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.

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, higher-order cognition, and spatial ordering are among the neurodevelopmental functions that play a role when children think with numbers. 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.

For children to succeed in mathematics, a number of brain functions need to work together. Children must be able to use memory to recall rules and formulas and recognize patterns; use language to understand vocabulary, instructions, and explain their thinking; and use sequential ordering to solve multi-step problems and use procedures. In addition, children must use spatial ordering to recognize symbols and deal with geometric forms. Higher-order cognition 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. Often, several of these brain functions need to operate simultaneously.

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.

Competence in mathematics is increasingly important in many professions (see sidebar). And it's important to remember that 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, both in the classroom setting, and later in the workplace.

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. 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.

**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.

Finally, **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.

Memory skills help children store concepts and skills and retrieve them for use in relevant applications. In turn, this kind of work relating new concepts to real-life contexts enhances conceptual and problem-solving skills. For example, a student may already know that 6 x 2 = 12. To solve the problem, "If there are six children, each with one pair of shoes, how many shoes in total?" the student will rely on memory of the multiplication fact and apply it to the particular case.

The 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.

Math is full of sequences. Almost everything that a child does in math involves following a sequence. 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. 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, being able to visualize -- or form mental images -- are aspects of spatial perception that are important to succeeding in math. The ability to visualize as a teacher talks about geometric forms or proportion, for example, 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.

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.

The fact that math skills are not necessarily learned sequentially means that natural development is very difficult to chart and, thus, problems are equally difficult to pin down.

Expected milestones for different grades are as follows

During this stage, children should begin to:

- count aloud
- compute the number of objects in a group
- understand that a particular number of objects has a fixed value despite the size or nature of those objects
- understand relative size and be able to sort objects by size and shape
- follow a sequence of two- and three-step commands
- be able to perform simple addition and subtraction computations

During this stage, children should:

- begin to perform simple addition and subtraction computations efficiently
- master basic math facts (such as, 3 + 2 = 5)
- recognize and respond accurately to mathematical signs
- begin to grasp the concept of multiplication (grade three)
- understand the notion of measurement and be able to apply this understanding
- improve their concepts of time and money

During this stage, children should:

- recall basic mathematical facts, including multiplication tables, with ease
- become competent with fractions, decimals, and percentages
- begin to understand the relationships among fractions, decimals, and percentages
- develop facility with word problems
- be adept at estimating quantities and rounding off numbers
- explain and illustrate mathematical concepts, rather than simply apply them
- plan and self-monitor during multi-step problem solving