Standards for School Mathematics
Introduction
Standards for school mathematics are defined as guidelines formulated by the National Council of Teachers of Mathematics (NCTM). The standards play a role in setting forth the recommendations for mathematics teachers and educators. Moreover, these standards form a vision for preschool through twelfth grade mathematics education in Canada and U.S. The process of formulating the standards was a consensus process involving mathematicians, classroom teachers and educational researchers. This paper will examine an elementary mathematics lesson with regard to application standards in mathematics. Standards in mathematics are meant to give instructors a basic guide to teaching mathematics.
Discussion
Each of the core subjects in school including mathematics is regulated by a set of state standards. The role of the standards is to regulate what is being taught in ach and every classroom and grade level. Moreover, it ensures that students learn what they are supposed to by the time they finish their education. It is evident that standards improve mathematical instruction due to the fact that they give teachers and instructors an overall guideline. They give these instructors a key guide that they are supposed to follow as students’ progress throughout their entire academic career. Instructors should use mathematics standards after assessing each and every student, and progress through the standards accordingly (National Council of Teachers of Mathematics, 2004).
There are various differences between traditional mathematics and constructivisttype programs in terms of addressing the standards. Traditional mathematics program is more like school mathematics in that what is learnt in mathematics plays a role in solving routine and familiar problems. The activities done in the classroom involve symbolic manipulation aimed at finding a specific result that has been predetermined by the teacher. In this case there is little or no negotiation of mathematical meaning and learners are not seen as active agents in their own learning process. The standards in traditional mathematics programs gave teachers the role as sole authorities in mathematics classrooms.
Contrary to traditional mathematics, constructivist type programs engage learners in the learning process. Students and teachers work together or cooperatively in order to constitute a community of inquiry and validation. The classroom activities done under this learning program include discussions, listening, mathematical justifications and dialogical encounters. Hence, students are viewed as active agents in a social and transactional learning process. Constructive type programs have incorporated standards that entail curricula change aimed at increasing students’ curiosity through utilization of inquirybased approaches to instruction (Fuller, 2001).
The limitation of traditional program is that students are not given a chance to demonstrate the knowledge and skills learnt in class. Moreover, they are not given an opportunity to ask questions and discuss concepts for better understanding. It would be much beneficial if the program incorporated group discussions and inquiry sessions in order to gain better understanding of difficult concepts.
One limitation of constructivist program is that students are not given the foundation knowledge and skills. Some critics have argued that instructors expect students to make sense out of abstract concepts without enough foundation skills. This becomes extremely challenging in the long run. Hence, the program would be more beneficial if students are given prior knowledge, concepts and understanding. This results to more activity that achievement.
The title of the observed elementary mathematics lesson was addition and subtraction game. One of the objectives of the mathematics lesson was for students to learn how to use accuracy in adding and subtraction. Moreover, students were expected to understand the concepts of adding and subtracting at the end of the mathematics lesson. The other objective was for the students to practice addition and subtraction facts and processes. The final objective was for the students to develop speed in adding and subtracting process.
One of the NCTM standards that were addressed is that of problem solving. This is due to the fact that students were asked by their teacher to solve some addition and subtraction problems. Each of the students was involved in a writing activity whose aim was to add and subtract numbers in order to obtain the highest and the smallest number respectively. The other process standard that was addressed is communication. During the entire mathematics lesson, the teacher constantly communicated with the students. Students were also asked to communicate the answers obtained to the entire class. Representation is the other process that was addressed in that students presented the various answers obtained after solving addition and subtraction problems (National Council of Teachers of Mathematics, 2004).
It is essential to have standards in mathematics since they provide an overall guideline for instructors to use during the instruction process. These standards also make sure that students learn all the informational and concepts required of them by the time they are through with the entire learning process. It is therefore evident that standards in mathematics are a basic guide to both students and instructors, and they make the entire learning process much easy and effective. They also make the learning process much interesting and involving to both students and teachers (National Council of Teachers of Mathematics, 2004).
The instructional approach used in the lesson was a teachercentered approach. This is an instruction method where the role of the teacher is to present information that is to be learned. Moreover, the teacher directs the entire learning process. The initial step is for the teacher to identify the lesson objectives and take the key responsibility for guiding the instruction by explaining information and modeling. This is then followed by students practicing what they have heard or learnt. The instructional methods that fall under this approach are direct instruction, demonstration, lecturediscussions and lecture (Van de Walle, 2010).
In the mathematics lesson observed, the teacher drew parallel lines and two intersecting lines on the blackboard and then explained to the students what was expected of them. This clearly shows that the role of the teacher was to guide students on what they were expected to do in the learning activity. As part of the lesson, students were required to fill numbers in squired drawn in their papers. The main objective of the activity was to obtain the highest number if adding and the lowest number if subtracting. While the students were performing the activity of putting their numbers into their paper, the teacher was also putting her numbers into the squires on the board. This activity illustrates that the demonstration method of teaching was used (Van de Walle, 2010).
Demonstration method of teaching entails the teacher showing students a process of procedure for doing a particular activity. Demonstration method can be less passive if students are allowed to participate in the activity involved. In the observed lesson, both the students and teacher were involved in the activity of filling numbers to squires after adding or subtracting. It is therefore clear that the demonstration process was more active than passive. It is recommendable to involve students in the learning process for better understanding (Johnson and Norris, 2006).
The other method that was used in the mathematics lesson observed is that of direct instruction. At the beginning of the lesson, the teacher explained to students what was supposed to be done and the aim of the activity. This is a direct instruction method since the role of the teacher is to instruct or guide student regarding the learning activity to be carried out. The role of direct instruction method is to help students to learnt specific skills and concepts that will help them to perform a given activity. The four main models of direct instruction are presentation of new information, introduction and review, independent practice and guided practice (Johnson, 2006).
The teacher did not differentiate instruction within a diverse classroom in the lesson observed. This means that the teacher lacked to recognize that the students were from diverse backgrounds including diverse cultures, ethnic groups and even diverse genders. The teacher might have differentiated instruction in a diverse classroom through the use of studentcentered instructional method. This method allows students to actively take part in the learning process. Moreover, the method appreciates the fact that different students have diverse backgrounds and diverse cultures. From this method, students can be able to learn from one other and learn the different aspects associated with individuals from different diverse groups (Lewis, 2005).
The other method of differentiating instruction in a diverse classroom is using culturally responsive teaching. In this type of teaching, culture forms an integral part in the learning process. This type of teaching responds to, acknowledges and celebrates fundamental cultures. It also provides equitable and full access to education of students from a diverse number of cultures (Johnson, 2006).
Technology was not used in the instruction of the mathematics lesson observed. It is evident that the field of mathematics had greatly benefited from technology throughout its history. Technological tools can be used in classrooms to enhance highlevel thinking in addition to highlighting links between taught concepts and their application in the real world. One of the technological tools that could be used in the observed mathematics lesson is the calculator. Students could use calculators to solve the addition and subtraction problems. This would in turn make computation much easy. The other technological tool that could be used in the observed mathematics lesson is computers. Computers could be used for demonstration and manipulative purposes (Lewis, 2005).
The teacher used manipulative objects in the observed mathematics lesson. The manipulative object used is a die. In the learning activity, the teacher rolled a die then the number rolled was placed in one of the squires at the top two rows. The die was rolled till the empty boxes in all the rows, except the bottom row were filled. It is therefore evident that the die, as a manipulative object was used effectively. The teacher was able to clearly demonstrate to students what the mathematics learning activity entailed. Moreover, students were able to understand the activity involving the use of die as a manipulative object.
One of the changes that I could have made to the lesson is to make it more studentcentered. This involves allowing students to take a more active role in the learning process. For instance, by letting the students work in groups in order to come up with different answers that could be debatable. The other way is through appreciating the concept of diversity. Through this teachers are able to acknowledge the fact that students are diverse and that the level of understanding among various students is difficult. It is essential to appreciate the fact that mathematics is among the most difficult subjects to standardize since the content builds from year to year (Van de Walle, 2010). Since students are different, it becomes quite challenging to apply the set mathematics standards. Hence, the standards should only give instructors an overall guideline for teaching mathematics.
Copy of the Lesson plan
Title: Addition and subtraction game
Grade Level: Grades 28
Overview: This is a group activity that reviews and drills in the format of the game for learning the facts involved in addition and subtraction. The activity appeals to multigrade and multilevel situations. The students may get so caught up in the game that they may consider it an exciting challenge instead of a review or drill.
Objective (S):
 To understand the concepts of adding and subtracting
 To practice addition and subtraction processes and facts
 To develop speed when adding and subtracting
 To use accuracy in adding and subtracting
Materials:
Activities and Procedures:
 The teacher draws three parallel lines then two intersecting lines on the board. The next step is to place a + or – sign next to the second parallel line, which makes a grid of empty boxes, with three boxes in each of the three rows.
 The students are to copy the grid onto their papers.
 The teacher will explain to the students that she is going to roll a die and the number rolled is to be placed into one of the squares in the top two rows and the bottom row is for the answer. The die is to be rolled until the empty boxes in all the rows, with the exception of the bottom row, are filled.
 Students are to work on the problem
 The aim of the game is to obtain the highest number on adding and the lowest number on subtraction.
 While students are putting their numbers, the teacher is to play by putting numbers into the squares on the board.
 The teacher should ask the students if anyone bets the answer obtained on the board. The best answer is written on the board and any student with a similar answer is awarded a point.
 The teacher should create smaller or larger grids in order to adapt to the students level.
The students’ papers should be checked for:
 Concepts learned
 Accuracy
 Speed
References
Fuller, J. (2001) An integrated hands on inquiry bases cooperative learning approach:
The impact of the PALMS approach on student growth. Paper presented at the Annual Meeting of the American Education Research Association.
Johnson, A., & Norris, K. (2006) Teaching Today’s Mathematics in the Middle Grades.
Boston: Pearson Education, Inc. (Chapters 13)
Lewis, A. C. (2005) Washington commentary: Endless pingpong over math education.
Parrot math. Phi Delta Kappan, 80(6). 434438.
National Council of Teachers of Mathematics (2004) Principles and standards for school
Mathematics.
Van de Walle, J. A. (2010) Elementary and Middle School Mathematics: Teaching
Developmentally. Boston: Pearson Education, Inc. (Chapters 1 & 2)
