Teacher Feature...
Math Principles and Standards
by Jay Davidson
Facility in math is recognized by educators as being key to later success in life. The National Council of Teachers of Mathematics (NCTM) has set ten content standards for the teaching and learning of mathematics from prekindergarten through twelfth grade.
The standards in this series refer to the entire range of grades. Examples, however, are for prekindergarten to second grade, which includes the grade I teach.
The NCTM publication Principles and Standards for School Mathematics has complete explanations of these. For more information, you may visit NCTM at www.nctm.org.
Bullet points are quotations from the publication. Underneath them are my suggestions for parents.
Data Analysis and Probability Standard
The instructional programs should enable all students to:
 Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
 Select and use appropriate statistical methods to analyze data.
 Develop and evaluate inferences and predictions that are based on data.
 Understand and apply basic concepts of probability.
With most children, this is their first experience with seeing a graph, which takes information and organizes it visually. At school, we take many surveys and depict the responses of the kids: which color apple is their favorite, what their favorite animal is, how many children are in their family, how they got to school that day.
The underlying idea is that they can get useful information from the graphs they create and find. This is a necessary skill that will be useful in later years as they try to analyze more complicated ways of collecting information.
It's the adults who can lead the children to understand that they can draw some conclusions from their studies in this area.
The key vocabulary is the understanding of the concept of likelihood. If six red blocks and six blue blocks are put in a bag and you cannot see which one you are choosing, what is the likelihood that you would choose a blue one? What about choosing a red one? Now, change the contents by putting in eleven blue blocks and one red one. What is the likelihood now of choosing each color? Why has it changed?
Problem Solving Standard
The instructional programs should enable all students to:
 Build new mathematical knowledge through problem solving;
 Solve problems that arise in mathematics and in other contexts;
 Apply and adapt a variety of appropriate strategies to solve problems;
 Monitor and reflect on the process of mathematical problem solving.
The ability to solve problems mathematically will depend largely upon each child's fluency with math terminology and situations. Ask a child how many coins he will have all together if he starts with four and then gets two more from his visiting grandfather. The words "all together" should tip him off that there is going to be adding.
Similarly, when we ask how much taller one person is than another, we demonstrate that one way we arrive at the answer is by subtracting the shorter amount from the larger one.
In preparing children to understand multiplying, we have to impress upon them that each object has to be standard in a certain way. In using the number two for example, we can use legs, arms, eyes, and ears, since this is standard. How many eyes are in our family? Two times the number of people. But we have to get across the idea that not everything is standard. To find out how many pockets are in the clothing being worn by the people in our family, we realize that not everyone has the same number of pockets, so we can't use multiplication to find out the answer for this.
If the family has supported this type of activity with discussion and the appropriate vocabulary, it will be much easier for children to solve problems such as these in school.
I have noticed that younger children in the family benefit tremendously from the mathematical discussions that their older siblings and parents have in the household.
We recognize that problemsolving is not taught as a separate skill but is part of an everyday approach to working with each other.
Reasoning and Proof Standard
The instructional programs should enable all students to:
* Recognize reasoning and proof as fundamental aspects of mathematics.
* Make and investigate mathematical conjectures.
* Develop and evaluate mathematical arguments and proofs.
* Select and use various types of reasoning and methods of proof.
The most important part of this standard is for a child to be able to explain the way she is thinking about something. The role of parents and teachers is to encourage each child to an understanding that what she sees makes sense. The family is nurturing by helping children to explain their perceptions.
With younger children, this occurs most frequently when they recognize patterns and find ways to classify objects.
When children are trying to group objects by similarities, the younger they are the more likely they are to focus on only one of the attributes of the objects.
In school, we have an activity in which I ask each child to take off one shoe. Then we put them in the center of a circle so that we can find ways to put them into groups. Some of the kids are focused on the colors, some on the types of fastening devices (laces, buckles, Velcro), and some on the brand names of the shoes. Many of the kids see only one attribute.
It takes a bit of doing to get them to sort and resort the same group of items using each of the attributes individually or more than one at the same time.
Communication Standard
The instructional programs should enable all students to:
 Organize and consolidate their mathematical thinking through communication;
 Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
 Analyze and evaluate the mathematical thinking and strategies of others;
 Use the language of mathematics to express mathematics ideas precisely.
This standard highly supports the use of language to express math concepts. It is through the language that children explain their thoughts. They can do this verbally, written (with drawings or words), and by listening.
Children will be able to express themselves verbally to the degree that adults around them have shown them how to communicate in similar circumstances. We understand that language facility is a precursor to being able to read. It is just as important and necessary in learning mathematics.
Parents are in an excellent position to talk onetoone or with just a few children about the way that they came up with mathematical answers. By doing this, children can clarify their own thinking.
In the event that the family speaks another language than English at home, it would be beneficial to children to have these concepts explained in both the home language and, if parents understand it, in English as well.
Parents raised in other cultures should also be aware of an approach taken in school that may not reflect the home culture: asking a question in which the answer is known by the questioner. Since this is a common technique for teachers to find out how children are thinking mathematically, it would be useful for parents to use a similar technique in family situations.
Connections Standard
The instructional programs should enable all students to:
 Recognize and use connections among mathematical ideas;
 Understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
 Recognize and apply mathematics in contexts outside of mathematics.
It is through this standard that children will form a relationship between mathematical ideas that they have learned on their own and those that are formally taught in school. It is particularly important because it is through this standard that they build an understanding that math is useful in their lives.
When I meet readers of my columns, I frequently get comments from families about the one I wrote just before Halloween a few years ago. It was about estimating pumpkin seeds the pumpkin that the family was carving. In this activity, I suggest that family members contribute their own guesses as to the number of pumpkin seeds from the pumpkin.
After all the guesses are made, they begin counting the seeds. After a while, though, the counting stops and everyone has an opportunity to change her or his guess. Contributing a new guess several times is an example of the use of this standard, as the family members are able to refer to the number that they have counted and use that information to refine their estimate.
Parents can help by making themselves aware of the countless opportunities available to use math vocabulary and see things mathematically in family activities: by comparing objects and quantities, by seeing the shapes and patterns in their environments, and, above all, by showing how math is useful in their daily lives.
Representation Standard
The instructional programs should enable all students to:
 Create and use representations to organize, record, and communicate mathematical ideas;
 Select, apply, and translate among mathematical representations to solve problems;
 Use representations to model and interpret physical, social, and mathematical phenomena.
When a child holds up two fingers to show how old he is, he is probably showing his first representation of a number with a physical object. Moving into the early school years, children learn to represent numbers in other ways as well: by speaking, writing, gesturing, and drawing. Some use numbers that we can easily recognize because they are standard; others invent their own symbols to get their point across.
Parents can be most helpful by listening carefully to what their children say about their mathematical observations. In the event that their children have math homework, parents can encourage their children to find a way to put their thoughts onto paper as they figure out math problems. Then, after this is completed, parents ask the kids to use what they wrote to explain their thought process to the adult.
When family members are trying to figure out a mathematical situation as a group, parents explain to the older siblings the need to respect the budding math talent of the younger ones. There has to be a safe environment in which all can explain the various ways of doing the math. Children at different stages of development will find ways to solve problems that reflect their own level of understanding.
For example, a thirdgrader who has learned times tables may easily understand and explain that 6 X 5 = 30. At the same time a first grader may show her understanding by writing 5 + 5 + 5 + 5 + 5 + 5 and then counting, "5, 10, 15, 20, 25, 30" and come up with the same answer, or by adding pairs of the fives into three separate tens, then counting, "10, 20, 30." Parents would do well to respect that there are many ways  not just one "right" way  to express oneself mathematically and come up with answers to questions in these situations.
