PST 201-F STUDY NOTESCHAPTER 1

TEACHING MATHEMATICS IN THE ERA OF NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS (NCTM) STANDARDS

Principles and Standards for School Mathematics

The six principles for school mathematics Equity high expectations and strong support for all students

Curriculum must be coherent, focused on important mathematics and well articulated.

Teaching understand math; understand how children learn math; select tasks and strategies to maximise learning.

Learning students must learn actively, building new knowledge from experience and prior knowledge.

Assessment support learning of math and provide info for both teacher and learner

Technology essential, it influences what is taught and enhances learning.

The five Content Standards (or strands of mathematics)

Number and operations Algebra Geometry Measurement Data Analysis and ProbabilityThe five process standards

Problem solving Reasoning and proof Communication Connections RepresentationThese refer to the process through which students should acquire and use mathematical knowledge.

N.B SEE TABLE 1.1, PAGE 4 OF TEXTThe Professional Standards for Teaching Mathematics

Teachers must shift from a teacher-centred to a child-centred approach in their instruction.

Five shifts in Classroom EnvironmentThe introduction to the professional standards lists five major shifts in the environment of the mathematics classroom that are necessary to allow students to develop mathematical power. Teachers need to shift

Toward classrooms as mathematics communities and away from classrooms as simply a collection of individuals

Toward logic and mathematical evidence as verification and away from the teacher as the sole authority for right answers

Toward mathematical reasoning and away from mere memorising procedures

Toward conjecturing, inventing, and problem solving and away from an emphasis on the mechanistic finding of answers

Towards connecting mathematics, its ideas, and its applications and away from treating mathematics as a body of isolated concepts and procedures.

The teaching standards

See appendix B of text!The reform movement in school mathematics

The reform movement and school mathematics is the movement away from teacher-centred methods of instruction and towards learner centred methods of instruction. It is strongly focused on problems centred teaching and learning approaches. It also focuses on the concept that all students can and should learner mathematics.

Forces driving the reform movement are:

The demands of society

The influence of technology

The direction of the National Council of teachers of mathematics

The Third International Mathematics And Science Study report

The five content standards and their relationship to the learning outcomes in our own curriculum

The five processes standards and what our own curriculum has to say about them

The five shifts in the classroom environmentCHAPTER 2

EXPLORING WHAT IT MEANS TO DO MATHEMATICS

Traditional views of mathematics

Traditional teaching, still the predominant instructional pattern, typically begins with an explanation of what ever idea is on the current page of the text followed by showing children how to do the assigned exercises. Even with a hands on activity, the traditional teacher is guiding students, telling them exactly how to use the materials in the prescribed manner. The focus of the lesson is primarily on getting answers. Students rely on the teacher to determine if their answers are correct.Mathematics as a science of pattern and order

Mathematics is the science of pattern and order. Science is a process of figuring things out or making sense of things. It begins with problem-based situations. Although you may never have thought of it in quite this way, mathematics is a science of things that have a pattern of regularity and logical order. Finding and exploring this regularity order and then making sense of it is what doing mathematics is all about.The world is full of pattern and order: in nature, in art, in buildings, in music. Pattern and order are found in commerce, science, medicine, manufacturing, and sociology. Mathematics discovers this order, makes sense of it, and uses it in a multitude of fascinating ways, improving our lives and expanding our knowledge.

What does it mean to do mathematics?Doing mathematics = engaging in the science of pattern and orderThere is a time in a place for drill but drill should never come before understanding.

The verbs of doing mathematics are:

Explore

Investigate

Conjecture

Solve

Justify

Represent

Formulate

Discover

Construct

Verify

Explain

Predict

Develop

Describe

Use

These are science verbs indicating the process of making sense and figuring out.What is basic in mathematics?

The most basic idea in mathematics is that mathematics makes sense! Everyday students must experience that mathematics makes sense.

Students must come to believe that they are capable of making sense of mathematics.

Teachers must stop teaching by telling and start letting students make sense of the mathematics they are learning.

To this end, teachers must believe in the students - all of them!

An environment for doing mathematics

The classroom must be an environment where doing mathematics is not threatening and where every student is respected for his or her ideas.

Students should feel comfortable taking risks, knowing that they will not be ridiculed if they are wrong.

The teachers role is to create this spirit of enquiry, trust, and expectation.

The focus is on students actively figuring things out, testing ideas and making conjectures, developing reasons and offering explanations.

No answer book

In the real world of problem solving outside the classroom, there are no teachers with answers and no answer books. Doing mathematics includes deciding if an answer is correct or and why. It also includes being able to justify your reasoning to others.CHAPTER 3

DEVELOPING UNDERSTANDING IN MATHEMATICS

The constructivist view of learning

Constructivism rejects the notion that children are blank slates. They do not absorb ideas as teachers present them. Rather, children of creators of their own knowledge.

The construction of ideas

Children construct their own knowledge The tools we use to build understanding are our existing ideas, the knowledge that we already possess.

The materials we act on to build understanding may be things we see, hear, or touchelements of our physical surroundings. Sometimes the materials are our own thoughts and ideas.

The effort that must be supplied is active and reflective thought. If minds are not actively thinking, nothing happens.

The construction of an idea is almost certainly going to be different for every learner, even within the same environment or classroom.

To construct and understand a new idea requires actively thinking about it. Children must be mentally active for learning to take place. Constructing knowledge requires reflective thought, actively thinking about all mentally working on an idea.

A new idea is constructed by using the ideas we already have. A network of connections between ideas is developed in the process; this is called a web of ideas. The more ideas used and the more connections made, the better we understand. Integrated networks, or cognitive schemas, are both the product of constructing knowledge and the tools with which additional new knowledge can be constructed. As learning occurs, the networks are rearranged, added to, or otherwise modified.

The general principles of constructivism are based largely on Piagets processes of assimilation and accommodation. Assimilation refers to the use of existing schemas to give meaning to experiences. Accommodation is the process of altering existing ways of viewing things or ideas that contradict or do not fit into existing schemas.

Construction in rote learning

Rote knowledge will almost never contribute to a useful network of ideas. Rote learning can be thought of as a weak construction. When mathematical ideas are used to create new mathematical ideas, useful cognitive networks are formed.

Understanding

Understanding can be defined as a measure of the quality and quantity of connections that an idea has with existing ideas. Understanding is never an all or nothing proposition. It depends on the existence of appropriate ideas and on the creation of new connections.One way that we can think about and individuals understanding is that it exists along a continuum. At one extreme is a very rich set of connections. Be understood idea is associated with many other existing ideas in a meaningful network of concepts and procedures. The two ends of this continuum are relational understanding the rich interconnected web of ideasand instrumental understanding ideas that are isolated and essentially without meaning. Note that knowledge learnt by rote is at the isolated end of the continuum; it is instrumental knowledge that is learned without meaning.

Benefits of relational understanding

It is intrinsically rewarding nearly all people enjoyed learning. This is especially true when new information connects with ideas already possessed. The new knowledge make sense; it fits; it feels good.

It enhances memory when mathematics is learned relationally, there is much less chance that the information will deteriorate. Connected information provides an entire web of ideas to reach full. If what you need to recall seems distant, reflecting on ideas that are related can usually lead you to the desired the idea eventually. There is less to remember constructivist to talk about teaching big ideas which are really just large networks of interrelated concepts.

It helps with learning new concepts and procedures an idea fully understood in mathematics is more easily extended to learn a new idea.

It improves problem solving abilities the solution of novel problems requires transferring ideas learned in one context to new situations. When concepts are embedded in originate work, transferability is significantly enhanced and, thus, so is problem solving.

It is self generative as networks grow and become more structured, they increase the potential for invention

It improves attitudes and beliefs when ideas are well and is didnt make sense, the learner tends to develop a positive self concept about his or her ability to learn and understand mathematics.

Concepts and procedures

Conceptual and procedural knowledge

Conceptual knowledge is knowledge that consists of rich relationships or webs of ideas.

Procedural knowledge is knowledge of the rules and procedures used in carrying out routine mathematical tasks and also the symbolism used to represent mathematics.Interaction of conceptual and procedural knowledge

Procedural knowledge of mathematics does have a very important role both in learning and in doing mathematics. Algorithmic procedures help us do routine tasks easily and, thus, free our minds to concentrate on more important tasks. But even the most skilful use of a procedure will not help develop conceptual knowledge that is related to that procedure. Procedural rules should never be taught in the absence of concepts. All mathematics procedures can and should be connected to the conceptual ideas that explain why they work.Classroom influences on learning

Effective teachers must help students construct their own ideas using ideas they already have. The following three factors influencing classroom learning are worth discussing: Reflective thought for a new idea you are teaching to be interconnected in a rich web of interrelated ideas, children must be mentally engaged. They must find the relevant ideas they possess and bring them to bear on the development of the new idea. A significant key to getting students to be reflective is to engage them in problems that force them to use their ideas as they search for solutions and create new ideas in the process.

Students learning from others reflective thought and, hence, learning are enhanced when the learner is engaged with others working on the same ideas Vygotsky theorised that social interaction is a key component in the development of knowledge. He referred to the transfer of ideas from those that are external to the individual ideas exchanged in the social settingto those that are internal, personal constructs, as internalisation. Internalisation only occurs within each learner zone of proximal development, a symbolic space created through the interaction of learners with more knowledge of all others and the culture that precedes them. Classroom discussion based on students own ideas and solutions to problems is absolutely foundational to childrens learning.Mathematical communities of learnersThe four features of a productive classroom culture for mathematics in which students can learn from each other as well as from their own reflective activity are:

1. Ideas are important, no matter whose ideas they are.

2. Ideas must be shared with the others in the class.

3. Trust must be established with an understanding that it is OK to make mistakes.

4. Students must come to understand the mathematics makes sense.

The Role of Models in Developing Understanding

Manipulatives, or physical materials to model mathematical concepts, are certainly important tools for helping children learn mathematics. But they are not the panacea that some educators seem to believe them to be. It is important that you have a good perspective on how manipulatives can help or fail to help children construct ideas.

Models for mathematical concepts The model for a mathematical concept refers to any object, picture, or drawing that represents the concept or onto which the relationship with a concept can be imposed.

It is important to include calculators in the near list of common models. The calculator models a wide variety of numeric relationships by quickly and easily demonstrating the effects of these ideas.

Models and constructing mathematics

To see a concept in a model, you must have some relationship in your mind to impose on the model. The teacher already has the correct mathematical concept and can see it in the model. A student without the concept sees only the physical object or perhaps an incorrect concept. A child would need to know the relationship before imposing it on the marble.

Models can play the role of a testing ground for emerging ideas. They can be thought of as a thinker toys, tester toys, and talker toys.

Models give learners something to think about, explore with, talk about, and reason with.

Models should always be accessible for students to select and use freely. Do not for students to use a particular model.

Expanding the idea of a model

There are five representations for concepts. Children who have difficulty translating a concept from one representation to another are the same children have difficulty solving problems and understanding computations. These representations are: Pictures

Written symbols

Oral language

Real world situations

Manipulative models

Incorrect use of models

The most widespread misuse of manipulative materials occurs when the teacher tells students, do as I do. 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.

Teaching Developmentally

Teaching involves decision-making. Decisions are made as you plan lessons and minute to minute in the classroom. The ideas explored in this chapter provide a foundation for making these decisions. These ideas are: Children construct their own knowledge and understanding; we cannot transmit ideas to passive learners.

Knowledge and understanding are unique for each learner.

Reflective thinking is the single most important ingredient for effective learning.

The socio cultural environment of the mathematical community of learners interacts with and enhances students development of mathematical ideas.

Models for mathematical ideas help students explore and talk about mathematical ideas.

Effective teaching is a student centred activity.

CHAPTER 4

TEACHING THROUGH PROBLEM SOLVING

In the problem solving approach, a thought provoking activity is used as a vehicle of learning.

A problem centred approach uses a non routine problem as a vehicle of learning.

Each of these approaches is known as a problem-based approach to teaching and learning.

Problem solving as a principal instructional strategy

Most, if not all, important mathematics concepts and procedures can best be taught through problem solving.

Problems and tasks for learning mathematics

A problem is defined here as any task or activity for which the students have no prescribed or memorised rules or methods, nor is there a perception by students and that there is a specific correct solution method.

A problem for learning mathematics also has these features:

It must begin when the students are must take into consideration the current understanding of the students.

The problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn.

It must require justification and explanations for answers and methods.

A shift in thinking about mathematics instruction

Teaching should begin with the ideas that children already have, the ideas they will use to create new ones.

Students learn mathematics as a result of solving problems.

Mathematical ideas are the outcomes of the problem solving experience from rather than elements that must be taught before problem solving.

Children are learning mathematics by doing mathematics.

The value of teaching with problems Problem solving places the focus of the students attention on ideas and sense making.

Problem solving develops a belief in students that they are capable of doing mathematics and that mathematics makes sense.

Problem solving provides on going assessment data that can be used to make instructional decisions, help students succeed, and inform parents.

Problem solving allows an entry point for a wide range of students.

A problem-based approach engages students so that there are fewer discipline problems.

Problem solving develops mathematical power.

It is a lot of fun.

A three-part lesson format

The before phase of the lesson

There are three related agendas for the before phase of the lesson:

1. Be sure that students understand the problem so that you will not need to clarify or explain to individuals later in the lesson.

2. Clarify your expectations to students before they begin working on the problem. This includes both how they will be working and what product you expect in addition to an answer.(think-write-pair-share)

3. Get students mentally prepared to work on the problem and think about the previous knowledge they have therefore be most helpful.(Begin with the simple version of the task; brainstorm solutions; estimate or use mental computation)The during phase of the lesson

Although this is the portion of the lesson when students work alone or with partners, there are clear agendas that you will want to attend to:

1. Let go! Give students a chance to work with our tour guide and all direction.

2. Listen actively to your students. This is a time for observation and assessment not teaching!

3. Cautiously provide appropriate hints. Be careful not to imply that you have the correct method of solving the problem.

4. Providing profitable or activity for students who finished quickly.

The after phase of the lesson

In the after phase of the lesson, your students will work as a community of learners, discussing, justifying, and challenging various solutions to the problem all have just worked on. The agendas for the after phase are:

1. Engage the class in productive discussion, helping students work together as a community of learners.

2. Listen actively without evaluation. Take this second major opportunity to find out how students are thinkinghow they are approaching the problem. Evaluating methods and solutions is the duty of your students.

3. Summarise main ideas and identify problems for future exploration.

Designing and selecting effective tasksA task is effective when it helps students to learn the ideas you want them to learn.

Your textbook

Good teachers use the text as a resource and as a basic guide to the curriculum. And

To use a tradition of textbook, examine a chapter or unit from beginning to end, and identify the 2 to 4 big ideas, the essential mathematics in the chapter. You can now do two things: adapt to the best or most important lessons in the chapter to a problem solving format; create or find tasks in the texts teacher notes and other resources that address the big ideas.

Good problems have multiple entry points

Access to the problem by all students demands that there be multiple entry pointsdifferent places to get on the ramp to reach solutions.

Childrens literature

Children so stories can be used to create reflective tasks at all grade levels.

A task selection guide

Step 1: How is the activity done?

Step 2: what is the purpose of the activity?Step 3: will the activity accomplish its purpose?

Step 4: what must you do?

The importance of student writing

The act of writing is a reflective process.

A written report is a rehearsal for the discussion period.

A written report is also a written record that remains when the lesson is finished.

Teaching about problem solving

Students need to be taught problem solving strategies and processes.

Strategies and processesStrategies for resolving problems are identifiable methods of approaching a task that are completely independent of the specific topic or subject matter.

Strategy and process goals

Develop problem analysis skills

Developer and selectors strategies

Justify solutions

Developing problem solving strategies

When important or especially useful strategies crop up, they should be identified, highlighted, and discussed. The following strategies are most likely to appear in lessons we mathematical content is the main objective.

Draw a picture, act it out, use a model.

Look for a pattern

Make a table or chart

Try a simpler form of the problem

Guess and check

Make an organized list

Metacognition

Metacognition refers to conscious monitoring (being aware of how and why you are doing something) and regulation (choosing to do something or deciding to make changes) of your own thought process. Students who learn to monitor and regulate their own problem solving behaviour do show improvement in problem solving.The metacognitive goal is to monitor and regulate actions to help students develop the habit and ability to monitor and regulate their strategies and progress as they solve problems.It is important to help students to learn to monitor and control their own progress in problem solving. A simple formula that can be employed consists of three questions: What are you doing? Why are you doing it?

How does it help you?

Disposition

Refers to the attitudes and beliefs that students posses about mathematics.

Attitudinal goals are:

Gain confidence and belief in abilities

Be willing to take risks and to persevere

Enjoy doing mathematics

CHAPTER 5

PLANNING IN THE PROBLEM BASED CLASSROOM

Planning a problem-based lessonChoices of tasks and how they are presented to students must be made daily to best fit the needs of your students and the objectives you are hired to teach.

Step 1: Begin with the math! Articulate clearly the ideas you want students to learn in terms of mathematical concepts.

Step 2: Consider your students. What they already know, and what is needed for them to build on that knowledge.

Step 3: Decide on a task. Keep it simple

Step 4: Predict what will happen.

Step 5: Articulate student responsibilities

Step 6: Plan the before portion of the lesson

Step 7: Plan the during portion of the lesson

Step 8: Plan the after portion of the lesson

Step 9: Write your lesson plan

Variations of the three-part lesson

Minilessons

A profitable strategy for short tasks in think-pair-share.

Workstations and games

It is often useful for students to work at different tasks or games at various locations around the room.

Dealing with diversity

Be sure that problems have multiple entry points.

Plan differentiated tasks plan a task with multiple versions; some less difficult, others more so.

Use heterogenous groupings capitalise on the diversity in your classroom.

Make accommodations and modifications for English language learners an accommodation is a provision of different environment or circumstances made with particular students in mind; a modification refers to a change in the problem or task itself. ( Equity is not the same as equality we must create accommodations that will help each child be successful)

Listen carefully to students

Drill or practice?

New definitions of Drill and Practice

Practice different problem based tasks or experiences, spread over numerous class periods, each addressing the same basic ideas.Drill repetitive, non-problem-based exercises designed to improve skills or procedures already acquired.

What drill provides

An increased facility with a strategy but only with a strategy already learned

A focus on a singular method and an exclusion of flexible alternatives

A false appearance of understanding

A rule oriented view of what maths is all about

Drill can only help students get faster at what they already know.

What practice provides

An increased opportunity to develop conceptual ideas and more elaborate and useful connections

An opportunity to develop alternative and flexible strategies

A greater chance for all students to understand, not just a few

A clear message that mathematics is about figuring things out and making sense.

When is drill appropriate?

Drill is only appropriate when:

An efficient strategy for the skill to be drilled is already in place

Automaticity with the skill or strategy is a desired outcome.

Automaticity means that the skill can be performed quickly and mindlessly.

Kids who dont get it

A conceptual approach is the best way to help students who struggle. Drill is simply not the answer.

Homework

Practice as homework

A problem-based task can be assigned for homework if the difficulty of the task is within the reach of most of the students.

On the follow day, begin immediately with a discussion of the task.

Some form of written work must be required so that students are held responsible for the task and are prepared for the class discussion.Drill as homework

Never assign drill as a substitute for practice before the concepts have been developed. If you do assign drill for homework;

Keep it short

Provide an answer key

Never grade homework based on correctness

Do not waste valuable classroom time going over drill homework

The role of the textbook

Suggestions for textbook use

Teach to the big ideas or concepts, not the pages

Consider the conceptual portions of the lessons as ideas or inspirations for planning more problems-based activities.

Let the pace of your lessons through a unit be determined by student performance and understanding rather than the artificial norm of two pages per day.

Use the ideas in the teachers edition

Remember there is no law saying every page must be done or every exercise completed.

CHAPTER 6

BUILDING ASSESSMENT INTO INSTRUCTION

Blurring the line between instruction and assessmentAssessment:

Should enhance students learning

Is a valuable tool for making instructional decisions.

What is assessment?

Assessment is the process of gathering evidence about a students knowledge of, ability to use and disposition towards mathematics and of making inferences from that evidence for a variety of purposes.

-Assessment can and should happen every day as an integral part of instruction.

The Assessment Standards

The mathematics standard assessment should reflect the mathematics that all students need to know and be able to do.The learning standard assessment should enhance mathematics and learning.

The equity standard assessment should promote equity The equity standard mandates that assessments respect the unique qualities, experiences, and expertise of every student.

The openness standard assessment should be an open process

The openness standard reminds us that students need to know what is expected of them and how they can demonstrate what they know.

The inferences standard assessment should promote their lead inferences about mathematics learning

The inferences standard requires that teachers reflect seriously and honestly on what students are revealing about what they know.

The coherence standard assessment should be a coherent a process

The coherence standard reminds us that our assessment techniques must reflect both the objectives of instruction as well as the methods of instruction.

Purposes of assessment

Monitoring student progressassessment should provide both teacher and students with ongoing feedback concerning progress towards those goals. This promotes growth.

Making instructional decisions teachers planning tasks each day to develop students understanding must have daily information about how students are thinking and what ideas they are using and developing. This improves instruction. Evaluating student achievement evaluation involves a teachers judgement. It should reflect performance criteria about what students know and understand; it should not be used to compare one student with another. This recognizes accomplishment.

Evaluating programmes assessment should be used as one component in answering the question, how well did this programme worked to achieve my goals?What should be assessed?

Appropriate assessment should reflect the full range of mathematics: concepts and procedures, mathematical processes, and even students disposition to mathematics.

Assessment tasks are learning tasks

Good tasks should permit every student in the class, regardless of mathematical prowess, to demonstrate some knowledge, skills, or understanding.

Rubrics and performance indicators: scoring not grading

Scoring comparing students work to criteria or rubrics that describe what we expect the work to be.

Grading the result of accumulating scores and other information about the students work for the purpose of summarising and communicating to others.

Rubric a framework that can be designed or adapted by the teacher for a particular group of students or a particular mathematical task. It consists of a scale of 3 to 6 points that is used as a rating of performance rather than a count of how many items are correct or incorrect. The rating or score is applied by examining total performance on a task as opposed to counting the number of items correct. Note that are rubric is a skill to judge performance on a single task, not a series of exercises.Performance indicators

Performance indicators are task-specific statements that describe what performance looks like at each level of the rubric and in so doing establish criteria for acceptable performance. A rubric and its performance indicators should focus you and your students on your goals.

Writing and journalsWriting is both a learning and an assessment opportunity.

The value of writing

When students write, they express their own ideas and use their own words and language. It is personal.

As an assessment tool writing provides a unique window to students thoughts and the way a student is thinking about an idea.

Journals

Journals are a place for students to write about such things as:

Their conceptual understandings and problem solving, including descriptions of ideas, solutions, and justifications of problems, graphs, charts, and observations.

Their questions concerning the current topic, an idea that they may need help with, or an idea they dont quite understand.

Their feelings about aspects of mathematics, their confidence in their understanding, or their fears of being wrong.

To grade journal writing defeats its purpose. It is essential, however, that you read and respond to journal writing.

Writing prompts and ideasStudents should always have a clear, a well defined purpose for writing in the journals. They need to know exactly what to write about and who the audience is, and they should be given a definite time frame within which to write.

Student self assessment

In a self-assessment, students make tell you:

How well they think they understand a piece of content.

What they believe or how they feel about some aspect of mathematics, perhaps what you are covering right now.

Tell your students why you are having them do this activity. Encourage them to be honest and candid.

Tests

Like all other forms of assessment, tests should reflect the goals of your instruction. Tests can be designed to find out what concepts students have and how the ideas are connected. Tests of procedural knowledge should go beyond just knowing how to perform an algorithm and should allow and require the student to demonstrate the conceptual basis for the process.

CHAPTER 16

DEVELOPING FRACTION CONCEPTS

Big ideas Fractional parts are equal shares or equal-sized portions of a whole or unit.

Fractional parts have special names that tell how many parts of that size are needed to make the whole.

The more fractional parts used to make a whole, the smaller the parts.

The denominator of a fraction indicates by what number the whole has been divided in order to produce the type of part under consideration. The numerator counts or tells how many of these fractional parts are under consideration.

Two equivalent fractions are two ways of describing the same amount by using different sized fractional parts.

Sharing and the concept of fractional parts

In constructing the idea of fractional parts of the whole, children eventually make the connections between the idea of fair shares and fractional parts, making sharing tasks a good place to begin the development of fractions.Sharing tasks

Generally posed in the form of a simple story problem.

Task difficulty changes with the numbers involved, the types of things to be shared, and the presence or use of a model.

Sharing tasks and fraction language

Children need to be aware of the components of fractional parts (1) the number of parts and (2) the equality of the parts.

Emphasise that the number of parts of the whole determine the name of the fractional parts or shares.

Models for fractionsModels can help students clarify ideas.

There are three types of models that are used for fractions:

Region or Area models circular pie pieces, paper grids

Length or Measurement models fraction strips, number lines, folded paper strips.

Set models the whole is understood to be a set of objects.

From fractional parts to fraction symbols

Top and bottom numbers

Top number the counting number. It tells how many shares or parts we have.

Bottom number this tells what is being counted, it tells what fractional part is being counted.

It is important to see the bottom number as the divisor and the top number as the multiplier.

Fraction number sense

Children should know about how big a particular fraction is and be able to tell easily which of two fractions is larger, this can be learned through the teaching of benchmarks.

Benchmarks of zero, one half and one

The most important reference points or benchmarks for fractions are 0, and 1.

Understanding why a fraction is close to one of these benchmarks is a good beginning for fraction number sense.CHAPTER 17

COMPUTATION WITH FRACTIONS

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