Mathematics Content for Elementary Teachers

Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson

• Editorial: Routledge
• Año: 2004
• ISBN:
  9781135633790
• Edición: 1
• Categoría: Education
• Categoría: Education General

THE book for elementary education mathematics content courses! Designed to help prospective teachers of elementary school mathematics learn content beyond the rote level, this text stimulates readers to think beyond just getting the problem right and fosters their development into thoughtful, reflective, self-motivated, life-long learners. It stresses the what and why of elementary school mathematics content. Hints are provided about how to teach the content but this is mostly left to courses and texts that are dedicated to that purpose.The text is organized around the National Council for Teachers of Mathematics' Principles and Standards for School Mathematics. The Standards dictate the basic sections of the text. Within each section, appropriate specific topics are developed, intertwined with technology, problem solving, assessment, equity issues, planning, teaching skills, use of manipulatives, sequencing, and much more. In addition, major focal points of the Standards are emphasized throughout: effective teachers of mathematics should be able to motivate all students to learn, should understand the developmental levels of how children learn, should concentrate on what children need to become active participants in the learning environment, and should be engaged in ongoing investigations of new mathematical concepts and teaching strategies. Mathematics Content for Elementary Teachers is based on several fundamental premises:
*The focus of mathematics education should be on the process, not the answer.
*Elementary teachers should know the mathematics content they are teaching, know more than the content they are teaching, and teach from the overflow of knowledge.
*It is important for teachers to be flexible in allowing students to use different procedures--teaching from the "overflow of knowledge" implies knowing how to do a given operation more than one way and being willing to examine many different ways.
*Teachers need to learn to carefully cover the topics to be taught, to reflect upon them, and to be able to organize them. To help prospective elementary teachers concentrate on the mathematics content they will be expected to teach and begin to build the foundation for the methods they will use, this text includes only elementary mathematics content and does not address middle school concepts. Pedagogical features:
*The text is organized according to NCTM Standards.
*An informal writing style speaks directly to readers and is geared to pre-service teachers.
*Focus is given to multiple methods of problem solving at four developmental levels.
*Questions, exercises, and activities are interspersed throughout each section rather than gathered at the end of each chapter.
*Complete solutions for exercises are provided.

1

Guiding Principles

Focal Points

• The Equity Principle
• The Curriculum Principle
• The Teaching Principle
• The Learning Principle
• The Assessment Principle
• The Technology Principle
• The Challenge
• Math Anxiety
• Where to From Here?

Although we hope you will be a lifelong learner, we know that your focus will soon turn from learning mathematics to the art of teaching mathematics to children. It is possible that this will be the last formal course you will take in mathematics content. Because you are enrolled in this course, we know that you have successfully learned a significant amount of mathematics content, either from your own elementary, middle school, high school, and college teachers or on your own.

Much of your study during this term will review material that is familiar to you. We want you to look at concepts from different points of view. In order to successfully help children learn mathematics, you will need more than the ability to do mathematics. You will need a profound understanding of the mathematics you will be expected to teach. You will need to know which concepts are easy to learn, which are difficult to learn, and just what makes each concept easy or hard. This means that you must accept the challenge to probe deeply into elementary school mathematics content, expanding and solidifying your knowledge into a powerful base of content knowledge from which you may draw as you help children learn and understand mathematics. We call this pedagogical content knowledge (Shulman, 1986) because it is a class of knowledge that is held almost exclusively by expert teachers.

In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics invites you to “imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction” (NCTM, 2000, p. 3). The environment in which such a dream can come true will exist only if you, as the classroom teacher, have a deeply reflective pedagogical content knowledge so you can provide rich curricula and help your students learn and understand the important concepts of elementary school mathematics. It is on these central concepts that your future students will rely as they find their ways through their higher educations. You must prepare for a future that will require an ever greater understanding of mathematical concepts in your personal life, workplace, and scientific and technical communities. In the Principles and Standards for School Mathematics, NCTM identified six principles for school mathematics (pp. 10–27).

The Equity Principle

Excellence in mathematics education requires equity—high expectations and strong support for all students. (NCTM, 2000, p. 12)

We can no longer consider mathematics to be an enrichment subject for the most able students. We cannot allow mathematics to be the separator between the haves and the have-nots. Individuals who hold a limited mathematical background and understanding are severely handicapped as they select a career. Furthermore, studies now indicate that individuals entering the 21st-century workforce will change careers up to five times over their working years. Changing careers implies a revamped education with each new selection. Persons who do not possess fundamental mathematical skills and understandings will have difficulty finding options that allow for a limited mathematical background. Every elementary school teacher must be ready to help all students learn and understand mathematics. In every classroom, you will find students ranging from those with deep and abiding interests and talents in all academic subjects to those with special educational needs in some or all subjects.

Traditionally, certain groups of students have been viewed with reduced expectations. Some of these children live in poverty, aren’t fluent in English, or have learning disabilities. Additionally, female and non-White students have been victims of lowered teacher expectations. Your pedagogical content knowledge must be sufficient to facilitate learning for all.

Simply being able to do arithmetic will be insufficient to the task you have set for yourself. Your goal must be to have the ability to teach mathematics as component parts and as an integrated whole, to identify each student’s academic need, and to provide the exact amount of support needed by that student. You will be responsible for helping each student obtain a common elementary foundation of mathematics. To do this, you must solidify your own pedagogical content knowledge so you can maintain high expectations and provide strong support for each and every one of your future students. In your mathematics methods course, you will learn a variety of techniques for putting your pedagogical content knowledge to use in lesson planning and teaching.

The Curriculum Principle

A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, and well articulated across the grades. (NCTM, 2000, p. 14)

To say that the mathematics curriculum in elementary schools is arithmetic is insufficient because it gives the impression that you only need to be proficient in arithmetic to teach it to children. The foundations of mathematics must be formed in elementary schools, which means that children will need the underpinnings of many interwoven strands of mathematics. Arithmetic is certainly one of those strands, but it is not the only one.

Some preprofessional elementary school teachers have told us that they want to teach kindergarten, first, or second grade because the early grades require less arduous mathematics. This is simply not true. The foundations of important mathematics strands, such as algebra and geometry, are formed very early in the lessons of the elementary school mathematics curriculum. Mathematics must not be broken into discrete, disconnected bits. You may not be involved in designing the curriculum for your school or grade level, but you will most certainly be responsible for organizing lessons that teach the power of interconnected strands of mathematics. You will find that you must understand and be proficient in the elementary school mathematics of all grade levels in order to be an effective teacher. A reflective knowledge of the concepts taught in the classes before and after the grade level to which you are assigned will help you focus on the important mathematics your students already know and will need to learn before they move on. You will be one rung of their ladder of learning and you must help them deepen and extend their understanding of mathematics.

Whether you are assigned to kindergarten or sixth grade, your curriculum will include foundational concepts such as the Base 10 numbering system, place value, sets of numbers, proportionality, and functions. These foundational concepts will support your students as they work to connect and extend their ideas through mathematical reasoning, conjecturing, and problem solving. Additionally, concepts such as symmetry will help your students see life’s beauty. Modeling problems on real-world phenomena will help them recognize the importance of quantitative literacy. You can see that the daily mathematics curriculum of your classroom will significantly depend on the extent of your pedagogical content knowledge.

The Teaching Principle

Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. (NCTM, 2000, p. 16)

Your decisions and actions in your classroom will determine what your children learn. Consider your own elementary school experiences: Which memories are rich and satisfying and which memories are disappointing and lacking? It is very likely that each memory is linked to a teacher and that teacher’s ability to challenge and support you as you learned. You may not remember whether your teachers possessed profound knowledge of the subjects they were teaching, but you learned the most from those teachers who knew the most. Your ability, confidence in that ability, and your attitudes toward mathematics were shaped by your elementary school teachers. Take a few moments to reflect on how you will shape the abilities, confidence, and attitudes of your students.

Many elementary school teachers spend a great deal of time trying to understand the mathematics they are teaching. With college courses and certification examinations completed, they find that they simply don’t have knowledge that is deep enough or flexible enough to teach mathematics to children. Some take years to become confident in their understanding of and ability to teach mathematics. These teachers reach high and go far by continuing to learn and grow in their knowledge of mathematics and mathematical pedagogy. Perhaps the best gift you can give your future students is to start today to be a self-reflective lifelong learner—a person who loves to learn and loves to help others learn.

What of the children who sit in a classroom while their teacher is striving to increase personal knowledge of the subject to become a more effective teacher? We cannot afford to ignore the needs of children while their teachers study to become proficient. We invite you to work during this term to construct and solidify a base of pedagogical content knowledge on which you can build during your mathematics methods course. That way, you will become a teacher who is capable of reaching the mathematical curricular needs of each student—and a lifelong learner who will continue to grow into an even stronger teacher.

The Learning Principle

Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. (NCTM, 2000, p. 20)

For many decades, there have been disagreements about what concepts in mathematics are important and how they should be taught. Traditionally, elementary school teachers have guided students in how to perform operations, isolating basic skills into discrete segments, and building incrementally toward higher order skills. This reflects a behavioral or information-processing point of view in which the teacher conveys the knowledge and the students record (supposedly memorize—without understanding what is going on) the steps involved. Whereas this may lead to proficiency in each skill, it does not necessarily lead to understanding or flexibility of knowledge. Students, who can follow memorized steps with precision and accuracy, often lack the ability to connect basic skills to problem situations; they simply don’t understand when or why to apply these basic skills. That is why many students find it so difficult to solve word problems in which the required operations are not explicitly defined.

The daily experiences provided by teachers influence what and how students learn. The mathematics content provided in a classroom depends on the content knowledge of the teacher. If the teacher doesn’t know the concepts required by the curriculum, those concepts may be poorly taught or even ignored. Additionally, teachers who are insecure about their own knowledge of mathematics often provide activities that are teacher centered—focusing only on following a given rule to get a single right answer. Students in such a classroom may learn algorithms without applications and will certainly learn teacher-dependent behaviors. Students adopt the teacher’s mathematical insecurities. In contrast, a teacher who is secure and confident of the content—not only what is being taught, but also that which precedes and follows the particular concept being addressed—encourages student independence and delivers less rule-based, more heuristically based lessons, enabling students to recreate their thinking and find connections to applications of mathematics. To create autonomous learners, a teacher must have the content knowledge and confidence to select tasks that are appropriate for encouraging students to tackle hard material, explore alternate paths, and become tenacious, independent, problem solvers.

Foundational concepts are best taught at the concrete, or hands on, level. Manipulatives are important tools for building understanding of foundational concepts. When you think about how you learn, you may realize that manipulatives are needed when learning something new, even within your university courses. In this text, manipulatives such as Base 10 blocks, algebra tiles, Cuisenaire rods, number lines, colored chips, buttons, and coins are discussed and used. Some of your study time will be spent learning to use these tools.

Anything, at any age, that helps you understand can be considered a manipulative. Concrete learning experiences are certainly not limited to elementary age children. Think about your experiences as you learned to drive a car. Of course, you learned the rules of the road first so that you would not be a danger to yourself or others. But to learn to drive you actually got into a car or a simulator, grabbed the steering wheel, put your foot on the pedals, and drove. A car as a learning tool? Sure! We hope that you will make it a point to look for learning tools whenever and wherever they might be useful.

The Assessment Principle

Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. (NCTM, 2000, p. 22)

One might think that a discussion of assessment has no place in a content course. Indeed, the discussion of how and when you will assess the mathematics learning of your students will be left for your methods course and classroom experiences. However, an early understanding of the purposes of assessment may help as you develop a focus on this important part of a teacher’s responsibility.

The assessments to which you are subjected as a college student are summative. They are designed to determine what you have attained with respect to the instructional goals of the course and may be used to establish your grade for the course. However, you can use all of the activities of the course as personally formative assessments. As you examine your progress, you can make decisions about which concepts need to become the basis of further investigations and which only need polishing efforts.

Feedback from formal and informal assessment tasks provided by your instructor is intended to assist you in setting your learning goals and becoming a more independent learner. Even if feedback from the instructor is not provided for a task, you can examine your own completed work to determine if it represents your best effort, if it completely satisfies the task requirements, and if improvements are desirable. A good strategy is to enlist the support of a critical friend who is involved in the course with you. The folks who are taking this course with you may very well become your colleagues and establishing a rapport with them now may provide a support system for you later. Critical friends encourage and challenge each other based on examinations of one another’s work and behaviors. In this way, you can gain formative...