Problem Solving and the Common Core

Activities

Structured Problem Solving

The general form of lessons I will use in this unit will be centered on structured problem solving. This is different when compared to a typical gradual release lesson format where students are taught a strategy explicitly, practice the strategy with heavy teacher guidance, and then eventually apply it through independent practice. In a problem-based format, students are presented a problem to be first worked on independently. Here students will apply their own mathematical knowledge in an effort to develop strategies to reach a solution. Once students have had time to work independently, their different strategies and solutions are shared and discussed in groups or as a class. During this time students will be able to see multiple approaches, discuss misconceptions, and come to new conclusions about the material. The discussion must be carefully facilitated in order to reach the desired outcomes for the lesson. If all student ideas have been exhausted and the strategies or understandings the lesson set out to achieve have still not been reached, explicit teaching will then take place. Finally the students are given a set of additional problems to apply and practice what they learned from the problem solving and the discussion. The key idea behind a structured problem solving approach is that students are first given the opportunity to reason and construct their own concepts and strategies, which leads to a deeper understanding of the content covered in a given lesson. In the pursuit of nurturing proportional reasoning, a new way of thinking for 6^th graders, this approach will be fundamental.

Structured problem solving is laid out in Akihiko Takahashi’s paper titled, “Characteristics of Japanese Mathematics Lessons”. Takahashi stresses that in addition to the attention devoted to extensive discussion, the selection of problems and activities needs to be carefully considered as well. (Takahashi 2006) For the purpose of developing proportional reasoning, the progression of problems in this unit is designed to bring out concepts and strategies that cohesively build on each other. In general, each lesson will present a new problem scenario for students to solve. This scenario will be engaging and include the use of visuals, multi-media, and props to ensure all students access and become invested in the problem. After the structured problem-solving process has taken place, students will be given a similar scenario if not the same but with different numbers. These exercises will give students the opportunity to process, apply, and generalize the previously discussed strategies and understandings.

The problems that I will use to center my lessons around require a greater explanation and use of images. For access to these, visit the website www.mrbingea.blogspot.com or email me at aaron.bingea@gmail.com.

Number Talks

A number talk is a five to ten minute routine that I will use to start most class periods. The purpose of this activity is to review prerequisite or fundamental concepts to the day’s lesson in an efficient and meaningful way. As stated earlier, proportional reasoning requires the mastery of many skills learned prior to sixth grade, and it is likely that my students will have large gaps in these areas that could hinder their learning about proportional relationships. I will use number talks to gradually remediate and develop these more basic understandings throughout the entire year. The routine follows a simple format that is centered on the purpose of bringing all students into the thinking and discussion around the lesson’s concepts.

A number talk begins by all students putting away materials to ready themselves for the prompt that will only require the use of mental math. The problem is presented on the board for all students. They then solve the problem mentally and put their thumbs up when they have reached a solution. Next, volunteers share their strategies as the teacher models each student’s line of thinking on the board. Finally the class discusses the accuracy and reasoning of each strategy. By the end of this discussion students can see multiple, valid approaches to solving a relatively simple problem. Mastery of any one strategy is never the goal for a single number talk; instead it is a daily practice for students to gradually acquire a wide range of skills with basic operations and number sense. For a more thorough description of number talks refer to the book written by Cathy Humphreys and Ruth Parker, “Making Number Talks Matter”. It details the procedure as well as how the teacher should facilitate student thinking and discussion. The book also gives example prompts and different strategies to help students develop. (Humphreys and Parker 2015)

The concept of focus in each number talk will be directly related to the lesson’s learning objectives. For example, when teaching the concept of equivalent ratios in a proportional relationship, the number talk will cover the concept of equivalent fractions. This sets up the opportunity to make connections to these foundational concepts in later class discussions. The following table shows several problems that will be used in number talks to prime lessons in this unit and the linked foundational concepts.

Each of these problems requires a skill or concept covered in grades three through five. Students will most likely know or be partially familiar one way to solve each problem. Through this routine students will be tasked with generating multiple strategies to solving these problems, making students more flexible with numbers and operations and ultimately being able to better access concepts within proportional reasoning.