What Is Mathematics?

The National Council of Teachers of Mathematics (NCTM), the world's largest organization devoted to improving mathematics education, is developing a set of mathematics concepts, or standards, that are important for teaching and learning mathematics. There are two categories of standards: thinking math standards and content math standards. The thinking standards focus on the nature of mathematical reasoning, while the content standards are specific math topics. Each of the activities in this booklet touches one or more content areas and may touch all four thinking math areas.

The four thinking math standards are problem solving, communication, reasoning, and connections. The content math standards are estimation, number sense, geometry and spatial sense, measurement, statistics and probability, fractions and decimals, and patterns and relationships. We have described them and then provided general strategies for how you as a parent can create your own activities that build skills in each of these areas.

Thinking Mathematics

Problem solving is key in being able to do all other aspects of mathematics. Through problem solving, children learn that there are many different ways to solve a problem and that more than one answer is possible. It involves the ability to explore, think through an issue, and reason logically to solve routine as well as nonroutine problems. In addition to helping with mathematical thinking, this activity builds language and social skills such as working together.

What parents can do:

Children are naturally curious about everyday problems. Invite your children to figure out solutions to everyday situations. You can do this by talking about the problem, asking your children for ways to solve it, and then asking how they came up with those solutions.
Encourage your children to suggest problems and ask questions, too. Your children will learn how to figure things out and will learn that many problems can be solved several different ways.

Communication means talking with your children and listening to them. It means finding ways to express ideas with words, diagrams, pictures, and symbols. When children talk, either with you or with their friends, it helps them think about what they are doing and makes their own thoughts clearer. As a bonus, talking with children improves their vocabulary and helps develop literacy and early reading skills as well.

What parents can do:

Talk with your children and listen to what they have to say.
Reading children's books that rhyme, repeat, or have numbers in them (available at your local library) is a great way to communicate using mathematics.
All communication doesn't have to be in words. You can represent math in ways other than talking. Your children can make diagrams or draw pictures to solve problems or represent numbers. They can use concrete objects like pieces of paper or even fingers to represent numbers.

Reasoning is used to think through a question and come up with a useful answer. It is a major part of problem solving.

What parents can do:

To promote reasoning, ask your children questions and give them time to think about the answer. By simply asking questions and listening to answers, you are helping your children learn to reason.
Ask your children to figure out why something is the way it is and then check out their ideas. Let them think for themselves, rather than try to figure out what answer you want to hear.

Connections: Mathematics is not isolated skills and procedures. Mathematics is everywhere and most of what we see is a combination of different concepts. A lot of mathematics relates to other subjects like science, art, and music. Most importantly, math relates to things we do in the real world every day. Connections make mathematics easier for children to understand because they allow children to apply common rules to many different things. What parents can do:

Ask children to think about and solve problems that arise in your everyday activities. For example, ask children to help you put the groceries away. They will practice sorting—the cereal boxes and the soup cans—and experiment with relative size and shape and how the big boxes take up more room than the smaller ones.
Look for mathematics in your everyday life and don't worry about what the particular aspect of mathematics might be. Something as simple as pouring water into different sized cups and thinking about which cup will hold more is a low-key activity that actually involves estimation, measurement, and spatial sense.

Content Mathematics

Patterns and relationships: Patterns are things that repeat; relationships are things that are connected by some kind of reason. They are important because they help us understand the underlying structure of things; they help us feel confident and capable of knowing what will come next, even when we can't see it yet. Patterns and relationships are found in music, art, and clothing, as well as in other aspects of math such as counting and geometry. Understanding patterns and relationships means understanding rhythm and repetition as well as ordering from shortest to longest, smallest to largest, sorting, and categorizing.

What parents can do:

Help your children find patterns in designs and pictures, as well as in movement and in recurring events such as the days of the week or the seasons of the year.
For a hands-on activity try stringing wooden blocks or pasta necklaces into a simple red-blue-red-blue pattern. As children get older they can reproduce and create more complex patterns.

Number sense and numeration: Number sense is much more than merely counting, it involves the ability to think and work with numbers easily and to understand their uses and relationships. Number sense is about understanding the different uses for numbers (describe quantities and relationships, informational tools). Number sense is the ability to count accurately and competently, to be able to continue counting—or count on—from a specific number as well as to count backwards, to see relationships between numbers, and to be able to take a specific number apart and put it back together again. It is about counting, adding, and subtracting. Counting and becoming familiar with numbers will help your children understand all other aspects of math.

What parents can do:

Count anything and everything! Count real things to help children use their own experience with objects to better understand numbers. Therefore, one of the best math activities you can do with your children is to have them count real objects.
To help children learn to count accurately and efficiently, up and down:
- Point out that counting lets them know how many things there are in a group.
- Point to the object as you recite each number name.
- Use fingers to count. Put up a finger one at a time as you count it: fingers are tools you always have with you.
- Help your children count without skipping numbers or counting something twice.
To help children learn that numbers are used to describe quantities and relationships, encourage them to:
- Sort objects looking for similarities in either color, shape, or size.
- Sort objects looking for differences, like which box is bigger.
Talk to your children about what numbers are used for, such as keeping score in a game, or finding an apartment or street address.

Geometry and spatial sense: Geometry is the area of mathematics that involves shape, size, space, position, direction, and movement, and describes and classifies the physical world in which we live. Young children can learn about angles, shapes, and solids by looking at the physical world. Spatial sense gives children an awareness of themselves in relation to the people and objects around them.

What parents can do:

You can help your children learn geometry by helping them learn to identify different angles, shapes, and three-dimensional figures. Ask them to identify and describe different shapes, to draw them in the air with their finger, to trace over them with their fingers, and to draw them on paper.
Children learn geometry best through hands-on experiences. Things they can touch and manipulate are particularly helpful, like blocks, boxes or containers, shape sorters, and puzzles. Even cutting the sandwiches they eat into different shapes and letting them fit them together or rearrange them helps children learn about geometry.
Geometry and spatial sense help children with directions and finding their way around. Let them climb in and out of boxes, on or around furniture, going under, over, around, through, into, on top of, and out of different things to experience themselves in space.
As children get older, they play games to find "hidden" triangles, squares, or rectangles that are turned in different ways or have odd shapes. Hidden shapes are ones that do not have flat bases, but are turned in different directions.

Measurement: Measurement is finding the length, height, and weight of an object using units like inches, feet, and pounds. Time is measured using hours, seconds, and minutes. Measurement is an important way for young children to look for relationships in the real world. By practicing measurement your child will learn how big or little things are and how to figure that out.

What parents can do:

Standard measurements like inches, feet, and yards are only one way to measure. Let children pick their own unit for measurement-- "Raymond is five cereal boxes tall."
Many daily activities involve measurement: cooking, gardening, grocery shopping, sewing, and woodworking are only a few examples. Keep your child involved in these chores by talking with them about what you are doing, and by asking them to help you when possible.
Young children won't understand the concept of time. However,
- They can learn that some activities take longer than others. Compare one activity with another to figure out what takes more time. Start by asking simple questions like "Who can stand on one foot longer?"
- Set end of time limits. "You can only play for 5 more minutes, then we have to go catch the bus." At first your children won't know what the minutes mean, but gradually the children will understand the idea of time passing.
- You can relate units of time to counting by using a watch to time events and counting the ticks on the second hand of the watch by saying, "1 second, 2 seconds, 3 seconds" without timing the event. This uses counting to help children develop a sense of the passing of time.
- Start time talk with ideas like "after lunch" or "after dinner" that provide solid milestones for children. It is not until children are older that they understand more abstract notions like yesterday, today, and tomorrow.

Fractions: Fractions represent parts of a whole. A very young child will see something cut into three pieces and will believe that there is more after cutting it than before it was cut. This is typical and should not cause alarm in parents. It is one example of how children and adults think differently!

To understand fractions, children need to think about:

what the whole unit is,

how many pieces are in the unit, and

if the pieces are the same size.

What parents can do:

Many sharing activities help children understand fractions by
dividing food, chores, or treats into equal portions. Cutting up pizzas
or sharing a pan of brownies are good ways to get children thinking
about fractions.

Estimation: To estimate is to make an educated guess as
to the amount or size of something. To estimate accurately, numbers
and size have to have meaning. Very young children will not be able to
estimate accurately, because they are still learning these concepts.
They first need to understand concepts like more, less, bigger, and
smaller. When children use estimation, they learn to make appropriate
predictions, to obtain reasonable results, and they learn math
vocabulary such as "about," "more than," and "less than."

It is important for children to learn:

how to use estimation,

when the technique is appropriate, and

when the solution is reasonable.

What parents can do:

Regularly use words such as "about," "near," "approximately," "in between," "around," "more than," and "fewer than."
To practice estimation, use any daily activity like eating
or going to the store. Ask your child to make an estimate of how much,
how long, or how many. When you find the actual answer, compare it with
the estimate you originally made. This will help your child learn to
make reasonable guesses.
Estimation activities may sound complicated, but they don't
have to be. Just take a guess about something—like which one of his
friends is the tallest—and then check it out for accuracy.
To help older children learn estimation, write down the
estimate and then the actual count. If you repeat the problem or give a
similar problem, children will eventually estimate closer to the real
count. It isn't important to get the "right" answer, but to see how
close children can come. Help them practice the technique to improve
their estimation skills.

Statistics and probability: Using graphs and charts,
people organize and interpret information and see relationships.
Graphing is another way to show and see information mathematically.
Charts, including calendars, can be used to organize everyone's weekly
activities. Even older children in elementary school may find it hard
to keep track of calendars, but, when adults use them with children,
calendars can be helpful tools to learning and understanding how we
organize information.

Statistics, like batting averages in baseball, tell stories about
our world. We know which player is having the best season and which
batter is most likely to hit a home run. Probability tells the
likelihood of something occurring.

What parents can do:

Practically everything you do is "chartable." For example, you can
take the stickers from bananas, apples, or pears and place them in
columns on a piece of paper. At the end of the week you can count them
up to see how many of each type of fruit you ate. Graphs help some
children reach a greater understanding of numbers because they can see quantities displayed on paper. This may help them understand math more than looking at numbers on paper.

Use color forms or stickers to record any regular daily
activity. Put a color form next to the phone each time it rings, or
have people put one next to the front door to record the number of
times someone enters.

数学发现正常路径。

假设这个事可以解决，然后必要有哪些条件，---然后反复修正。。。。最终证明。