Friday, July 6, 2012

Essay on NCTM

National Council of Teachers of Mathematics Standards

The following report consists of an analysis of various lesson plans found on four different websites. The lessons of discussion have something in common, which is their concentration on the National Council for Teachers of Mathematics (NCTM) Standards for Number Sense and Operations. The selected lesson plans will be analyzed according to several criteria, including the use of NCTM Standards, the use of differentiated instruction, and the use of manipulatives. A short introduction to these factors in the preface will serve as a foreshadowing of the analysis of each of the websites and its corresponding lesson plans.

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Preface (Guidelines and Criteria)
The NCTM encourages instructors to allow students to use their natural insights when discovering number sense and operations. According to their standards, students will much more willingly explore concepts by figuring it out each time rather than memorizing formulas or using a calculator. Instead of learning how to add or divide, children are exercising their number sense and operation sense. Different students have different strengths, weaknesses, abilities, and handicaps. The use of differentiated instruction is a means to combat weaknesses and handicaps and capitalize on strengths and abilities. Differentiated instruction refers to the simultaneous availability of alternative methods of instruction in each lesson.

Manipulatives are often used within a math lesson in order to illustrate a concept to a class. Instead of learning math, the student is doing math, an idea supported by the NCTM . A more hands-on approach allows students to understand in which ways mathematical principles can be used practically. In the instruction of Number Sense and Operations, for example, manipulatives such as base building blocks can be used to illustrate addition, subtraction, fractions, and decimals. Manipulatives can be physical or virtual in nature, and they can come in many forms, depending largely on the creativity of the instructor.

Website 1
This lesson plan was gathered from a website called Teach the Children Well ( The implementation of the lesson involves using technology to help instruct the inverse relationship between addition and subtraction. The lesson plan is designed for students in late grade 1 and early grade 2. By the end of this lesson, students should be more readily able to identify patterns in numbers and understand how subtraction and addition are related.

The lesson should be given in a computer lab where there is access to computers and earphones for the students, and a projector and speakers for the teacher. These tools allow the students to solve and check problems. The idea is that the students should be able to create true number sentences, using +, -, and =, with the help of ‘fact families’, which are families of information relating to a certain mathematical relationship.

This lesson allows students to realize how technology can aid them in their studies of mathematical concepts. It is crucially important, as up until recently, calculators were the primary technology associated with math learning. Given the findings of the NCTM, this would provide a negative contribution to developing enthusiasm for math among youngsters. With websites such as the Fact Family, students gain insight into fun educational activities that could help them in their future.

In this lesson plan, there is not much involvement of traditional manipulatives. In a role-play exercise, however, students are assigned to wear specific numbers are used as manipulatives in forming variations of one fact family. Each student is assigned a number and another student must organize his or her peers into a number sentence that makes sense. Other manipulatives could have been included such as blocks or beads that would act as countable objects.

The lesson plan explores the need for differentiated instruction thoroughly. Different types of learners are accounted for by different activities. For example, the introductory activity addresses kinesthetic learners by allowing students to participate in a role-play exercise with number sentences. The availability of computer aid makes it possible to add sound files for auditory learners and those who are visually impaired. The graphics used in the lesson are culture-sensitive and include elements of multiculturalism. There is a special needs consultant who provides assistance to the physically handicapped and there are multiple levels of assessment for different ability levels.

Website 2
This website was created by the Educational Broadcasting Corporation. It is based on a television series called Cyberchase, which teaches students grades 3 to 5 mathematical concepts in a design linked to NCTM standards. The lesson plan chosen covers both mathematics and social studies. It is called Donut Dinero and it is designed to introduce the concepts of Bartering and Money Systems. The lesson teaches about the history of money through videos, quizzes, a time machine, and a specimen of a bill. It makes the concept of money relatable, as the currency used throughout many of the exercises is donuts. The students are required to perform mathematical operations throughout this lesson, including working with fractions, decimals, and percents. The students use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.

Differentiated Instruction
The activity that resembles the use manipulatives in this lesson plan the most is when students are handed 10 plain donut cutouts. As the donuts are the currency of the class, they are used to assign value to various objects. Some objects are worth more than others and some are worth only a fraction of a whole donut. The cutouts are used in combination to trade items via bartering and monetary transactions. The lesson could have involved base ten blocks in order to exemplify the differing values of various currency, such as coins versus bills. Where a cube normally represents ones, a rod represents tens, and a flat represents hundreds, in this exercise, a cube would represent a penny, a rod would represent a dime, and a flat would represent a dollar. Other forms of currency, such as a nickel or a 20-dollar bill could be represented visually once this concept was understood. The same could be done with attribute logic blocks.

The lesson plan included many forms of instruction such as videos, literature including poems and picture books, and exercise worksheets. However, each activity serves a specific purpose and there is not many alternatives for each activity. This means that if one student has trouble digesting a video due to a visual or hearing impairment, he or she may have trouble participating in the following exercise, as it requires an understanding of the concepts brought forth previously. Although the lesson plan offers different possible instruction methods, there is no clear structure, which ensures that each concept has a more than one designated form of instruction.

Website 3
This lesson is the first of four lessons included in a Unit entitled All about Multiplication created by a subsidiary website of the NCTM called Illuminations. The lesson, entitled Hopping on the Number Line is designed to highlight the measurement of multiplication and encourages students to predict products and answer puzzles using the number line model. According to the NCTM, multiplication is a focal point that is explored at grade 4 levels. At this level, they are expected to understand the various meanings and effects of multiplication and division.

The students use individual number lines to explore order properties and understand the communitive property of multiplication The activities involve a number line and counters such as chips and markers to create “hops”, based on certain multiples. The instructor communicates with the class hops of 5 and 3, and the possibilities that are associated with these two multiples.

This lesson is very important to the students’ development of an NCTM focal point, understanding of multiplication and division theory. It presents a unique representation of these operations that helps form an overall picture in the minds of the students. It is also important because it serves as a stepping stone for further lessons including Exploring Equal Sets, Modeling Multiplication With Streets and Avenues, and Balancing Beam Discoveries.

Manipulatives Used
After the introductory activity, the students are presented with a virtual manipulative tool called Number Line Bars. The manipulative provides an opportunity for the students to model new and customizable multiplication problems on the number line. The hop size and the times hopped are completely up to the user, along with possibility of color customization for easy identification.

There is no differentiated instruction presented in this lesson plan. Alternative methods of instruction could include the use of tiles, drawing pictures, and creating word problems with objects. Students with handicaps could benefit from video or audio-based instruction.

Website 4
The following lesson plan was taken from and is called Banana Math Scrabble. It is designed to improve mental math speed and accuracy as well as develop greater understanding of numbers and operations. It is targeted to middle school and early high school students. Students exercise mental math speed by using reason and logic to discover patterns and quantitative relationships through addition, subtraction, multiplication and exponents.

A timer, some number cards, symbol cards, and goal cards are handed out to each student. Using the cards in aggregate, students attribute numbers to symbols in the aim to fulfill a goal written on a goal card. The objective of the game is to construct equations that will make them comfortable with order of operations. The game consists of 7 steps each round, with two-minute rounds that repeat in a circular process. Each round the student should be able to compute information quicker and more easily.

The lesson is a great way to instill confidence in the students in their own mental math abilities, which is very important moving forward. As concepts get more complex and difficult, students should understand that practice and repetition will ensure that they do not have an unnecessarily difficult time with calculations in the future.

The manipulatives used in this activity are the three types of cards. Students have the freedom to choose the equation they execute and the goal of each equation. Cards are a great manipulative that can be used by most students. As a group exercise, the students use the cards together to achieve a common goal. Although students are competing against one another, they are interacting which gives them the opportunity and possibility to help each other throughout the lesson.

The lesson plan offers differentiated instructions for students who struggle with working with the cards. In this case, the instructor would go over the same lesson structure but this time working on paper. The instructor provides the student with similar data, numbers and symbols and assigns a task, similar to the goal card in the regular activity. This approach is more personal and allows for the instructor to cater to the students’ personal needs.
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