### Essay on Observing Math Instructions:

**Standards in Mathematics**

Standards in **mathematics** are crucial to the process of instruction as they improve methods of instruction in several ways. In any subject, standards help to regulate the things that the student learns in each classroom, limiting the content to what the student is required to know at that point in his or her education. Standards give instructors a proper framework to operate in terms of assessment, curriculum, evaluation, and professionalism. Standards called curriculum focal points give insight as to what are the most important topics that need to be covered at each level of mathematics instruction. The National Council of Teachers of Mathematics (NCTM) provide resources such as the handbook entitled “Principles and Standards for School Mathematics” (PSSM), which was published to help instructors create an effective program for learning. Principles covered in this handbook include equity, curriculum, teaching, learning, assessment, and technology. These principles are covered in depth to help understand how students’ learning and knowledge should grow throughout the education process.

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**Traditional vs. Constructivist Instruction**

The traditional method of mathematics instruction refers to the most common form of instruction of mathematical concepts. The idea behind traditional mathematics, also known as classical math education, is to teach concepts using direct instruction and standard methods. Concepts are taught using isolated formulas, one by one. A text book that follows traditional mathematics methods would have, in print, the formula for the area of a triangle, which students could copy and apply to the specifics of a math problem. There have been many criticisms of traditional math programs, under the basis that this type of instruction gives too much emphasis on repetition and memorization while neglecting the inventive and imaginative aspects of mathematics. Tests have shown that students who experience traditional mathematical methods have a harder time with problem solving and conceptual understanding.

Constructivist mathematicians believe that creating situations where a student can develop his or her own constructions does not require lecturing, explaining, or transferring knowledge. This method gives the student the freedom of choice of how they want to process the information. The focus here is on determining how people learn and on engaging the learner in his or her own application of mathematical models. Critics of this method of instruction say that techniques such as learning by doing are not suitable for novice learners. Many believe that those with no prior knowledge to any mathematical concept cannot excel in unstructured constructivist teaching methods.

**Objective of Lesson Plan**

The objective of the lesson that was observed was to teach the students the equivalent forms of fractions and decimals by constructing models that would represent these fractions and decimals.

**National Council of Teachers of Mathematics Standards**

The NCTM curriculum standards were addressed and followed in many aspects. The students were expected to have prior instruction and knowledge of concepts such as percentage, decimals- one, tenths, hundredths, and proper fractions- tenths and hundredths. The instructor benefited greatly from having the confidence that the students already knew the basic principles of fractions, decimal, and percentages because time was not wasted on unnecessary repetition of these topics. Teaching and learning standards were exemplified as well. The PSSM states that while learning the basics is important, procedural facility and conceptual understanding is vital as well. The teacher of this class was able to choose a high-quality and hands-on method of instruction by utilizing professional expertise and intimate knowledge of the students. When it came to assessing the students’ understanding, the teacher used a personal and formative approach, evaluating and attending to each student’s level of understanding of the topic. The topic, numbers and operation with regards to understanding decimals and fractions, is a focal point for the fourth grade level, according to NCTM standards. The students in this class were aged 9-10 and the subject matter was appropriate in theory and in practice.

**Methods of Instruction**

The use of several instruction methods by the teacher were observed, including brainstorming, the Inductive-Deductive method, and the Learning by Doing method. In order to introduce the subject, the teacher asked the class to brainstorm about their ideas of what the Base 10 Models that were distributed to them represented. The ideas gathered by the instructor, written on the board, and were used to explain the idea of equivalence with words and relate it to the idea of equivalent numbers.

Aspects of the Inductive method were visible when the teacher asked the pupils to measure the amount of strips it took to cover one mat, in order to communicate the concept of tenths. The process was repeated when the purpose was to have the class show a representation of 27/100 or .27 using the Base 10 Models. Later on, the teacher used the board and an overhead projection to help the students formulate a rule based on their experiences and observations. The teacher utilized the ‘Learning by Doing’ method throughout the class by instructing the pupils to use Base 10 models. It was evident by the students’ participation that they felt interested in the simple experiment. The exercise had elements of the Deductive method also, as the students were encouraged to learn individually and not as a group.

**Instruction Differentiation**

The instruction was standard across the whole class and the students did not require much special treatment. However, the teacher provided extra assistance during the assessment phase to those students who were having difficulty making the models and understanding the concepts. Although the class was filled with students who have different learning styles and interests, the various methods of instruction, which included writing on the board, using the overhead, and self-administered exercise, provided a rainbow of activity. Therefore the students were able to consume the information in the way they felt most comfortable. If the pupils had varying abilities on a larger scale or suffered from some sort of handicaps, then instruction differentiation would have been necessary. For instance, if the students had physical handicaps, the instruction could therefore have involved audio and video programming including animations and text to teach the concepts and a less hands on approach to application of the topic.

**Use of Technology**

Technology was used in the instruction of the math lesson. The instructor used an overhead projector near the end of the exercise in order to solidify the principles being taught. The physical Base 10 Models had an overhead counterpart which was displayed on the overhead projector. The teacher used drawings of squares, lines, and smaller squares to represent the mats (one), strips (tenths), and units (hundredths). The students followed the projection by comparing it to the models they had made individually. They were able to identify resemblances to the work they had completed correctly and differences in the parts they had misunderstood.

Technology that wasn’t utilized was the use of an electronic presentation to provide a demonstration of the activity. If the pupils saw the physical model prepared and saw a short video clip of other pupils performing the same exercise they were about to perform, it may have instilled more self-confidence in the class and more enthusiasm for the activity. This could have been an effective way to explain the process of making a model before distributing the Base 10 Models.

**The Use of Concrete Objects and Manipulations**

The Base 10 Models are concrete objects which can be manipulated via organization into various shapes. They were the focus of the lesson and were used effectively. The students perceived them as fun playful objects and received the instructions well. However, some students started to play with the items before any instructions were given. The teacher distributed the models before giving instructions and it was harder for her to get the message across because of this.

**Assessment**

The assessment of the learning in the classroom was done using two assessment methods. During the class exercise, the instructor circled the classroom to track progress of each individual student. Anyone who required extra assistance in making the models of the different fractions and decimals that the teacher requested received help at that point. This allowed the teacher to understand whether or not the class understood the concepts as they proceeded through the exercise. The lesson plan also included an activity sheet that requires the students to draw models similar to the ones they made, which represent a corresponding fraction or decimal given to them within the activity sheet. After they draw the models, they must answer whether or not the numbers are equivalent to one another.

**Recommended Changes**

Overall, most of the pupils of the class were successful in achieving the objective. However, it was evident that some students did not attain the same level of understanding as others. Some students felt demotivated because they did not understand the instructions, so they passively played with the Base 10 Models instead of participating. The main reason for this was the distribution of the models before instruction was given. The class could have benefited from an introduction video and would have had more enthusiasm if the tasks were executed in a group. Moving from teacher-student assessment and individual work to peer-review and group work seems like a plausible modification. Furthermore, the instructor could still circle the class and help when needed, as well as assess the knowledge and comprehension using the activity sheet.

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