The National Council of Teachers of Mathematics created Principles and Standards of School Mathematics to serve as a coherent guide for mathematics teachers around the country. The publication contains general principles that should govern effect math programs as well as more specific content and process standards. The vision of the Principles and Standards is to support the development of flexible users of mathematics who can solve problems, use reasoning skills, communicate about mathematics, connect it to their everyday lives, understand how to use technology, and so much more. By grounding their instruction in the Principles and Standards, math teachers should aid in the development of students who are able to do all of these things. Teachers of mathematics need to be committed to the program and must be eager and willing to change their methods of instruction to better support their students fulfill the vision of the NCTM.

At the K-2 level, students and teachers should be able to have numerous opportunities to incorporate measurement into the classroom. The concepts of measurement cover a wide horizon of skills and uses numbers to cover areas such as geometry and number recognition for everyday life skills. When students are at this age, they tend to have a difficult time putting things in perspectives. Recognizing that one object is bigger than another or that something is long, wide and deep helps the child to start making connections to a more realistic size comparison. When making measurements in a classroom it is also important that the teacher provides a wide variety of materials to measure items with throughout the class.

The most important and effective way to help a child understand measurement is to have hands-on activities that help develop these skills. Actually touching, looking at, comparing, and manipulating objects gives the child a real sense of how big or small something is and it’s dimensions. Learning about these attributes helps the child to compare and relate these objects to everyday life. At this level students should be recognizing size by height, length, width, time, and area. However, it is also important that the teacher provides materials other than rulers to make measurements. Applying daily used objects as measuring tools is a great way for students to make connections as well. There should also be objects in or around the room to be measured that are larger than the measurement tool itself. For example if the students are to measure their classroom chalkboard and doorways, they should try to apply the ruler to see how many times they have to move it and add up the continuing measurements.

One of the most difficult steps is to make sure children can explain the outcome of their actions when making measurements. This step allows the teacher to incorporate some assessment and room to plan for the next level. Although math is seen primarily through numbers, there are great ways to include good conversation into math and develop a mathematical vocabulary. Teachers should encourage the students to estimate and predict what may happen when measuring and help figure out any misunderstanding or confusion. Probing questions are good to allow students to make inquiries and think about things themselves instead of being fed new ideas and concepts. It’s important that the teacher directs the focus to the students when asking what tool of measurement would be the most appropriate and why when measuring so that there is a clear reason for making that choice. A teacher might want to ask what would happen to a measurement that was recorded if the tool that was used to measure the object was half the size. These types of questions allow the students to perceive things in a different light.

Teachers should also be sure to include weighing objects at this level with a balance to include comparison to other objects and to all them to discover equality of weight. This helps students to apply numerical value to an object. Along with measurement, students should start to recognize the concept of time. Using the classroom schedule is a great way to make reference to their day and relate periods of time for a particular activity.

The most important key element at this level of understanding is to allow the children as much hands-on activity as possible. Measurement is an element of mathematical development that can very easily be used in a direct hands-on manner. The students can physically manipulate objects and use a variety of tools to gain a better understanding. This is very important when considering how to teach this concept of math. Today, the classroom should be filled with a variety of activities for students of all levels of understanding. When applying these concepts it’s a matter of relating to everyday life experiences that allow the students to feel as though these mathematical ideas will be important. Fun and interesting things can be done when using measurement in a classroom, and these activities steer away from the basic paper and pencil methods that can often lead to boredom and lack of interest for students.

The NCTM has three major goals regarding connections for all of the grade levels. Students should: "recognize and use connections among mathematical ideas, understand how mathematical ideas interconnect and build on one another to produce a coherent whole, and recognize and apply mathematics in contexts outside of mathematics" (NCTM, p. 131). The connection standard for grades K-2 focuses on the importance of connecting mathematical ideas with previously learned knowledge, and connecting math learned in the classroom to the outside world. Most importantly, students should be able to connect mathematical concepts with their informal experiences, thereby truly experiencing mathematics and building on their personal experiences. Understanding and displaying the ability to connect and apply mathematical concepts to the real world helps students learn and enjoy mathematics. Making connections is absolutely essential to mathematical understanding.

This standard is particularly important to the current trend in teaching and learning mathematics. The previous, more traditional approaches focused on rote memorization. Connections allow for a much more meaningful and more effective method of teaching mathematics. By linking school-taught math to the outside world, teachers provide students with a variety of tasks and problems not found on the commonly used worksheet. Teachers also help students think mathematically by enabling them to see how mathematics occurs in different real-life situations. Teachers must constantly help students see connections, and they need to talk with their students to assess connections that students have already made to help design further instruction.

In grades K-2, children are still very young and may not have many previous formal experiences to which they can link mathematics. However, they make connections to informal, early experiences. Simple connections can occur, for example linking concrete objects to mathematical ideas, such as counting (i.e. physically counting objects and understanding that the final number counted represents the total number of items in a set). Activities that require the use of concrete manipulatives can be extremely beneficial for student learning if teachers make sure that students are understanding what they do and can relate it to the particular concept. If used correctly, manipulatives help students to connect and link mathematical concepts to their own worlds.

Students should also be able to recognize and connect mathematical ideas. Students will not always automatically understand connections among mathematical concepts, so it is important for teachers to focus on connecting new ideas with other concepts and skills, prior experiences, and the real world. To help elementary students make sense of what they learn, mathematics should be integrated with other subjects, such as science, language arts, and physical education. So much of what children do in other subject areas involves mathematics, and teachers can help make mathematics more real for their students by showing them how mathematics relates to their everyday lives. It is necessary for students to make connections when learning new material for true understanding to occur.

The Illinois State Standards have many apparent differences from the NCTM Standards. Once both standards are studied it becomes apparent that the Illinois State Standards are much more specific than the NCTM standards. The Illinois State Standards include currency and temperature as important units of measurement to be taught in the elementary level. Conversely, the NCTM Standards do not mention either units of measurement within their expectations. Comparisons of measurements are significant in both the Illinois State Standards and the NCTM Standards. However, the state standards involve comparing an estimated measure to an actual measure that involves tangible measuring tools, whereas the NCTM Standards place more emphasis on comparing amounts of specific objects without the use of tools first. For example, in Principles and Standards for School Mathematics it mentions that students "should begin to develop an understanding of attributes by looking at, touching, or directly comparing objects" (103). Therefore, the NCTM Standards place more significance on comparing "sizes of piles of objects" by the way they look and how heavy they feel. On the other hand, the Illinois State Standards seem to place more importance on measuring with tools, even when estimating is used.

Some of the similarities between the Illinois State Standards and the NCTM Standards also seem to have differences at some point. For instance, the NCTM Standards and the Illinois State Standards place emphasis on using nonstandard units of measurement before standard units of measurements. However, the state standards use geoboards, and tiles as nonstandard units, while the national standards mention using paperclips or shoes as nonstandard units. It is readily apparent that although both standards place importance on using nonstandard units of measurement, the national standards consider objects that are readily available such as shoes, pencils, and paperclips as nonstandard units (103 & 105). In contrast, the examples of nonstandard units mentioned in the state standards are not as readily available as the national standard’s examples of nonstandard units. One of the very obvious differences between the two standards are that the Illinois State Standards have an explanation of why each goal, such as measurement, is important to know in real life situations, while the NCTM Standards does not.

Overall, it is apparent that the Illinois State Standards and NCTM Standards have more differences than similarities. However, the differences between the two sets of standards do not make one set of standards more useful than the other. Although both sets of standards are different in their expectations as to what the children should learn in the elementary (K-2) level, both sets of standards are very effective guides for teachers. They allow teachers to know what needs to be taught on a specific subject, such as measurement. For example by looking at both sets of standards for measurement at the early elementary level, a teacher can see that he or she will have to include estimation, comparisons, experiments, etc. into a unit on measurement. As a result all students statewide and nationally are basically learning the same material; therefore, no one child or group of children will be left behind from the rest of the students in the state or nation.

The concept of making connections is central to the NCTM standards and the Illinois Learning Standards. In order for mathematics instruction to be effective, students must be able to connect what they learn to their prior knowledge and experiences, other concepts within mathematics, and situations in their everyday lives. The NCTM standards devote an entire section to connections, providing rationale for the necessity of making connections and highlighting examples of specific instructional practices that focus on connections. The Illinois standards address the importance of making connections in the section on the introduction to the mathematics standards and also briefly for each of the five goals.

Although both sets of standards address the importance of making connections to facilitate understanding, the NCTM standards place much more emphasis on the idea. They go into much more detail explaining what kinds of connections students should make, the role that the teacher should play in helping students understand, and what some of these connections should actually look like. In addition, they talk about connections as they relate specifically to early elementary students. Younger students will tend to connect what they learn to their early informal experiences with mathematics, while older students continue to make connections between prior mathematical instruction and new material as they build on their formal conceptions of mathematics and see relationships between the various mathematical fields.

The Illinois standards for mathematics are much more specific. The introduction addresses such process goals as solving problems, communicating, using technology, working on teams, and making connections, but it includes only a few sentences about each of these areas. The main focus of the Illinois standards is on the five main concept areas: number sense, estimation and measurement, algebra and analytical methods, geometry, and data analysis and probability. For each of these five goals, there is a heading entitled "Why This Goal Is Important." In each of these sections, there is information about how the goal relates to the lives of the students and, therefore, how they could be helping to facilitate connections. These sections, however, are not nearly as detailed as in the NCTM standards, and the information that they include is not specific to any particular grade.

Both sets of standards address the importance of students needing to connect what they are learning to their everyday lives. The specificity of the NCTM standards in this area really serves to highlight the importance of connections in mathematics instruction much more effectively. By creating a standard devoted solely to connections and by including so many age-appropriate examples, the NCTM standards present a more practical guide to teachers for facilitating connections. The Illinois standards, although they mention making connections, do not talk enough about how or why these connections can and should be made. They should definitely be more practical in order to assist teachers in helping their students develop a meaningful understanding of mathematics.

The University of Chicago School Mathematics Project, Everyday Mathematics, is the current math program at Yankee Ridge Elementary School. The program is accompanied by an elaborate teacher’s resource package which includes a lesson guide, an assessment handbook, a minute math book, a learning goals poster, various games, home links, a resource book, math masters worksheets, and individual children’s math journals. It also provides activities for students to take home and work on with their parents to practice the skills they learn at school. Another asset is that it gives the teacher an idea of what the children have learned in past grades and where they need to be as they advance to the next grade. Within this program, there are two stages the first graders take part in. The entire class progresses simultaneously throughout the program. Everyday Mathematics has specific learning goals for each chapter, which seem to be quite similar to the NCTM Principles and Standards. This allows individual classrooms that follow this program to reach the same goals. Therefore, when students progress to the next grade, they should ultimately be at similar levels.

The measurement chapter of the text begins with various goals for the children to meet and the lessons which accompany each goal.
Goal 1: Use standard units for measuring length
One lesson that helps the children reach this goal is "Personal Foot and Standard Foot." Children in groups of two use their feet to measure commonly used objects in their classroom. Because each child’s foot is a different size, the partners will come up with different estimations. This will lead to using the standard U.S. foot to measure the same objects to get the same measurements between partners. This activity emphasizes the need for using standard units. This measurement skill is one which the students will need in order to take part in everyday activities such as cutting material for sewing, measuring wood for building, or measuring a piece of paper for an art project.
Goal 2: Find simple sums and missing addends
An activity that accompanies this goal is "Reading a Thermometer." Each morning during calendar time, the children read the thermometer placed outside the window of the classroom (if possible). This is obviously an everyday skill which the children will need in order to choose what they are going to wear each day, measure their body temperature when they are ill, or see if a roast is done in the oven. In addition to being able to read the actual number temperature, the children are taught how to relate the number to a meaning, such as hot/cold.
Goal 3: Tell time to the nearest half hour
"Telling Time to the Quarter Hour," emphasizes estimation of time. This connects to everyday life in that adults will not say, "It is 6:27." They instead say, "It is 6:30." Children need to learn that this type of estimation is acceptable in society. When introducing quarter hours, the program relates the 15 minute periods of time to quarters of a pizza. This way, the children are not just hearing numbers, they understand increments of time and make connections to what they already know. Also, the teacher should draw attention to a 15-minute time span so the children are aware of how long 15 minutes actually is. To connect their study of time to real life situations, the teacher and students should remind classmates when a day to day activity is going to occur. For example, a child could remind the class that there are 15 minutes left of center time before they need to leave for music.

As a whole, the measurement unit of this program is very age appropriate. The children deal with items and situations they relate to and understand. They are able to see their math lessons when they go shopping with parents, cook at home, go to the bank, get measured at the doctor’s office, or visit their parents’ work. The parents are also aware of what their children are studying and can therefore support and give examples at various times spent with the child.

Everyday Mathematics is very demanding in that it requires many additional materials and resources for the teacher, students, and parents to provide. This can be a negative aspect because many districts are currently experiencing financial hardships. Even though math is important, the money is not always there to support it as many teachers would like or need. However, there are ways to get around this. Parents and teachers can get together and make manipulatives from everyday items, or children can make their own materials in art class, at home, or during free choice time. If teachers look hard enough and ask for help in the community, many times adults who do not even have children in the school district are more than willing to donate items. One positive aspect of the program is that once it is purchased, all of the necessary resources in order to teach the program are included. Only the extra items to enhance the lessons are not included.

Overall, we felt that the chapter on measurement succeeding in fulfilling the NCTM Learning Standards. For example, the children are able to understand the process of measurement, systems of measurement, and appropriate techniques, tools, and formulas to determine measurements. In the lessons we reviewed, it was apparent that these goals were met. In addition, the entire program successfully conveys the message to children that measurement is used in their everyday life. They are able to leave the classroom and apply the skills they have learned at home. We feel that we would be able to effectively teach children the skills and the importance of mathematics in their everyday lives, while meeting the NCTM Standards using Everyday Mathematics.

worksheet

book, How Big is a Foot?

children’s shoes

pencils

various objects to measure

Students should be able to understand standard units of measure. Students should also be practicing cooperative learning with student groups.

Split students up into groups of four or five throughout the classroom. Before the lesson begins the teacher reads aloud the story, How Big is a Foot. Each child in the group should then take off one shoe for measurement purposes. Various objects such as windowsills, doorways, table tops or desks are good objects to consider for the children to measure. Designate a particular object to each group and have them each measure it with the length of their shoe. Students will then record their data on the worksheet provided. Once the children finish measuring and recording data, they will finish the questions on the worksheet and come together as a group to discuss their findings.

The provided worksheet and discussion time act as an assessment tool for this activity. Leading questions for discussion can start with the answers on the worksheet and continue with questions or comments made by the children.

Rubric:

All recordings and measurement are correct………………………………..3 2 1

All questions on worksheet are correct…………………………………… 3 2 1

Little or no confusion presented to students………………………………..3 2 1

Group clearly worked well together and everyone participated……………..3 2 1

TOTAL: /12