Mathematics Lesson Observation: Representing and Writing with Mixed Numbers
Grade: Two
Topic: Representing and Writing
Mixed Numbers (part 1)
(Introductory Lesson}
Introduction:
After spending the last few months observing a grade two class, I am delighted and surprised to known how much the math program has changed since I was a child. Now, most lessons are hands]on and student focused compared to the previous teacher focused lessons. Those unproductive teacher focused lessons tended to dominate many elementary classrooms a few years ago. Children now have opportunities to use technology, computers, manipulatives, drawings, and other devices that make the learning process much easier. As a new teacher about to begin teaching, it is reassuring that all these resources are available to help convey mathematical concepts.
Throughout this document I will discuss the components of a grade two math lesson that I had the opportunity to observe. Some areas .of interest that will be covered are concepts and skills, questions posed, reflective inquiry, and use of technology, along with many other details.
Objectives, Concepts, Skills, and Standards:
The concepts and skills primarily addressed in this lesson pertain to representing and writing mixed numbers such as fractions. According to the New York State Education Standards, "students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas." I believe it is important for children to develop these skills early because it is the foundation for further mathematical advancement in the classroom and in the real world.
Prior to the main focus of the lesson the children participate in a small activity that reviews calendar date, clock time, temperature, counting money, numerical patterns, and counting by different numerical intervals. These small activities are done every day so the children become very accustom to the routine and are extremely accurate when responding.
This lesson was taking directly from the Saxon Math Publishers, and satisfies the New York State Educational Standards. The specific standards that are evident in this lesson are:
Elementary Standard 3 ] Mathematics
Key Ideas:
Materials and Procedure:
The entire lesson is included with this document and lists the procedure as well as the materials. The teacher that I was observing followed this lesson very closely.
Conceptual Development:
When learning mathematics it is important to acknowledge the conceptual knowledge as well as the operational knowledge. Only after repetitive application and exposure to the same math concepts are children able to apply proper functioning operational knowledge. It is unreasonable to expect children to grasp a complete understanding of a topic during the first lesson. That is why there are usually several consecutive lessons pertaining to the same concept. This allows the children to apply their knowledge until they have mastered a firm operational knowledge of the subject.
The lesson that I observed was the students' first exposure to mixed numbers. Previously, the students had learned about fractions and how they are represented, so they had some operational knowledge about when it is appropriate to use a fraction. Yet, they were somewhat confused about the whole number being mixed in with a fraction. It was obvious that the students were not going to grasp this idea from a verbal explanation, so the teacher handed out pattern blocks to each child.
In each separate bag of manipulatives there were pieces that represented one
whole (yellow), one half (red), one third (blue), and one sixth (green). All the numerically smaller pieces were created so that when they were assembled, they would be the same size as the "one whole" piece. For example, when the children put six one]sixth pieces together it would be the same size as the "one whole" piece. After the children were able to handle the pieces for a few seconds, the teacher began the lesson.
She started by quizzing the students about the fractional name for each pattern block. As she called out the fraction, the children were expected to hold up the corresponding piece. Then the children were required to cover the three different whole pieces with the one]half, one]third, and one]sixth pieces so they would have a visual representation of how a whole piece can be created using smaller pieces. This was completed fairly quickly by the class, however it is difficult to tell whether they understood why this worked, or how it worked. The teacher was not overly concerned about giving explanations either. She kept the lesson moving quickly, and only wanted to see what the lesson plan called for.
As the lesson continued, some children quickly observed the relationship between pattern blocks and mixed numbers, while other students were quite content forming patterns and stacks of blocks. It was difficult to keep the entire class on track and expect them to learn in the same manner. On one occasion a child had discovered that three one-sixth pieces could also be considered one]half, and when he applied that to a question that required one]half the teacher reprimanded him and forced him to use the one]half piece. This was disappointing because the teacher was not accepting of a diverse way of thinking. I was upset, but understanding because I know that she was just trying to keep everyone at the same pace.
Perhaps it would have been a good idea to separate the children into groups or pairs to perform this task so that the stronger students could have helped the weaker ones. I also believe that having the fraction written on the block might have helped the children understand more clearly, and given them the basis for developing a conceptual understanding of mixed numbers. It is difficult to say what might have helped this situation because of the diversity of students in the room. Some students have learning disabilities, some have attention deficit disorders, while other children are constantly coming and going from the room to receive special needs assistance. Thus, it is extremely difficult to keep everyone moving at the same pace and to know if they really understand. Not to mention the fact that manipulatives make it hard to keep a student's attention. It is just another thing to play with., when they are suppose to be paying attention.
Reflective Inquiry:
Afterobserving this lesson it is difficult to say how many children where able to actually reflect on the process of problem solving with the manipulatives. Due to the teaching style of the teacher, the children were restricted to text book answers and were not encouraged to "think outside the box." According to Van De Walle, "The most widespread misuse of manipulative materials occurs when the teacher tells students, 'Do as I do. "(31) In this lesson the teacher did not actually say this, but she still did not let the students experiment and find new answers to the problem. Thus, restricting their problem solving skills and emphasizing the importance of just getting the correct answers. "A natural result of overly directing the use of models is that children begin to use them as answer]getting devices rather than as thinker toys."(3 1) Van Dc Walle is in favor of letting the children play with the manipulatives in order to expand their reflective thought process. This active learning process is the only way children will experience growth and comprehend mathematical concepts.
Apart from the fact that the children were limited to minimal exploration with the pattern blocks, the teacher did make sure that the lesson was student focused. Each child was given the opportunity to use the pattern blocks and was expected to participate during the exercises. The children were also encouraged to ask questions because there were three adults in the room and it was easy to quickly resolve the problem and move on. Most of the children's questions pertained to correct usuage of the blocks, and confirmation of the mixed number they were required to show. The students get excited when they are able to use manipulatives, so they are easily distracted and require constant guidance. During this time the students also conferred with each other and discussed mathematical concepts related to the lesson. Most of the time they were not even aware that they were discussing math related issues. I believe this is due to their young age and their immature conceptual development. As time passes and they are exposed to this concept again, they will be more aware of the math connection in their dialogue.
As the hands-on aspect of this lesson came to an end, the teacher passed out several hand]outs for the children to complete. The first was a written assignment that was directly related to the exercises done with the pattern blocks. I believe this first paper was to assess whether the children understood the concept of mixed numbers. The next two papers were a review of the continuing topics that are reviewed everyday before a new topic is introduced. The students were expected to work on these papers and then finish them for homework. In the morning they hand the work in to be corrected by the teacher. I am not sure that the students are getting the full benefit of homework, Wit is always handed in for marking. I think it might be a good idea to take homework up in class periodically. By doing this the children will understand why they got questions wrong and will be able to correct them properly. This also establishes a better communication link between the teacher and the students. It allows the students to ask more questions and gives the teacher a better indication of which students need more help, and if the technique used while teaching the lesson was effective.
Connections:
In order for children to fully comprehend any concept in education, a connection between what is being taught and the real world must be established. Personally, I feel this is extremely important in the subjects of mathematics and technology. As an elementary and high school student I was constantly questioning the reason why we had to learn certain concepts. If a connection between the real world and the concept I was learning was explained, I was more incline to understand. I also put more effort in to comprehending the concept.
With regards to the lesson I observed, I feel as though there was a lack of realworld connections made during this lesson. The teacher was more concerned with following the Saxon lesson plan, than with incorporating examples from the world around us. It may have been easier for the students to develop a conceptual understanding of mixed numbers had she given them an example using pizza or cookies.
Maybe next time the teacher will be better prepared and have props to assist her as she explains mixed numbers to the students.
Technology:
During this lesson technology was not used. The only non]traditional aparatice used in the lesson were the pattern blocks. These were very helpful to explain the mixed number concept, however some children were distracted by them and lost focus of the task at hand. Otherwise, the children only used paper and pencils.
In the classroom there are only two computers. One is for the teacher and the other is generally used for the children to do reading tests on. I never see any other program run on the student computer. The only other time the children use computers is when they have a scheduled computer class. While in this class, the technology specialist plans lessons for the children that include literacy, some mathematics, and social studies. It is a very structured and interesting class that allows the children to learn how to properly use and care for a computer. The children look forward to this class and are extremely well behaved for the teacher because they consider this class to be a privilege.
Unfortunately, it is rare for the math lesson to be incorporated in to the computer class because the teacher I observed is not that computer savvy, and probably would not have been able to successfully create a lesson that used computers. Even with professional training days that help teachers incorporate computers, most of them still do not feel confident enough to try.
It is difficult for me to decide whether a computer would have been helpful during this lesson because I am not up to date with the many math programs available. I imagine there is a program that might have been insightful and stimulating for the children to use. In an age where technology is used on daily basis, it is very important to expose children to the benefits of using computers and other technological devices.
Conclusion:
When observing a fellow teacher engage a class in a lesson it is important to be open]minded. I also found that the more I reflected on different aspects of the lesson, the more I learned. As a result of this inner reflection I concluded that their were some portions of the lesson that I would repeat in a class of my own, and some that I would omit in order to teach a worthwhile and interesting lesson.
I was very fond of the use of pattern blocks. I think these manipulatives help the children visualize mixed numbers, and assist them as they form concepts needed to build a firm understanding of basic mathematics. I also thought that the problem sheets given out at the end of the lesson were helpful in discerning whether the students understood the lesson.
On the other hand, I think I would avoid teaching the lesson to the class as individuals and arrange the children in groups in order to promote mathematical dialogue. This would hopefully keep the children all at the same pace, and minimize time spent repeating directions. I would also try to incorporate real]world examples prior to handing out the pattern blocks in order to get the children thinking. Another aspect I would try to incorporate in to the lesson would be a technological portion that could be applied during the computer class. Perhaps the children could do a short activity the following day on the computers to expose them to the mixed number concept again. Thus allowing them time to reflect, and then apply conceptual knowledge to an operational task.
Overall, the lesson was a successful attempt to introduce a new concept, and it met the New York State Standards. While the children were learning a new skill, I was as well. I am happy to have had the opportunity to observe this lesson, and judge for myself what I think will make learning math exciting and effective.
References:
Van De Walle, J.A. (2004). Elementary and Middle School Mathematics (fifth edition). Boston: Pearson Education, Inc.
New York State Educational Standards (Elementary Standard 3 Mathematics)
Saxon Publishing (Math Lesson Ill ] Grade 2)