INTRODUCTION:

What does it mean to do mathematics? When I was a child in public school, I learned the traditional way of doing math. This was a teacher-centered classroom in which I learned to memorize formulas and rules without really understanding the role of math in my everyday life. Today, teaching math is no longer just about math facts and memorizing those facts. The National Council of Teachers of Mathematics (NCTM) state that "children must from the earliest grades and throughout their school experiences, be made to feel the importance of personal success in solving problems, figuring things out, and making sense of mathematics" (Van de Walle, p. 5). They cannot do this by just learning rules, instead students need to become problem solvers and logical thinkers who reason and prove, and apply math in the real world. Today's math is child-centered and promotes the use of technology, problem solving and reflective inquiry.

For the past several months I have been observing a split grade 1-2 classroom at a school in Canada. In this paper I will discuss my observations of a math lesson on measurement directed for the grade one level. I will begin my paper by describing the lesson to give you an idea of what I observed. The rest of my paper will focus on how the teacher's lesson met (or did not meet) the components of conceptual development, reflective inquiry, connections, and use of technology.

GRADE LEVEL: Grade 1

TOPIC: Measuring

Measuring length is a fairly new topic for the students in grade 1. Students have learned to compare two or more lengths when working on worksheets about "long, longer, longest"; "short, shorter, shortest"; and "longer, shorter, same".

EDUCATIONAL OBJECTIVES:

The measurement strand of mathematics has not received much emphasis in traditional curriculum, yet measuring things allows students to associate numbers with real quantities. A good place to start in the primary grades is by measuring length, where ãthe measure of an attribute is a count of how many units are needed to fill, cover, or match the attribute of the object being measured.ä (Van de Walle, p. 278). According to Van de Walle, there are three steps to measuring something, (1) decide on what attribute is being measured (length); (2) select a unit that has that attribute (measuring tape); and (3) compare the units by filling, covering or matching with the attribute of the object being measured. By measuring the length of a room, this lesson should introduce grade one students to what it means to measure, units of measure, selection and use of measurement instruments, systems of measurement, and formulas (addition).

KEY IDEAS:

Measurement

Length

STANDARDS:

The objectives and key ideas of the observed lesson did correspond to Ontario Mathematics curriculum and the Measurement strand. Students were be able to:

-demonstrate an understanding of and ability to apply measurement terms (length)

-demonstrate that a non-standard unit is used repeatedly to measure

-measure and record the linear dimensions of objects using non-standard units

-use math language to describe dimensions

-solve problems related to their day-to-day environment using concrete experience of measurement

-represent the results of measurement activities using concrete materials and drawings

MATERIALS:

Measuring Tape

Paper

Pencil

Worksheet

PROCEDURE:

1. The teacher selected only the top five math students from grade one to participate in the math activity.

2. The teacher paired four of the students into pairs and left one student to work on her own.

3. She handed out worksheets to the students and told one group to measure around the inside of the gym, another group to measure a hallway from end to end, and the individual student to measure around the inside of the kitchen. I was to supervise and assist them.

4. Students measured their designated areas in the school with a measuring tape and I guided them as they asked questions. For instance, I told them to keep count of how many measuring tapes it took to reach the end, and not to focus on the actual numbers on the tape. I also suggested drawing a picture of the area being measured.

5. After completing the activity, students worked on worksheets by describing how many measuring tapes they needed in total. To do this they had to add the number of times the measuring tape length was used.

6. The teacher assessed what the students learned by checking their completed worksheets.

CONCEPTUAL DEVELOPMENT:

As part of the NCTM Principles and Standards for School Mathematics (2000), students are to learn math with understanding, that is to have math skills and the ability to think and reason math to solve problems. Mathematics educators distinguish between two types of mathematical knowledge. The first being conceptual knowledge when children understand math relationship to other ideas. The second is the operational knowledge of math rules and symbols (Van de Walle, p. 31). In observing the measurement lesson I am not quite certain if the students involved really fully understood the task conceptually, but were more focused on the procedure and operations involved. This is what I will discuss in this section.

Students can learn math with understanding but they need an environment that stresses problem solving, reasoning and math instruction that is hands on and active! Although this is usually a child-centered classroom, when it comes to math instruction students tend to work individually on math worksheets. I think the teacher does this as part of a classroom management strategy as she has a mixed grade with different levels of learning abilities and behavioral needs. So in this lesson I think the teacher only allowed five students to participate in the measurement activity because of this reason. However, all students in the math community should have had the opportunity to develop their math ideas by learning and doing math actively instead of working individually on a worksheet.

To increase student's relational understanding student interaction should be frequent by placing students into small cooperative groups or pairs. During this lesson constructivist learning was happening for the five children involved because they were active participants in the development of their own understanding, and the teacher did not transmit her ideas to passive learners. However, I found that the individual student working in the kitchen should have been grouped with a partner or one of the other groups because she had trouble deciding on what to do and with holding down the measuring tape. One thing I noticed was that as children constructed their own knowledge, different learners were sharing different ideas to give meaning to the same new idea. For instance, the group in the gym measured along the floor but when they were faced with an obstacle they offered suggestions to each other on how to continue measuring and decided to measure around it. Grouping students in this way during math instruction helps students to make connections with others ideas and concepts, enhances their memory, helps them with learning new concepts and procedures, improves their problem-solving abilities, helps kids to invent new ideas of their own, improves students' self-confidence to learn and to do math, and is fun! (Van de Walle, p. 5).

In this lesson five students measured length using an iteration process by measuring with a single measuring tape by moving it from position to position and keeping track of which areas the tape covered, but this seemed very difficult for children this young. For example, it took them a long time to figure out how to get started and after I helped guide them they still could not seem to get the hang of how to use the measuring tape. Also as the first graders measured the length of the room by laying 1 meter measuring tape from end to end sometimes it would not be at the right marker or the tape was not pulled tight. So did the students really understand the concept of length as an attribute of the room? Did they understand that each strip of 1 meter has the attribute of length? I think what they probably learned to understand was that they were supposed to be making a line of measuring tapes stretching from wall to wall. Thus they were performing a procedure instrumentally without a conceptual basis.

To better develop conceptual knowledge during this measurement activity the teacher could of had the students compare objects on the basis of some measurable attribute such as placing one length directly in line with another. For example, students could have made their own rulers out of strips of paper, measured length once with their strip of paper, and then a second time with actual unit models such as the measuring tapes. Students could have also used physical models of measuring units using informal units such as their own feet or straws compared to the formal units of the 1-meter tape. Comparing measurement with formal unit models and informal instruments could have helped the children understand two ways of getting the same results. Informal units such as these when used to measure length are beneficial because they make it easier to focus directly on the length being measured; the size of the numbers are kept reasonable; it avoids conflicting objectives (learning to measure is different from learning about the standard units to be measured); provides a good rationale for standard units; and they are fun! (Van de Walle, p. 279-280). To extend the topic, as students develop more precision children should use the measuring strips they made with subunits as fractional parts. After the task, the teacher should have discussed with the students to help them understand that the unit used was important as the attribute being measured, and smaller units produce larger numeric measures and vice versa.

Another observation I had about the lesson was that it was lacking estimation as part of the measurement process to help children develop an understanding of relative measures. An idea would be to estimate and measure the same area with different sized units, for example the length of their bodies, string or a strip of paper. Furthermore each group could estimate several things in succession using the same unit, such as each group measuring the gym, hallway and kitchen and then comparing their predictions and results. If different groups of students measured the same distance and got different results, then they could have discussed the differences. This would teach why units such as the measuring tape need to be stretched to their full length and end to end for precise measurement calculations.

REFLECTIVE INQUIRY:

Reflective thinking is an important ingredient for effective math learning because students need to reason to help them in solving problems and decide if and why their answers make sense. The new reform suggests that students need to be challenged to make sense of new math ideas through explorations and projects, preferably in real contexts to help them become logical thinkers. In this classroom environment the teacher provides a learning environment for doing math and students are respected for their ideas, however I do not know if all students in the class feel comfortable with taking risks. For example, in the measurement lesson, the students involved were actively figuring things out, testing ideas and making conjectures, but did the rest of the class feel less confident about their ability to do math because they were not chosen to participate?

Effective math teaching is a child-centered activity where the teacher should engage all students by posing good problems and creating a class atmosphere of exploration and sense making. In this activity the teacher only selected five students out of a class of 13 grade ones and 12 grade twos. I think all students could have benefited from this lesson. According to the NCTM, in a math classroom all students should learn to value math, become confident in their ability to do math, become mathematical problem solvers, learn to communicate math, and learn to reason mathematically (Van de Walle, p. 4). All students in the class should have an equal opportunity to learn the measurement concept. Perhaps the teacher needs to set higher expectations for all students instead of only a few to make sure that all students feel they are capable of making sense of math.

Before the math task, the teacher got the five students mentally ready to work on the problem by giving them a worksheet and telling them their instructions. She also checked to make sure they understood the task before going out on their own. During the activity students communicated mathematically with each other as they worked together to hold the measuring tape and record their results. I was there to provide hints and suggestions such as drawing a picture of the room they were measuring. I also encouraged each group to write their findings and explanations. I listened to find out how different children were thinking, what ideas they were using, and how they approached the problem. I found that all students had difficulty with developing reasons and offering explanations, and they did not ask reflective questions but instead they wanted to be told how to do the activity like learned helplessness. I tried to encourage the students to figure it out on their own but looking back I should have asked more open-ended questions. For instance, when students asked if they were correct, I should have asked them "Do you think that might be right?" instead of jumping in too quickly to help them with the answer.

Communication is also important so that students should talk about, write about, describe and explain to help them reflect on their math ideas. To do this, students need to be active in a verbal environment to increase their social interaction and math exploration. Although the partners working together shared and discussed their ideas, after the task there was no engagement of discussion among the groups or with the teacher. I think written and oral communication should have been encouraged and supported during this activity, as students should be encouraged to compare and talk about their findings by listening to each other and deciding which solutions make the most sense and why. The teacher should have also encouraged reflection and extensions to this new math concept by giving feedback to help the students establish goals and become independent learners, and then conclude the lesson by summarizing the main points to be learned. After discussion and reflection then the teacher should supplement her math instruction with the math worksheets.

As with conceptual understanding, estimation should have been used in this lesson to help the children to think or reflect on what number might tell how long the hallway is. The teacher should have asked the students to estimate how many times or numbers of measuring tape they expected to count to help them learn what is meant by ãaboutä. For example, the teacher could of asked the students for a comparison estimate such as ãIs the length of the hallway more, less, or about the same as 15 tapes?ä Approximate language such as ãaboutä and ãless thanä can help develop the idea that all measurement includes some error, as well as about the notion of precision when measuring. Moreover, just measuring and recording data is not very effective because there is no reason for students to be interested in or think about the result. Instead to begin teaching the math concept of measurement so that students think reflectively, informal measuring units could have been introduced first, followed by estimating whether or not the formal units of the measuring tape would be the same or different as the informal instrument.

CONNECTIONS:

One of the five process standards of the NCTM is connections made within and among math ideas, to the real world, and to other disciplines (Van de Walle, p. 8). With respect to the curriculum, students must be helped to see that math ideas are linked to one another. I think this lesson did make math connections to other strands of the curriculum such as numbers, geometry and statistics. For instance students drew the shapes of the area being measured (geometry), and they used number sense and numeration to write down the numbers on paper as well as use their addition skills.

Secondly, measurement is all around us in our everyday lives, in that almost every job in today's society uses math in some way. For instance, in this lesson I pointed out to the students that carpenters, interior designers, and architects might need to know how to measure the length of a room in a school. Thirdly, this lesson can be connected to other disciplines such as social studies and the history of measuring things, and language arts in being able to write to explain their results.

TECHNOLOGY:

Technology such as computers, manipulatives and calculators are essential tools that enhance math learning. For example, using technology such as models in the classroom can help children develop new concepts or relationships; help children make connections between concepts and symbols; can be used to assess children's understanding, and it extends the range of problems that students can explore their ideas in different ways (Van de Walle, p. 35). Moreover, the NCTM fifth process standard of representation states that students should be able to physically represent their findings to communicate ideas in visual form such as symbols, graphs, charts and diagrams (Van de Walle, p. 8).

In this classroom, technology is available for mathematical demonstrations but the teacher never uses it to engage students' interests and intellect. For instance, there is an availability of manipulative materials such as pattern blocks, square tiles, and Cuisenaire rods in the classroom but they are only used for play. There are also three computers in the classroom with Internet connection and software such as Kid Pix, Excel, Microsoft Word and Geometer's Sketchpad, but again they are only used for play if students are done their work early. Perhaps in this instance the teacher could have had the five students use geometric software to manipulate and investigate relationships. For example, the students could have used graphing technology to draw a floor plan of the measured perimeter on the computer, explore algebraic concepts, and organize data using drawings or charts. As per the Ontario standards Number Sense and Numeration strand, calculators should have been used to explore counting and to solve problems beyond the required pencil-and-paper skills like adding the units of measurement in this lesson. Sadly, it appears that worksheets are the only technology really used in this classroom, but I think worksheets should only be used to supplement instruction not be the basis of math learning. Although the teacher knows that math instruction and learning should be enhanced with technology such as manipulatives, her reason for its lack of use in her math environment is because she does not have time to use them. However, the days that I am in the room I think she takes advantage of an extra pair of hands and has students work with me in pairs using manipulatives. For example, one day I worked with two students at a time to learn about probability using spinners that each student made. Another day I worked with pairs again with a toy clock to help teach the concept of time.

CONCLUSION:

Length is usually the first attribute students learn to measure, but children in primary grades do not always immediately understand it. With respect to the lesson on measurement, children should begin learning about measurement with informal units and over time be introduced to standard units and standard measuring tools. First-grade students need a lot of experience with a variety of informal units of length. Also, estimation should be included in almost all measurement activities because it helps students focus on the attribute being measured and the measuring process, provides intrinsic motivation to measurement activities, and when standard units are used, it helps develop familiarity with the unit.

"Purposeful mental engagement or reflective thought about the ideas we want students to develop is the single most important key to effective teaching. Without actively thinking about the important concepts of the lesson, learning will not happen" (Van de Walle, p. 36) Was this effective teaching demonstrated in this lesson? I think not because only a few students were allowed to participate, procedural knowledge seemed to be the emphasis, there was no chance for discussion and reflection, and math learning was not connected or supplemented with technology. What I did learn from observing this lesson is that as a future teacher my classroom should be a constructivist one in which all students are actively involved in math to construct their own meaning through mentally engaging experiences. So I will try to create a mathematical environment where I give students worthwhile math tasks that allow them to inquire why things are and make connections to other math ideas and the real world. I will use cooperative learning groups to promote social interaction and enhance math learning, and my students will use technology such as manipulatives, models, calculators and computers as thinking tools to supplement their learning. I will require all students to explain their answers or findings by encouraging my students to discuss orally and write their ideas on paper as part of their reflection, and I will listen actively to what students area saying to assess what they are learning and what level they are at to plan my instruction. I learned a lot by observing and participating in this lesson, but most importantly I learned to reflect on my own lessons to assess and evaluate whether my instruction is effective in teaching math effectively to all students in order for them to grow up to be productive and successful citizens in a rapidly changing and competitive world.

 References

Van de Walle, J.A. (2001). Elementary and Middle School Mathematics (fourth edition). New York: Addison Wesley Longman.