Foundations for The Math Workshop

Understanding Problem Solving | Understanding Differentiation
Understanding State and National Standards

Understanding Problem Solving

Problem Solving is the cornerstone of mathematics. It is full of definitions, vocabulary, algorithms, and proofs. But it is the quest that humans have for solving seemingly unsolvable problems or puzzles that keeps the science of mathematics evolving into the master of all sciences. (Contreas, 2005) Everyday life is full of problems to solve. Every day, there is something new to solve, some new dilemma to unravel. Most people thrive in an environment ruled by problems to solve. Children especially like to be mini-detectives, going in and out of all kinds of problems. Their young inquisitive nature and natural curiosity fuels a passion for solutions.

When children get into school they tend to lose their natural curiosity and desire for problem solving. Textbooks and the unnatural contexts that they attempt to create, leave children feeling confused and frustrated. In order to combat those feelings, educators have been working with investigations, and more life like problems to get to the core of children’s thinking and understanding. Marilyn Burns has spent years creating problems that will accelerate children’s understandings of concepts and also creating intriguing contexts those children understand and enjoy. Burns says: “Rapid advances in knowledge are occurring in our society. Children must be prepared for a technological future that will require them to face a myriad of yet undetermined needs.” (Burns, 1992) She contends that hands-on teaching, and the use of problem solving will give children the confidence they need to be successful when confronted with proofs, algorithms and theorems.

Another body of research that has children solving problems with manipulatives and interesting contexts is Cognitively Guided Instruction (Carpenter and Fennema, et al 1999.) These researchers found that children were able to construct their own strategies when solving problems in addition, subtraction, multiplication and division. By creating problem types to elicit certain strategies, children can learn the algorithms on their own and therefore have a greater understanding of the concepts within the problems. Problems are also designed with familiar contexts making it more realistic and relevant for children. “ Until recently, we have not clearly recognized how much young children understand about basic number ideas, and instruction in early mathematics too often has not capitalized on their rich store of informal knowledge. As a consequence, the mathematics we have tried to teach in school often has been disconnected from the ways that children think about and solve problems in their daily lives.” (Carpenter, Fennema, et al 1999)

The Math Workshop takes a similar stance to problem solving as these and other researchers. The activities are designed with multiple ways to get to the solution. One piece of The Math Workshop has the children solving logic problems. These problems have multi-step clues and children need to use a process of elimination and proving in order to correctly solve the problem. Puzzles and Shapes is another problem solving piece The Math Workshop has incorporated into its program. Students develop their spatial sense by manipulating tangrams into shapes by solving the problems being posed. The Math Workshop values the innate curiosity that children bring into a mathematics classroom. Instead of trying to discourage and frustrate these abilities to problem solve, The Math Workshop allows them to follow their instincts and develop the important concepts they will need for life.

Understanding Differentiation

Whether you teach a single grade level or in a multiage classroom, every classroom has a wide range of students with varying abilities, strengths, backgrounds, and needs. Students’ math abilities and developmental levels differ as well. Differentiation (Tomlinson, 1999) reminds us that students need instruction and tasks that are based on developmental readiness. Students learn more when they make sense out of information on their own, rather than when information is imposed on them. Sometimes students need more time to grasp a concept. Sometimes students need more challenging work to continue their increasing knowledge.

The Math Workshop was designed based on this philosophy and knowledge. The twelve math strands’ activities of The Math Workshop are tiered, focusing on the same concepts and objectives but at different levels of complexity. Each strand’s activities are similar in structure so that more than one level of the program can be used simultaneously to meet the different levels of mathematicians in any classroom. As students develop mastery of a math strand at one level, that level can be increased to meet developmental needs. The Math Workshop provides repeated opportunities for students to make sense of math concepts with their own timelines and at their own level. Learning new content requires a lot of practice over time for students to grasp the knowledge and “it is only after a great deal of practice that students can perform a skill with speed and accuracy” (Marzano, 2001). The Math Workshop helps teachers differentiate their math instruction to meet the abilities, strengths, backgrounds, and needs of all students for the entire school year.

Understanding National/State Standards
The move to Standards based education in this country has been a long one. After the book, A Nation at Risk, (National Commission on Excellence in Education, 1983) educators began the gentle, slow-moving change from textbook based instruction, to the comprehensive, long-term movement of Standards based education.

The Math Workshop was designed to help teachers face the monumental task of using standards to raise student achievement. Each of the twelve activities was designed to cover the most essential, and difficult skills students will need to be successful in mathematics.

The Regional Educational Laboratory Network has many articles discussing Standards based education. One of the many notions to come out of the Network and its studies is that standards based education requires a long term commitment on the part of educators. Standards move from grade to grade in a continuum designed to enhance sequential steps in learning. The Math Workshop designed its program with a similar format. Each skill is repeated in each grade level with new steps added as students progress through the levels.

The Math Workshop researched National Standards and found performance objectives that are most crucial to developing students understanding of mathematics. Those concepts are repeated throughout the program with an emphasis on mastery. The National Council of the Teachers of Mathematics (NCTM) has published a series called Principles and Standards for School Mathematics Navigation Series (NCTM, 2002). This series discusses the sequential nature of the National Standards and delves into the types of differentiated skills used at each level.

With problem solving, differentiation and Standards based education as foundations, The Math Workshop has created a program that will encourage students to become more independent as well as teach fundamental skills in mathematics so that they will be successful throughout their educational careers.