# Mathematics

## Mathematics Basics

Mastering math
Mathematics is often thought of as a subject that a student either understands or doesn't, with little in between. In reality, mathematics encompasses a wide variety of skills and concepts. Although these skills and concepts are related and often build on one another, it is possible to master some and still struggle with others. For instance, a child who has difficulty with basic multiplication facts may be successful in another area, such as geometry. An individual student may have some areas of relative strength and others of real vulnerability.

The importance of early intervention
Because math is so cumulative in nature, it is important to identify breakdowns as early as possible. Children are more likely to experience success in math when any neurodevelopmental differences that affect their performance in mathematics are dealt with promptly - before children lose confidence or develop a fear of math.

## The Developing Math Student

Math Milestones
Some math skills obviously develop sequentially. A child cannot begin to add numbers until he knows that those numbers represent quantities. Certain skills, on the other hand, seem to exist more or less independently of certain other, even very advanced, skills. A high school student, for example, who regularly makes errors of addition and subtraction, may still be capable of extremely advanced conceptual thinking.

Because math skills are not necessarily learned sequentially, natural development is difficult to chart; problems are equally difficult to pin down. Nevertheless, educators identify sets of expected milestones for a given age and grade as a means of assessing a child's progress. Learning specialists pay close attention to these stages in hopes of better understanding what can go wrong and when.

During this stage, children should begin to:

• count with understanding and recognize "how many" in sets of objects;
• use multiple models to develop initial understandings of place value and the base-ten number system;
• develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections;
• develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers;
• connect number words and numerals to the quantities they represent, using various physical models and representations;
• understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.

During this stage, children should:

• understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals;
• recognize equivalent representations for the same number and generate them by decomposing and composing numbers;
• develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers;
• use models, benchmarks, and equivalent forms to judge the size of fractions;
• recognize and generate equivalent forms of commonly used fractions, decimals, and percents;
• explore numbers less than 0 by extending the number line and through familiar applications;
• describe classes of numbers according to characteristics such as the nature of their factors.

During this stage, children should:

• work flexibly with fractions, decimals, and percents to solve problems;
• compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
• develop meaning for percents greater than 100 and less than 1;
• understand and use ratios and proportions to represent quantitative relationships;
• develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;
• use factors, multiples, prime factorization, and relatively prime numbers to solve problems;
• develop meaning for integers and represent and compare quantities with them.

During this stage, children should be able to:

• develop a deeper understanding of very large and very small numbers and of various representations of them;
• compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions;
•  understand vectors and matrices as systems that have some of the properties of the real-number system;
• use number-theory arguments to justify relationships involving whole numbers.

Additional information about milestones and K-12 math curriculum is available on The National Council of Teachers of Mathematics Web site. NCTM's Principles and Standards for School Mathematics outlines grade-by-grade recommendations for classroom mathematics instruction for both content matter and process.

Competence in mathematics is increasingly important in many professions. This competence draws on more than just the ability to calculate answers efficiently. It also encompasses problem solving, communicating about mathematical concepts, reasoning and establishing proof, and representing information in different forms. Making connections among these skills and concepts both in mathematics and in other subjects is something students are more frequently asked to do in the classroom setting and later in the workplace. For specific information about the range of skills and concepts in school mathematics, please visit the Principles and Standards for School Mathematics on the National Council of Teachers of Mathematics Web site.

## Neurodevelopmental Functions and Math

In recent years, researchers have examined aspects of the brain that are involved when children think with numbers. Most researchers agree that memory, language, attention, temporal-sequential ordering, spatial ordering, and higher-order thinking are among the neurodevelopmental functions that play a role in mathematics. These components become part of an ongoing process in which children constantly integrate new concepts and procedural skills as they solve more advanced math problems.

Memory may have a significant impact on thinking with numbers. As Dr. Mel Levine points out, "Almost every kind of memory you can think of finds its way into math."

• Factual memory in math is the ability to recall math facts. These facts must be recalled accurately, with little mental effort.
> Try it yourself. Experience a problem with basic facts.
• Procedural memory is used to recall how to do things - such as the steps to reduce a fraction or perform long division.
• Active working memory is the ability to remember what you're doing while you are doing it, so that once you've completed a step, you can use this information to move on to the next step. In a way, active working memory allows children to hold together the parts of math problems in their heads. For example, to perform the mental computation 11 x 25, a child could say, "10 times 25 is 250 and 1 times 25 is 25, so adding 250 with 25 gives me 275." The child solves the problem by holding parts in his or her mind, then combining those parts for a final answer.
> Try it yourself. Experience a multi-step problem.
• Pattern recognition also is a key part of math. Children must identify broad themes and patterns in mathematics and transfer them within and across situations. When children are presented with a math word problem, for example, they must identify the overarching pattern, and link it to similar problems in their previous experience.
• Memory for rules is also critical for success in math. When children encounter a new problem, they must recall from long-term memory the appropriate rules for solving the problem. For example, when a child reduces a fraction, he or she divides the numerator and the denominator by the greatest common factor - a mathematical rule.
> See it now. Math Video: Mathematics and Memory

Language demands of mathematics are extensive.

• Children's ability to understand the language found in word problems greatly influences their proficiency at solving them.
•  In addition to understanding the meaning of specific words and sentences, children are expected to understand textbook explanations and teacher instructions.
• Math vocabulary also can pose problems for children. They may find it confusing to use several different words, such as "add," "plus," and "combine," that have the same meaning. Other terms, such as "hypotenuse" and "to factor," do not occur in everyday conversations and must be learned specifically for mathematics. Sometimes a student understands the underlying concept clearly but does not recall a specific term correctly.

Attention abilities help children maintain a steady focus on the details of mathematics.

• For example, children must be able to distinguish between a minus and plus sign - sometimes on the same page, or even in the same problem.
• In addition, children must be able to discriminate between the important information and the unnecessary information in word problems.
• Attention also plays an important role by allowing children to monitor their efforts; for instance, to slow down and pace themselves while doing math, if needed.

Ordering plays an important role in mathematics. Temporal-sequential ordering involves appreciating and producing information in a particular sequential order. Spatial ordering involves appreciating and producing information in an appropriate form.

• Sequencing ability allows children to put things, do things, or keep things in the right order. For example, to count from one to ten requires presenting the numbers in a definite order.
• Math is full of sequences --  almost everything that a child does in math involves following a sequence.
• When solving math problems, children usually are expected to do the right steps in a specific order to achieve the correct answer.
• Recognizing symbols such as numbers and operation signs and being able to visualize - or form mental images - are aspects of spatial perception that are important to succeeding in math.
• Visualizing as a teacher talks about geometric forms or proportion can help children store information in long-term memory and can help them anchor abstract concepts. In a similar fashion, visualizing multiplication may help students understand and retain multiplication rules.
> Try it yourself. Experience making 3-D inferences.

Higher-order thinking helps children to review alternative strategies while solving problems, to monitor their thinking, to assess the reasonableness of their answers, and to transfer and apply learned skills to new problems.