Unit 1: Why the push for problem-solving?

Learning Objectives

· Describe the need for students to become better problem solvers and to be able to solve problems in more than one way.

· Identify Common Core Standards for Mathematical Practices in videos of middle school exemplary lessons.

Mini-lecture

One of the premises of the Common Core State Standards is to provide a set of clear standards in which students are expected to master. These standards are said to be easy for teachers and parents to decipher as well so that they know what they must do to help students deeply understand the concepts. The implementation of the Common Core also serves the purpose of helping students become college and career ready once they leave high school. In recent years, it has become more common for businesses in the United States to employ people from other countries who have the necessary skills for particular jobs. To reverse this trend, the creation of a pool of United States citizens meeting specific requirements is important. To reach this goal, the common core aims for students to be avid problem solvers and to be able to work with others to accomplish a common goal. Students need to reach a deep level of understanding of each of the concepts that they are being introduced to so that they will be able to apply the mathematics to real-life situations and they need to be able to utilize a variety of methods and strategies to arrive at an accurate conclusion.

Since the focus of this unit is problem solving, we are going to take a look at the common core’s first and fourth standards for mathematical practice.

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

This mathematical practice stresses the importance of students being able to understand the relationship between different representations that are used for different problems such as tables, equations, and graphs. Also, students need to be able to use different approaches to solve problems in a different way. Students need to be able to analyze whether or not the alternative approach makes sense by using mathematical reasoning and how the results correspond with other approaches.

Watch the Joe Condon’s video (6th grade) -> http://www.insidemathematics.org/index.php/standard-1

CCSS.Math.Practice.MP4 Model with mathematics

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

It is evident that math educators need to relate concepts to real-life applications. If students grasp the connections of what they are learning to real situations, they are likely to become more motivated to learn because they can visualize a purpose. Also, these real-life applications can help to engage students and develop them into better problem solvers.

Watch Cecilio Dimas’ video (7th/8th grade) -> http://www.insidemathematics.org/index.php/standard-4

Learning Activities

Now that you have been provided with a case for students to improve their problem solving skills to apply mathematical concepts to real-life situations, examined two standards of mathematical practice, and have watched clips of two exemplary middle-school lessons, it is time to tie this material together and review what you have learned.

1. How do you feel about the common core and the need for students to be able to solve real-world problems?

2. Do you feel as though it is more important to gear your lessons towards students arriving at the correct answer for a real-life application problem or using the problem to develop skills the students need to be able to solve the problem?

3. From the first video clip, which involves the students creating strategies to compare unit rates, how does this clip connect to the first standard for mathematical practice? How does the second video clip, which takes a real-life problem about DVD rental plans and requires to use multiple representations, connect to the fourth standard for mathematical practice? Use specific evidence from the videos to support your answer.

4. What are some strategies that you incorporate/will incorporate in your classroom which connect to the standards of mathematical practice?

Conclusion

When using Common Core Standards, the stress is places on the process of arriving at a solution rather than all of the importance being placed on a singular result. By using this approach to learning, the students should develop problem solving skills over time that will allow them to grow into competent adults who will be valuable assets to the American work force.