Math:



Welcome to Ms. Worthing’s Mathematics class! As a parent, I hope you are ready for your child to have a great year gaining an understanding of important mathematical concepts including number sense, geometry, algebra, measurement and data analysis all based on cultivating reasoning, problem solving and communication skills. Let me describe to you what a typical mathematics class will look like for your child.

Students will be given a carefully designed problem to work on individually or in small, heterogenous groups. As they reason through a problem, they are free to use manipulatives and calculators and will record their work, showing the steps they took to arrive at an answer. This will assist them in articulating their reasoning about the problem during discussion. At the end of work time, students will come together as a class and discuss different reasoning and methods behind solving problems.

The goals for discussion are that students will effectively communicate their own mathematical reasoning, be exposed to and work with different ways of solving problems and clarify ideas proposing questions to one another. Ultimately, discussion will provide students time for reflection and feedback from their peers and myself related to their understanding of math concepts. We will work through problems relating to a concept over multiple days so that students have time to explore concepts’ relationships before learning traditional algorithms associated with them.

This mathematics day is likely very different than the way you and I learned mathematics. The way your child will learn mathematics is created to develop a strong mathematical foundation based on conceptual understanding, recognizing relationships between concepts and development of reasoning and problem-solving skills. The process entails learning mathematics by doing mathematics, not just by memorization and drill. Although this may be new to you, it will be exciting for your child and my goal is that he or she develops not only a strong mathematical foundation but a love of mathematics and see it’s use and practicality in everyday life.

The theory behind my teaching of mathematics is based on the constructivist theory. According to constructivist theory, learning occurs through doing mathematics, putting students in an active learning role and emphasizing reflection. (Van de Walle). Learning is therefore inquiry and problem based, discussion centered and contextual allowing students to develop their own mathematical understanding through exploring concepts, solving problems and discussing findings in meaningful contexts (Steen). Research has shown that placing math in meaningful, interesting contexts facilitates students’ intrinsic motivation and contributes to functional knowledge of mathematics which is useful in everyday situations (Middleton, Spanias, 1999).

Traditional mathematics is historically the dominant theory of mathematics and most likely the way you and I learned mathematics. In traditional mathematics learning takes place through direct instruction and practice to learn rules to solve problems in order to get a correct answer. An external source, such as rules or what I say is correct, is the heart of knowing math in traditional mathematics. On the other hand, constructivism places individual reasoning at the center of knowing math instead of getting correct answers, which is consistent with math as a formal discipline (Lampert). In class, often we will learn traditional algorithms, which are a major focus of traditional math, but not until students come to their own understanding of concepts. Similarly, there are some fluency concepts including knowing basic mathematical facts, such as multiplication, which should be automatic for students and will involve drill and practice. However, the major emphasis in our class will be that students create and understand interconnected math concepts.

An important tenant of student learning in my classroom is that it is problem based. You may be wondering how students will be solving problems to further their mathematical understanding. Important means of solving problems in our class will include the use of manipulatives, calculators and working in groups.

MANIPULATIVES:

I believe students should have open access to manipulatives. Manipulatives are concrete objects such as beans, base-ten counters, and geoboards assist students in understanding and representing abstract mathematical concepts. (McNeil, Jarvin, 2007). If you walk into the classroom it may look like students are simply “playing” with blocks, rods and shapes. However, by physically working with objects they become familiar with concrete properties and attributes of numbers, shapes and other mathematical concepts which will enhance their understanding. As I described, learning mathematics is a process of doing mathematics and using manipulatives is an integral aspect of that process as students physically construct their own understandings (Johnson). Research supports the use of manipulatives in learning mathematics because students have different learning styles and using manipulatives provides an additional mode of learning through touch which goes beyond visual or auditory learning and will assist additional students in their learning (McNeil, Jarvin). Similarly, placing learning in real-world contexts as is often done with manipulatives improves students’ understanding and performance.

A common concern for parents and educators is that manipulatives are used incorrectly, not as a learning tool. However, after students have initial time to become familiar with a new manipulative, I will demonstrate effective manipulative use by carefully constructing problems designed for specific manipulatives. In this way, students will learn how to appropriately choose and use manipulatives to benefit their learning. We will also record our representations with manipulatives so that students can reflect upon and use them again. My goal is that manipulatives are seen by students as a learning tool that they can effectively manage and use to aid in their understanding of abstract concepts and solving problems. Therefore, manipulatives will always be accessible to students in any mathematical work students are doing in the classroom.

CALCULATORS:

Calculators are another useful tool for solving problems and can assist students in furthering their mathematical understanding. Calculators can be used to develop an understanding of concepts, assist in solving problems when computation is not the focus and can save time which is best used in conceptual development (Van de Walle). Similarly, I cannot restrict student usage of calculators at home or in real world contexts such as at the bank or a job, nor would I want to because calculators are commonly used and beneficial in society. Thus, allowing students to become comfortable with calculators in the classroom and learn to use them appropriately, not as a substitute for conceptual understanding, will benefit them. For example, once students have developed a conceptual understanding of multiplication of large digit numbers and place value, if they are working on a different conceptual problem, such as area, which involves the computation 568 x 753, using a calculator to determine this large number computation will allow them to focus on the conceptual development of area instead of solving the complex computations involved. That said, there will be times when I will restrict calculator access because we are focusing on developing computational skills, such as addition, subtraction, multiplication and division. These computations can easily be solved with a calculator but students initially need to gain an understanding of the concept or the why and how behind the computations. The National Council for Teachers of Mathematics supports the use of technology and calculators in the classroom and state that calculator usage should support, not substitute, conceptual development (NCTM, 2000).

GROUPING:

Students will explore concepts and problems individually and in heterogenous groups. There will be a definite focus on cooperative learning in our math class and I hope to create an environment where students feel they can openly share their reasoning and thoughts, ask questions of one another and assist each other in clarification and development of ideas. Often, I will initially allow students time to work on a problem on their own so they have the opportunity to develop and record their own reasoning. However, it will also be common for students to begin work on problems or with manipulatives in small heterogenous pairings and move into this type of grouping after individual work to share and discuss. Providing students opportunities to share their ideas with others, receive feedback and cooperatively learn or explain concepts to one another will assist students at all ability levels. Studies show that many students begin mathematics with positive attitudes about their abilities. However, as they are placed in homogenous “ability” groupings, they begin to believe they are either innately good or bad at math and that it is a special subject in which some students are meant to succeed while others will fail (Middleton & Spanias). That is not the case! All students can learn math and by working in mixed ability groups every student will receive the message that he or she can understand math.

ASSESSMENT:

Informal assessment will be common in our classroom as students work through many problem based learning tasks. This means I will listen to student reasoning regarding concepts to assess where students have strong understanding and where additional instruction is needed. Informal assessment is critical to improving and differentiating instruction for individual students and making sure all students are learning and meeting mathematic standards and grade level expectations. Another informal assessments I will be using is a math journal. A math journal is a place a student reflects on learning, clarifies ideas and looks back on previous material making connections between math topics. We will use journals to record conceptual understandings, reasoning, questions and feelings about math. I will respond to journals regularly and provide feedback for students on their ideas and progress. However, these will not be graded in the traditional sense. You are probably wondering what is going to be graded. I am deviating from traditional grading scales and using rubrics and performance indicators to score work and provide feedback. Grades will be the result of many scores and other aspects of student performance such as effort put forth and improvement which will assist in communicating to you and the student about his progress. By using rubrics in class students will have clear expectations from the outset of what is expected. Also, rubrics account for total performance including the reasoning process, answer, justification and effort (Van de Walle). As performance indicators describe understanding, students will be provided with descriptive feedback of where they can improve instead of just the number of problems they correctly answered. I will have some traditional tests in which students will be expected to show work and reasoning. Grading this way reflects the values of my classroom including emphasis on process, problem solving and reasoning and not just getting correct answers.

WHAT YOU CAN DO:

At times your child may find certain problems exceptionally challenging, however, resist the urge to provide correct answers or tell him how to perform an operation. Although this is difficult, an important way your child is learning is by creating his own understanding. He will learn through succeeding as well as getting stuck and then discussing problems with other students, challenges are great learning opportunities! Encourage your student to talk through his understanding of concepts or problems with you and show you how he arrived at solutions. This will assist him in developing logical reasoning skills and reinforce that having a “right answer” isn’t the point of math. You can also discuss and include your child in math that comes up in everyday life including making budgets, shopping, arranging furniture, etc. This will reinforce the idea that math is useful and meaningful in the real world.

I look forward to working with your child this year as he learns and grows in his understanding of mathematics. It will be an exciting time as he develops conceptual understandings, connections, reasoning and problem solving skills.

References

Johnson, L. (2005). Manipulatives. Retrieved June 6, 2009, from .

Lampert, M. (1990). When the problem is not the question and the solution is not he answer: Mathematical knowing and teaching. American Educational Research Journal. 27 (1), 29-63.

McNeil, N., & Jarvin, L. (2007, October 1). When Theories Don't Add Up: Disentangling the Manipulatives Debate. Theory Into Practice, 46(4), 309-316. (ERIC Document Reproduction Service No. EJ780955) Retrieved June 6, 2009, from ERIC database.

Middleton, J.A., Spanias, P.A. (1999). Motivation for achievement in mathematics education: What matters in coming to value mathematics. Journal for Research in Mathematics Education, 30, 65-85.

Steen, L.A. (2002). Achieving mathematical proficiency for all: Moving beyond the math wars. The College Board Review. 196, 4-11.

Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally. (6th ed.). Boston: Pearson Education.

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