicit teaching.” This idea has been central in Roger Howe’s seminar “Problem Solving and the Common Core”. The concept stems from the premise that a large portion of student learning happens due to the order in which content is presented and the variety of problem types and contexts or lack thereof students are exposed to. To expand on this idea I will explain how order, problem type and context can impede the development of proportional reasoning and then outline how my unit will take a different approach.

In reflecting on my own practice, this concept of implicit teaching and learning is something that I often lose site of when designing lessons and analyzing curriculum. A general illustration of students implicitly learning from a lack of context is in the teaching of a triangle. A common mistake primary educators make is when they teach the basic elements of a triangle only using examples that are equilateral with a horizontal base. If this is the only manner in which students first see triangles, they are likely to implicitly learn that triangles are shapes that are limited to three equal side lengths with a horizontal base. Even though this was not explicitly taught, the majority of students will likely take on this narrow view of triangles due to the lack of context. Analogous to this example, if ratios and proportions are not taught in proper order and without a variety of problem types and contexts, narrow and insufficient proportional reasoning will likely develop. This unit will provide a purposeful sequence of concepts and a diversity of problem types that promote the essential understandings of ratios and proportions.

Order

The typical progression of concepts learned in a similar unit is as follows:

1. Students learn different ways to write ratios.
2. Students learn that proportions are two equivalent ratios and can be found by multiplying both quantities by the same value.
3. Students learn how to find unit rates by dividing both quantities by the second quantity.
4. Students then are taught the cross-multiplication algorithm to solve proportion problems with missing values. (Lobato et al. 2010)

This order roughly outlines the progression that my district-mandated curriculum uses. There are several flaws with this order. Students are first introduced to the concept of ratios by identifying basic relationships such as red marbles to blue marbles or boys to girls with the purpose of teaching what is and how to set up a ratio. See two examples of such problems:

Example D: There are 6 blue marbles and 4 red marbles. Write a ratio comparing red to blue marbles and blue to red marbles.

Example E: There are 15 boys and 8 girls in class. What is the ratio of boys to the total number of students in class?

In the above problems aim to address the concept of ratio in isolation, however there is no purposeful context. Meaning, there is no proportional relationship for which these ratios might be the constant rate of proportionality. These problems simply prompt students to order quantities in the form of a ratio without reasoning about the relationship between the quantities. This approach could potentially lead to misconceptions about the fundamental idea of ratio and makes the concept seem arbitrary and abstract.

Another and even more harmful impact of this typical sequence is how early the cross-multiply and divide method is presented. When students are given this algorithm too early in the development of concepts around proportional reasoning, it turns off the thinking switch when solving problems. I have experienced this in my own classroom when presenting this strategy to solve proportion problems. Shortly after the procedure is given, it is the only method used. Kids find the algorithm to be quick, comfortable, and fool-proof. Very little thinking happens thereafter. “Proportional reasoning involves much more than setting two ratios equal and solving for a missing term.” (Lesh et al. 1988) This procedure is efficient and one that students should learn to utilize eventually. However, it should be avoided until students develop a conceptual understanding of proportions.

The sequence of my unit will have two principal objectives: (1) unlike traditional progressions, the sequence will keep the idea of a proportional relationship at the fore by developing the concepts of ratio in conjunction with the concepts of proportion, not in isolation, and (2) the sequence will introduce skills and strategies that make the concepts of ratios and proportionality more accessible before teaching any procedural algorithms.

Context of Problems

When first developing proportional reasoning, students need to work with problems that help students visualize and grasp the desired outcomes. If students are given problems with scenarios in which they cannot easily model for themselves, they are more likely to look for procedural solutions and abandon the task of reasoning. They should be exposed to simple, familiar contexts and extend to more complex and/or unfamiliar ones as proportional reasoning develops. (Cramer and Post 1988)

Lessons in my unit will first consist of problems that are accessible and of high interest to my students. The contexts will be tangible so that students can use their own experiences to deepen their understanding of concepts. Examples of these contexts are “chocolateness” of chocolate chip cookies, strength of fruit juice, cost per item, length of time it takes to run around a track, and crowdedness of a room. These scenarios are easy to visualize and will provide an entry point for all students in whole-group discussion. In addition, they demand proportional reasoning and give purpose to comparing the relationship of different quantities.

Problem Types

The variety and order in which different problem types are presented to students plays a significant role in the development of proportional reasoning. Students need to be exposed to a range of problem types so that they develop reasoning that can be generalized to all situations involving ratios. As is true with all areas of math, if a student’s problem solving is limited to a subset of problem types and if the key issues are never presented or discussed, they will likely develop a narrow and less flexible understanding of the area’s concepts. The curriculum that I have used in the past has taught specific proportional reasoning strategies in isolation and dealt only with specific problem types. For example, problems that require students to find a missing value are traditionally used to teach the cross-multiply and divide algorithm. If students do not see a substantial variety of problems, they tend to stop analyzing and reasoning through problems. Seeing the same problem type repeatedly elicits the same strategy repeatedly, and this leads to a fixed and inflexible use of reasoning when presented with a problem type that could require a different strategy or approach. To avoid this, my unit will consist of mixed problem sets that contain a combination of many types of problems so that students have space to apply their different forms of reasoning under varying conditions.

There are three different problem types that require the use of proportional reasoning. They are: (1) missing value, (2) numerical comparison, and (3) qualitative prediction and comparison. (Post, 1988) The following sections provide a description and example of each type.

Missing Value

In the various curriculums I analyzed, missing value problems constituted a large majority. A problem of this type gives one complete ratio with two given quantities and another incomplete but proportional ratio with only one given value. Consider the following problem:

36 chocolate chips are used to make 6 chocolate chip cookies. If 12 cookies are made with same number of chips per cookie, how many chips are needed?

Here the complete ratio of 36 chips to 6 cookies is given with the task of finding the missing number of chips for 12 cookies in a proportional relationship. The degree of difficulty changes with different numerical contexts. The problem is generally easier if the relationship between the ratios or within the ratio is integral, as in the previously mentioned cookie problem. When the relationship between or within is fractional, the problems of course become more difficult. Because this unit aims to introduce and develop proportional reasoning for the first time, whole number relationships will be used to start with. Once students are comfortable with this idea of proportion, students will be exposed to proportional relationships with larger whole numbers, unit fractions, and general fractions with gradually increasing numerators and denominators. Using this strategy will allow different abilities to be challenged at an appropriate level throughout the unit.

Numerical Comparison

Numerical comparison problems require the evaluation of two given ratios. Consider the following problem:

Mixing flavored powder with water makes Kool Aid juice. Grandma made Kool Aid by mixing 3 cups of water with 9 tablespoons of powder. Grandpa made Kool Aid by mixing 6 cups of water with 12 tablespoons of powder. Which Kool Aid will taste stronger?

In order to reasonably answer the question, students must compare the two given ratios, 3 cups to 9 tablespoons and 6 cups to 12 tablespoons. This problem type requires students to apply proportional reasoning and determine a greater or lesser value in relation to the question. Although these problems do not explicitly ask the solver to express a proportional relationship, one must reason with the implicit ones to answer the question. Here, a proportional relationship was implied in each of the recipes. Grandma would use 3 tablespoons of powder for every cup of water and Grandpa soul use 2 tablespoons to every cups of water.

Qualitative Prediction

My analysis revealed that qualitative prediction is the most unrepresented problem type in middle school curriculum. It consists of a scenario that requires a comparison that is not dependent on numerical data. Consider the following problem:

Two friends each hammered a line of equally spaced nails into different boards  using the full length of each board. Bill hammered more nails than Greg. Bill’s board was shorter than Greg’s. On which board are the nails hammered closer? (Cramer and Post 1988)

The problem requires the comparison of the board length and number of nails. The absence of numbers forces the solver to focus on the relationship of two different ratios in order to answer the question. The implicit proportional relationship that must be considered is the length of a row of evenly spaced nails to the number of nails. This type of problem is valuable because it demands proportional reasoning and provides no opportunity to apply a procedure or algorithm. Although I have found that this type of problem is not typically assessed, it will be a valuable exercise in developing a complete understanding of proportion.