The Basics of Mathematics Pedagogy

Representations
Representations are the tools of symbols and materials. They are used to help students understand mathematics by giving them visual guides in their thinking. For example, the conventional symbols that indicate addition, subtraction, equality, and so on (into the higher realms of symbols used in geometry, algebra, and calculus) tell students, at a glance, the process that is being calculated. Materials that are used as representations are called manipulatives. These can be small plastic objects or pictures for the students to count, line up, or otherwise use to solve a problem. Representations make abstract concepts become concrete. They put mathematics into the students' hands as well as heads, and the result is improved learning. Using familiar manipulatives with new problems helps the student to make connections and feel more confident and capable of expanding their skills.

Concepts Taught in Kindergarten Before Introducing Numbers
In kindergarten, children can be prepared for the study of mathematics by practicing certain concepts such as:
position – top, middle, bottom, above, below, before, after, between, under, inside, outside, left, and right
visual attributes – same and different colors, shapes, and sizes; identifying items that are out-of-place or don't belong
sorting – by size, color, type, or shape; identifying an equal number, more, or fewer of a given item
graphing – the use of picture graphs and using data from graphs
patterns – identifying, copying, extending, and making patterns; finding patterns that are different or alike, making predictions from patterns
measurements – longer and shorter; how much they weigh, heavier and lighter; how much an item can hold

Problem-Solving Strategies for Mathematics and Steps for Solving Word Problems
For any problem, the following strategies can be used according to their appropriateness to the type of problem or calculation:

i) Use manipulatives or act out the problem, ii) draw a picture, iii) look for a pattern, iv) guess and check, v) use logical reasoning, vi) make an organized list, vii) make a table, viii) solve a simpler problem, and ix) work backward.

In order to solve a word problem, the following steps can be used:

1. Achieve an understanding of the problem by reading it carefully, finding and separating the information needed to solve the problem, and discerning the ultimate question in the problem.
2. Make a plan as to what needs to be done to solve the problem.
3. Solve the problem using the plan from step 2.
4. Review the word problem to make sure that the answer is the correct solution to the problem and makes sense.

Building Number Sense Among Students
It is important to think flexibly to develop number sense. Therefore, it is imperative to impress upon students that there is more than one right way to solve a problem. Otherwise, students will try to learn only one method of computation, rather than think about what makes sense or contemplate the possibility of an easier way. Some strategies for helping students develop number sense include the following:

1. Frequently asking students to make their calculations mentally and rely on their reasoning ability. Answers can be checked manually afterwards, if needed.
2. Having a class discussion about solutions the students found using their minds only and comparing the different approaches to solving the problem. Have the students explain their reasoning in their own words.
3. Modeling the different ideas by tracking them on the board as the discussion progresses.
4. Presenting problems to the students that can have more than one

Using Manipulative Materials in Mathematics Classrooms

1. As with all classroom supplies, the students must understand that there are rules for their use, including how to store the materials when they are not in use. In addition:
2. The teacher should discuss with the students the purpose of the manipulatives and how they will help the students to learn.
3. The students should understand that the manipulatives are intended for use with specific problems and activities; however, time for free exploration should be made available so students are less tempted to play when assigned specific tasks
4. A chart posted in the classroom of the manipulatives with their names will help the students to gain familiarity with them and develop mathematical literacy skills.
5. Loans of manipulatives for home use with a letter of explanation to the parents about the purpose and value of the manipulatives will encourage similar strategies with homework.

Implications of Mathematics Today
Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences, such as economics. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.

Connecting Math and Science to Real Life for Gifted Students
Since math and science are so often heavy in calculations and facts, students are usually not taught how these subjects relate to the real world. All students, especially gifted students, need to find meaning in an academic subject and understand how it applies in life. We may not appreciate enough that math and science are not like people whose opinions we can disagree with; they provide hard, objective, unchangeable facts. Teachers can show students the consequences of ignoring facts with examples like these: mathematicians and engineers advised not launching the Challenger space shuttle, but management overruled them, thereby leading to the deadly explosion. Pop singer Aaliyah died in a plane crash after pilot and crew ignored the mathematics indicating airplane overload and flew regardless. A mathematician proved racial bias in jury selection by calculating that the mathematical probability of fair selection was approximately 1 in 1,000,000,000,000,000.

Analyzing the Use of Appropriate Mathematical Concepts, Procedures, and Vocabulary when Evaluating Student Solutions
When evaluating student solutions, it is important to analyze the use of appropriate mathematical concepts, procedures, and vocabulary for a variety of reasons. First and foremost, we must be sure that we have provided adequate practice and instruction of important concepts before assessing them. Once we have established that instruction is sufficient, we must ensure that students are following the appropriate procedures when faced with various tasks and that those procedures are executed correctly.
Finally, we must hold students accountable for using high-level vocabulary to ensure that students are able to read, understand, and communicate their mathematics thoughts at age- and grade-appropriate levels.

Correcting Student Errors
A student is asked to find the area of a rectangle measuring 8 feet by 4 feet. The student is able to break the rectangle into 32 squares but states that the area is 24 square feet. The student calculated the perimeter of the rectangle rather than the area. The perimeter of a rectangle is the sum of its sides. Here, the rectangle's sides measure 8 feet, 8 feet, 4 feet, and 4 feet, which total to 24 feet. To calculate area, the student must multiply the length by the width  which equals 32 square feet. To help the student correct his or her error, the teacher can explain that the student was on the right track when he or she broke the rectangle into 32 squares. Because area is the amount of unit squares that can be contained in a two-dimensional figure, the student could have also opted to simply count the unit squares he or she created, which would have also led him or her to the correct answer of 32 square feet.

Exploring Problem Structures with Unknowns in All Positions
It is important to provide students with the opportunity to explore and find solutions for problem structures to develop their higher-level thinking skills and problem-solving strategies. By exposing students to problems with unknowns in all positions, students are forced to not only memorize procedures but also to analyze the framework of a problem, make connections for relationships within the problem, and develop strategies for solving problems that may not follow specific rules or procedures. Problems that include put-together/take-apart scenarios are excellent problem structures for teachers to use in an effort to develop higher-level thinking skills. In addition, giving students the opportunity to use arrays to model their solutions provides teachers with a glimpse into the thought processes of each student's solution.

Unknown Addends
A teacher tells her students that she has five pieces of fruit in her refrigerator. Two are apples, and the rest are oranges. She asks the students how many oranges are in her refrigerator. In this type of problem, there is an unknown addend. When dealing with unknown addends and numbers up to 20 (typically in the elementary classroom), this type of problem is known as a put-together/take apart problem. Ideally, the teacher's goal is for the students to visualize this the scenario as   Depending on the students' approach (drawing a picture, subtracting, number facts, modeling, etc.), they may choose to put together or take apart the problem in a variety of ways. To analyze the students' process, the teacher should check that each student has developed a strategy that will result in correctly identifying the missing addend in a way that could be applied to other problems similar in nature, continually resulting in the correct answer.

Evaluating Validity of Mathematical Model or Argument when Analyzing a Solution
When assessing a student's proficiency, simply arriving at the correct answer does not validate true mastery of a mathematical concept. To determine that a student has truly mastered a skill or concept, that student must be able to explain or defend his or her process as well as the solution. By requiring a model or compelling students to defend their answers, teachers can truly assess the validity of each solution and determine whether or not each student has a true grasp of the concept being assessed. A teacher can be confident that a student has truly mastered a concept when that student can describe his or her process, explain the meaning behind his or her solution, and defend why that solution is logical and appropriate.

Analyzing Validity of a Student's Mathematical Process
A teacher is assessing a student's ability to multiply two-digit numbers. A student in the class arrives at the correct answer without using the traditional algorithm. In mathematics, there is often more than one way to arrive at the correct answer. The focus of the teacher should be to analyze the validity of the student's mathematical process to determine if it is a process that could be applied to other multiplication problems, which would result in the correct answer. If the student has modeled his or her thought process, or can argue that the inventive strategy is applicable across all multiplication problems, then that student should be considered to have mastered the skill of multiplication. However, if the student's argument or model is not applicable to other multiplication problems, that student should be provided with more instruction and opportunity for improvement.

Using Individual Student Mathematics Assessment Data to Guide Instructional Decisions and Differentiate Instruction
Teachers should constantly assess student learning. These assessments should be quick in nature and offer immediate feedback to both the teacher and student. Assessments such as quick 'exit tickets' at the end of each lesson can tell a teacher whether or not students achieved the objective of the day. If the majority of the class achieved mastery, the teacher should look to modify and reteach that lesson to the students who did not achieve mastery. On the other hand, if the majority of the class did not achieve mastery, the teacher should reflect on the way in which the lesson was presented and try another whole-class approach before moving on. By monitoring student learning on a constant basis, teachers are able to improve their teaching skills as well as identify and differentiate for struggling learners more quickly and effectively.

Creating Structured Experiences for Groups According to Cognitive Complexity of Tasks
Depending on the complexity of the task, teachers should modify the delivery of instruction. For less complex tasks, the teacher may opt for whole-group instruction, in which the entire class is introduced to a new concept together. Whole-group instruction usually includes a connection to prior knowledge, direct instruction, and some form of media. For more complex tasks, the teacher may choose small-group instruction, where students are grouped based on ability levels. The more accelerated learners are given a quicker, more direct form of instruction, whereas the struggling learners work independently at a math center or station activity. After a rotation, the struggling learners are given a modified form of instruction by the teacher, which has been differentiated and allows for more gradual release based on individual needs and abilities.