Elementary mathematics

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Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.

Topics[edit]

The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on probability and statistics and on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.[1]

In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.[2]

A mastery of elementary mathematics is necessary for many professions, including carpentry, plumbing, and automobile repair, as well as being a prerequisite for all advanced study in mathematics, science, engineering, medicine, business, architecture, and many other fields.

United States[edit]

In the United States, there has been considerable concern about the low level of elementary mathematics skills on the part of many students, as compared to students in other developed countries.[3] The No Child Left Behind program was one attempt to address this deficiency, requiring that all American students be tested in elementary mathematics.[4]

Common core[edit]

Released in 2010, the Common Core State Standards Initiative is an education initiative in the United States that details what K-12 students should know in English and math at the end of each grade.

The stated goal of the mathematics Standards[5] is to achieve greater focus and coherence in the curriculum (page 3). This is largely in response to the criticism that American mathematics curricula are "a mile wide and an inch deep".

The mathematics Standards include Standards for Mathematical Practice and Standards for Mathematical Content.

Mathematical practice[edit]

The Standards mandate that eight principles of mathematical practice be taught:

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

The practices are adapted from the five process standards of the National Council of Teachers of Mathematics and the five strands of proficiency in the National Research Council’s Adding It Up report.[6] These practices are to be taught in every grade from kindergarten to twelfth grade. Details of how these practices are to be connected to each grade level's mathematics content are left to local implementation of the Standards.

As an example of mathematical practice, here is the full description of the sixth practice:

6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Mathematical content[edit]

The Standards lay out the mathematics content that should be learned at each grade level from kindergarten to Grade 8 (age 13-14), as well as the mathematics to be learned in high school. The Standards do not dictate any particular pedagogy or what order topics should be taught within a particular grade level. Mathematical content is organized in a number of domains. At each grade level there are several standards for each domain, organized into clusters of related standards. (See examples below.)

Four domains are included in each of the six grades from kindergarten (age 5-6) to fifth grade (age 10-11):

  • Operations and Algebraic Thinking;
  • Number and Operations in Base 10;
  • Measurement and Data;
  • Geometry.

Kindergarten also includes the domain Counting and Cardinality. Grades 3 to 5 also include the domain Number and Operations--Fractions.

Four domains are included in each of the Grades 6 through 8:

  • The Number System;
  • Expressions and Equations;
  • Geometry;
  • Statistics and Probability.

Grades 6 and 7 also include the domain Ratios and Proportional Relationships. Grade 8 includes the domain Functions.

In addition to detailed standards (of which there are 21 to 28 for each grade from kindergarten to eighth grade), the Standards present an overview of "critical areas" for each grade. (See examples below.)

In high school (Grades 9 to 12), the Standards do not specify which content is to be taught at each grade level. Up to Grade 8, the curriculum is integrated; students study four or five different mathematical domains every year. The Standards do not dictate whether the curriculum should continue to be integrated in high school with study of several domains each year (as is done in other countries, as well as New York and Georgia), or whether the curriculum should be separated out into separate year-long algebra and geometry courses (as has been the tradition in most U.S. states). An appendix[7] to the Standards describes four possible pathways for covering high school content (two traditional and two integrated), but states are free to organize the content any way they want.

There are six conceptual categories of content to be covered at the high school level:

Some topics in each category are indicated only for students intending to take more advanced, optional courses such as calculus, advanced statistics, or discrete mathematics. Even if the traditional sequence is adopted, functions and modeling are to be integrated across the curriculum, not taught as separate courses. In fact, modeling is also a Mathematical Practice (see above), and is meant to be integrated across the entire curriculum beginning in kindergarten. The modeling category does not have its own standards; instead, high school standards in other categories which are intended to be considered part of the modeling category are indicated in the Standards with a star symbol.

Each of the six high school categories includes a number of domains. For example, the "number and quantity" category contains four domains: the real number system; quantities; the complex number system; and vector and matrix quantities. The "vector and matrix quantities" domain is reserved for advanced students, as are some of the standards in "the complex number system".

Examples of mathematical content[edit]

Second grade example: In the second grade there are 26 standards in four domains. The four critical areas of focus for second grade are (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes. Below are the second grade standards for the domain of "operations and algebraic thinking" (Domain 2.OA). This second grade domain contains four standards, organized into three clusters:

Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Add and subtract within 20.
2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
Work with equal groups of objects to gain foundations for multiplication.
3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Domain example: As an example of the development of a domain across several grades, here are the clusters for learning fractions (Domain NF, which stands for "Number and Operations—Fractions") in Grades 3 through 6. Each cluster contains several standards (not listed here):

Grade 3:
  • Develop an understanding of fractions as numbers.
Grade 4:
  • Extend understanding of fraction equivalence and ordering.
  • Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
  • Understand decimal notation for fractions, and compare decimal fractions.
Grade 5:
  • Use equivalent fractions as a strategy to add and subtract fractions.
  • Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
In Grade 6, there is no longer a "number and operations—fractions" domain, but students learn to divide fractions by fractions in the number system domain.

High school example: As an example of a high school category, here are the domains and clusters for algebra. There are four algebra domains (in bold below), each of which is broken down into as many as four clusters (bullet points below). Each cluster contains one to five detailed standards (not listed here). Starred standards, such as the Creating Equations domain (A-CED), are also intended to be part of the modeling category.

Seeing Structure in Expressions (A-SSE)
  • Interpret the structure of expressions
  • Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Functions (A-APR)
  • Perform arithmetic operations on polynomials
  • Understand the relationship between zeros and factors of polynomials
  • Use polynomial identities to solve problems
  • Rewrite rational expressions
Creating Equations.★ (A-CED)
  • Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities (A-REI)
  • Understand solving equations as a process of reasoning and explain the reasoning
  • Solve equations and inequalities in one variable
  • Solve systems of equations
  • Represent and solve equations and inequalities graphically

As an example of detailed high school standards, the first cluster above is broken down into two standards as follows:

Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.★
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2y2)(x2 + y2).

Modern Elementary mathematics[edit]

Modern elementary mathematics is the theory and practice of teaching elementary mathematics according to contemporary research and thinking about learning. In applying modern elementary mathematics, teachers may use new and emerging media and technologies such as computer games, interactive whiteboards and social networks as learning tools.

References[edit]

  1. ^ Gary L. Musser, Blake E. Peterson, and William F. Burger, Mathematics for Elementary Teachers: A Contemporary Approach, Wiley, 2008, ISBN 978-0-470-10583-2.
  2. ^ Timothy J. McNamara, Key Concepts in Mathematics: Strengthening Standards Practice in Grades 6-12, Corwin Prss, 2006, ISBN 978-1-4129-3842-6
  3. ^ Liping Ma, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning.), Lawrence Erlbaum, 1999, ISBN 978-0-8058-2909-9.
  4. ^ Frederick M. Hess and Michael J. Petrilli, No Child Left Behind, Peter Lang Publishing, 2006, ISBN 978-0-8204-7844-9.
  5. ^ mathematics Standards
  6. ^ Garfunkel, S. A. (2010). “The National Standards Train: You Need to Buy Your Ticket.” UMAP J 31 (4): 277 – 280.
  7. ^ appendix