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 Probability The five process standards Problem solving Reasoning and proof Communication Connections Representation These refer to the process through which students should acquire and use mathematical knowledge. N.B SEE TABLE 1.1, PAGE 4 OF TEXT The Professional Standards for Teaching Mathematics Teachers must shift from a teacher-centred to a child-centred approach in their instruction. Five shifts in Classroom Environment The 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 1 Even with a hands on activity. Mathematics as a science of pattern and order Mathematics is the science of pattern and order. in music. and sociology. medicine. 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 environment CHAPTER 2 EXPLORING WHAT IT MEANS TO DO MATHEMATICS Traditional views of mathematics Traditional teaching. Pattern and order are found in commerce. It is strongly focused on problems centred teaching and learning approaches. still the predominant instructional pattern. Students rely on the teacher to determine if their answers are correct. 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. science. What does it mean to do mathematics? 2 . its ideas. and its applications and away from treating mathematics as a body of isolated concepts and procedures. in buildings. 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. The focus of the lesson is primarily on getting answers. Toward mathematical reasoning and away from mere memorising procedures Toward conjecturing. and problem solving and away from an emphasis on the mechanistic finding of answers Towards connecting mathematics. in art. Finding and exploring this regularity order and then making sense of it is what doing mathematics is all about. Mathematics discovers this order. It begins with problem-based situations. the traditional teacher is guiding students. manufacturing. inventing. makes sense of it. Science is a process of figuring things out or making sense of things. Although you may never have thought of it in quite this way. and uses it in a multitude of fascinating ways. The world is full of pattern and order: in nature. mathematics is a science of things that have a pattern of regularity and logical order. improving our lives and expanding our knowledge. telling them exactly how to use the materials in the prescribed manner. It also focuses on the concept that all students can and should learner mathematics. Doing mathematics includes deciding if an answer is correct or and why. teachers must believe in the student’s . knowing that they will not be ridiculed if they are wrong. No answer book In the real world of problem solving outside the classroom. What is basic in mathematics? The most basic idea in mathematics is that mathematics makes sense! Everyday students must experience that mathematics makes sense. there are no teachers with answers and no answer books. Students must come to believe that they are capable of making sense of mathematics. The focus is on students actively figuring things out. 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. It also includes being able to justify your reasoning to others. Teachers must stop teaching by telling and start letting students make sense of the mathematics they are learning. developing reasons and offering explanations. Students should feel comfortable taking risks. testing ideas and making conjectures. trust. The teacher’s role is to create this spirit of enquiry. 3 . and expectation.Doing mathematics = engaging in the science of pattern and order There is a time in a place for drill but drill should never come before understanding.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. To this end. or otherwise modified. children of creators of their own knowledge. The construction of an idea is almost certainly going to be different for every learner. It depends on the existence of appropriate ideas and on the creation of new connections. or touch—elements of our physical surroundings. Understanding is never an all or nothing proposition. or cognitive schemas. Understanding Understanding can be defined as a measure of the quality and quantity of connections that an idea has with existing ideas. Rote learning can be thought of as a weak construction. this is called a web of ideas. Sometimes the materials are our own thoughts and ideas. added to. 4 . actively thinking about all mentally working on an idea. the knowledge that we already possess. A new idea is constructed by using the ideas we already have. Children must be mentally active for learning to take place. The effort that must be supplied is active and reflective thought. The materials we act on to build understanding may be things we see. The more ideas used and the more connections made. They do not absorb ideas as teachers present them. useful cognitive networks are formed. The construction of ideas Children construct their own knowledge The tools we use to build understanding are our existing ideas. Construction in rote learning Rote knowledge will almost never contribute to a useful network of ideas. hear. Assimilation refers to the use of existing schemas to give meaning to experiences. the better we understand. If minds are not actively thinking. When mathematical ideas are used to create new mathematical ideas. Rather. As learning occurs. nothing happens. Constructing knowledge requires reflective thought. A network of connections between ideas is developed in the process. Accommodation is the process of altering existing ways of viewing things or ideas that contradict or do not fit into existing schemas.CHAPTER 3 DEVELOPING UNDERSTANDING IN MATHEMATICS The constructivist view of learning Constructivism rejects the notion that children are blank slates. even within the same environment or classroom. are both the product of constructing knowledge and the tools with which additional new knowledge can be constructed. Integrated networks. To construct and understand a new idea requires actively thinking about it. the networks are rearranged. The general principles of constructivism are based largely on Piaget’s processes of assimilation and accommodation. This is especially true when new information connects with ideas already possessed. If what you need to recall seems distant. children must be mentally engaged. so is problem solving. There is less to remember – constructivist to talk about teaching big ideas which are really just large networks of interrelated concepts. they increase the potential for invention It improves attitudes and beliefs – when ideas are well and is didn’t make sense. 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. But even the most skilful use of a procedure will not help develop conceptual knowledge that is related to that procedure. thus. Algorithmic procedures help us do routine tasks easily and. They must find the relevant ideas they possess and bring them to bear on the development of the new idea. there is much less chance that the information will deteriorate. Connected information provides an entire web of ideas to reach full. Procedural knowledge is knowledge of the rules and procedures used in carrying out routine mathematical tasks and also the symbolism used to represent mathematics. reflecting on ideas that are related can usually lead you to the desired the idea eventually. The new knowledge make sense. It enhances memory – when mathematics is learned relationally. It improves problem solving abilities – the solution of novel problems requires transferring ideas learned in one context to new situations. transferability is significantly enhanced and. Procedural rules should never be taught in the absence of concepts. At one extreme is a very rich set of connections. Interaction of conceptual and procedural knowledge Procedural knowledge of mathematics does have a very important role both in learning and in doing mathematics. the learner tends to develop a positive self concept about his or her ability to learn and understand mathematics. Classroom influences on learning Effective teachers must help students construct their own ideas using ideas they already have. All mathematics procedures can and should be connected to the conceptual ideas that explain why they work.One way that we can think about and individuals understanding is that it exists along a continuum. it fits. it feels good. Concepts and procedures Conceptual and procedural knowledge Conceptual knowledge is knowledge that consists of rich relationships or webs of ideas. 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. Note that knowledge learnt by rote is at the isolated end of the continuum. It helps with learning new concepts and procedures – an idea fully understood in mathematics is more easily extended to learn a new idea. When concepts are embedded in originate work. free our minds to concentrate on more important tasks. 5 . it is instrumental knowledge that is learned without meaning. thus. Benefits of relational understanding It is intrinsically rewarding – nearly all people enjoyed learning. It is self generative – as networks grow and become more structured. The two ends of this continuum are relational understanding – the rich interconnected web of ideas—and instrumental understanding – ideas that are isolated and essentially without meaning. Be understood idea is associated with many other existing ideas in a meaningful network of concepts and procedures. Classroom discussion based on students’ own ideas and solutions to problems is absolutely foundational to children’s learning. Internalisation only occurs within each learner zone of proximal development. 4. A child would need to know the relationship before imposing it on the marble. Mathematical communities of learners The 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. Children who have difficulty translating a concept from one representation to another are the same children have difficulty solving problems and understanding computations. tester toys. The calculator models a wide variety of numeric relationships by quickly and easily demonstrating the effects of these ideas. picture. 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. Expanding the idea of a model There are five representations for concepts. These representations are: Pictures Written symbols Oral language 6 . Students learning from others – reflective thought and. you must have some relationship in your mind to impose on the model. are certainly important tools for helping children learn mathematics. no matter whose ideas they are. It is important to include calculators in the near list of common models. Models for mathematical concepts The model for a mathematical concept refers to any object. Models and constructing mathematics To see a concept in a model. They can be thought of as a thinker toys. and reason with. personal constructs. He referred to the transfer of ideas from those that are external to the individual – ideas exchanged in the social setting—to those that are internal. Ideas are important. 2. or physical materials to model mathematical concepts. Models can play the role of a testing ground for emerging ideas. hence. explore with. Models should always be accessible for students to select and use freely. Students must come to understand the mathematics makes sense. 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. a symbolic space created through the interaction of learners with more knowledge of all others and the culture that precedes them. as internalisation. Trust must be established with an understanding that it is OK to make mistakes. The teacher already has the correct mathematical concept and can see it in the model. and talker toys. Ideas must be shared with the others in the class. Do not for students to use a particular model. A student without the concept sees only the physical object or perhaps an incorrect concept. The Role of Models in Developing Understanding Manipulatives. or drawing that represents the concept or onto which the relationship with a concept can be imposed. Models give learners something to think about. 3. talk about. a thought provoking activity is used as a vehicle of learning. if not all. A problem centred approach uses a non routine problem as a vehicle of learning. 7 . Effective teaching is a student centred activity. 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. The ideas explored in this chapter provide a foundation for making these decisions. Models for mathematical ideas help students explore and talk about mathematical ideas. The socio cultural environment of the mathematical community of learners interacts with and enhances students’ development of mathematical ideas. Problem solving as a principal instructional strategy Most. Real world situations Manipulative models Incorrect use of models The most widespread misuse of manipulative materials occurs when the teacher tells students. 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. 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. important mathematics concepts and procedures can best be taught through problem solving. “do as I do”. CHAPTER 4 TEACHING THROUGH PROBLEM SOLVING In the problem solving approach. Decisions are made as you plan lessons and minute to minute in the classroom. Teaching Developmentally Teaching involves decision-making. These ideas are: Children construct their own knowledge and understanding. nor is there a perception by students and that there is a specific correct solution method. Each of these approaches is known as a problem-based approach to teaching and learning. It must require justification and explanations for answers and methods. the ideas they will use to create new ones. A shift in thinking about mathematics instruction Teaching should begin with the ideas that children already have. Reflective thinking is the single most important ingredient for effective learning. Knowledge and understanding are unique for each learner. we cannot transmit ideas to passive learners. 2. Problem solving develops mathematical power. Mathematical ideas are the outcomes of the problem solving experience from rather than elements that must be taught before problem solving. Problem solving allows an entry point for a wide range of students. Children are learning mathematics by doing mathematics. Providing profitable or activity for students who finished quickly. Be sure that students understand the problem so that you will not need to clarify or explain to individuals later in the lesson. Problem solving provides on going assessment data that can be used to make instructional decisions. Get students mentally prepared to work on the problem and think about the previous knowledge they have therefore be most helpful. your students will work as a community of learners. This includes both how they will be working and what product you expect in addition to an answer. Problem solving develops a belief in students that they are capable of doing mathematics and that mathematics makes sense. Students learn mathematics as a result of solving problems. It is a lot of fun. help students succeed. Cautiously provide appropriate hints. The agendas for the after phase are: 1. Engage the class in productive discussion. 2. This is a time for observation and assessment – not teaching! 3. A three-part lesson format The before phase of the lesson There are three related agendas for the before phase of the lesson: 1. 2. Let go! Give students a chance to work with our tour guide and all direction. brainstorm solutions. Listen actively to your students. helping students work together as a community of learners. The after phase of the lesson In the after phase of the lesson. Evaluating methods and solutions is the duty of your students. and challenging various solutions to the problem all have just worked on. Listen actively without evaluation. 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. The value of teaching with problems Problem solving places the focus of the students’ attention on ideas and sense making. Summarise main ideas and identify problems for future exploration.(think-write-pairshare) 3. 3. A problem-based approach engages students so that there are fewer discipline problems. Take this second major opportunity to find out how students are thinking—how they are approaching the problem. 8 . Clarify your expectations to students before they begin working on the problem. there are clear agendas that you will want to attend to: 1.(Begin with the simple version of the task. 4. justifying. discussing. Be careful not to imply that you have the correct method of solving the problem. and inform parents. Designing and selecting effective tasks A task is effective when it helps students to learn the ideas you want them to learn. Teaching about problem solving Students need to be taught problem solving strategies and processes. create or find tasks in the text’s teacher notes and other resources that address the big ideas. 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. the essential mathematics in the chapter. Your textbook Good teachers use the text as a resource and as a basic guide to the curriculum. and identify the 2 to 4 big ideas. examine a chapter or unit from beginning to end. You can now do two things: adapt to the best or most important lessons in the chapter to a problem solving format. Strategies and processes Strategies for resolving problems are identifiable methods of approaching a task that are completely independent of the specific topic or subject matter. A written report is also a written record that remains when the lesson is finished. And To use a tradition of textbook. Strategy and process goals Develop problem analysis skills Developer and selectors strategies Justify solutions Developing problem solving strategies 9 . Good problems have multiple entry points Access to the problem by all students’ demands that there be multiple entry points—different places to get on the ramp – to reach solutions. Children’s literature Children so stories can be used to create reflective tasks at all grade levels. A written report is a rehearsal for the discussion period. Draw a picture. 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. 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 lesson Choices 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.When important or especially useful strategies crop up. Step 2: Consider your students. What they already know. The following strategies are most likely to appear in lessons we mathematical content is the main objective. Students who learn to monitor and regulate their own problem solving behaviour do show improvement in problem solving. Step 3: Decide on a task. 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 10 . Keep it simple Step 4: Predict what will happen. 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. use a model. highlighted. act it out. Step 1: Begin with the math! Articulate clearly the ideas you want students to learn in terms of mathematical concepts. and discussed. and what is needed for them to build on that knowledge. 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. It is important to help students to learn to monitor and control their own progress in problem solving. they should be identified. spread over numerous class periods. 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. non-problem-based exercises designed to improve skills or procedures already acquired. Use heterogenous groupings – capitalise on the diversity in your classroom. Dealing with diversity Be sure that problems have multiple entry points. 11 . others more so. Drill is simply not the answer. not just a few A clear message that mathematics is about figuring things out and making sense. some less difficult. Kids who don’t get it A conceptual approach is the best way to help students who struggle. 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. ( 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. 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. Drill – repetitive.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. a modification refers to a change in the problem or task itself. Plan differentiated tasks – plan a task with multiple versions. Make accommodations and modifications for English language learners – an accommodation is a provision of different environment or circumstances made with particular students in mind. each addressing the same basic ideas. Automaticity means that the skill can be performed quickly and mindlessly. 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. Some form of written work must be required so that students are held responsible for the task and are prepared for the class discussion. CHAPTER 6 BUILDING ASSESSMENT INTO INSTRUCTION Blurring the line between instruction and assessment Assessment: Should enhance students’ learning Is a valuable tool for making instructional decisions. What is assessment? Assessment is the process of gathering evidence about a student’s knowledge of. Drill as homework Never assign drill as a substitute for practice before the concepts have been developed. 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. Use the ideas in the teacher’s edition Remember there is no law saying every page must be done or every exercise completed. The Assessment Standards 12 . On the follow day. If you do assign drill for homework. 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. -Assessment can and should happen every day as an integral part of instruction. not the pages Consider the conceptual portions of the lessons as ideas or inspirations for planning more problemsbased activities. ability to use and disposition towards mathematics and of making inferences from that evidence for a variety of purposes. begin immediately with a discussion of the task. The rating or score is 13 .The mathematics standard – assessment should reflect the mathematics that all students need to know and be able to do. It should reflect performance criteria about what students know and understand. mathematical processes. 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. or understanding. 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. 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. “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. 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 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. and expertise of every student. Grading – the result of accumulating scores and other information about the students’ work for the purpose of summarising and communicating to others. to demonstrate some knowledge. regardless of mathematical prowess. and even students’ disposition to mathematics. Purposes of assessment Monitoring student progress—assessment should provide both teacher and students with ongoing feedback concerning progress towards those goals. 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. Assessment tasks are learning tasks Good tasks should permit every student in the class. This recognizes accomplishment. Rubric – a framework that can be designed or adapted by the teacher for a particular group of students or a particular mathematical task. Evaluating student achievement – evaluation involves a teacher’s judgement. This promotes growth. experiences. skills. This improves instruction. Evaluating programmes – assessment should be used as one component in answering the question. Rubrics and performance indicators: scoring – not grading Scoring – comparing students’ work to criteria or rubrics that describe what we expect the work to be. it should not be used to compare one student with another. Encourage them to be honest and candid. Writing prompts and ideas Students should always have a clear. 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. Their feelings about aspects of mathematics. They need to know exactly what to write about and who the audience is. including descriptions of ideas. students make tell you: How well they think they understand a piece of content. To grade journal writing defeats its purpose. Tests Like all other forms of assessment. however. solutions. their confidence in their understanding. Tests can be designed to find out what concepts students have and how the ideas are connected. and observations. What they believe or how they feel about some aspect of mathematics. Note that are rubric is a skill to judge performance on a single task. Their questions concerning the current topic. Student self assessment In a self-assessment. It is essential. The value of writing When students write. graphs. Journals Journals are a place for students to write about such things as: Their conceptual understandings and problem solving. 14 . As an assessment tool writing provides a unique window to students’ thoughts and the way a student is thinking about an idea. they express their own ideas and use their own words and language. Writing and journals Writing is both a learning and an assessment opportunity. and they should be given a definite time frame within which to write.applied by examining total performance on a task as opposed to counting the number of items correct. or their fears of being wrong. A rubric and its performance indicators should focus you and your students on your goals. It is personal. an idea that they may need help with. tests should reflect the goals of your instruction. not a series of exercises. Tell your students why you are having them do this activity. perhaps what you are covering right now. 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. charts. that you read and respond to journal writing. a well defined purpose for writing in the journals. and justifications of problems. or an idea they don’t quite understand. The numerator counts or tells how many of these fractional parts are under consideration. and the presence or use of a model. Set models – the whole is understood to be a set of objects. folded paper strips. Models for fractions Models can help students clarify ideas. 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. It tells how many shares or parts we have. 15 . Sharing tasks Generally posed in the form of a simple story problem. There are three types of models that are used for fractions: Region or Area models – circular pie pieces.CHAPTER 16 DEVELOPING FRACTION CONCEPTS Big ideas Fractional parts are equal shares or equal-sized portions of a whole or unit. Task difficulty changes with the numbers involved. paper grids Length or Measurement models – fraction strips. 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. making sharing tasks a good place to begin the development of fractions. Fractional parts have special names that tell how many parts of that size are needed to make the whole. Bottom number – this tells what is being counted. it tells what fractional part is being counted. children eventually make the connections between the idea of fair shares and fractional parts. The more fractional parts used to make a whole. the smaller the parts. It is important to see the bottom number as the divisor and the top number as the multiplier. From fractional parts to fraction symbols Top and bottom numbers Top number – the counting number. The denominator of a fraction indicates by what number the whole has been divided in order to produce the type of part under consideration. number lines. the types of things to be shared. Emphasise that the number of parts of the whole determine the name of the fractional parts or shares. Understanding why a fraction is close to one of these benchmarks is a good beginning for fraction number sense. this can be learned through the teaching of benchmarks. one half and one The most important reference points or benchmarks for fractions are 0.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. CHAPTER 17 COMPUTATION WITH FRACTIONS 16 . ½ and 1. Benchmarks of zero.