Share your Opportunity. Advertise with us. Submit an Article. Forgot Your password? Academic Conferences. Scholarships and Fellowships. Exchange Programs. Competitions and Awards. Model United Nations. College Students. High School Students. For example,. Another example is a multiple-choice problem in which students were asked to estimate The choices were 1, 2, 19, and Fifty-five percent of the year-olds chose either 19 or 21 as the correct response. Simply observing that and are numbers less than one and that the sum of two numbers less than one is less than two would have made it apparent that 19 and 21 were unreasonable answers.
This level of performance is especially striking because this kind of reasoning does not require procedural fluency plus additional proficiency. In many ways it is less demanding than the computational task and requires only that basic understanding and reasoning be connected. It is clear that for many students that connection is not being made. A second kind of item that measures adaptive reasoning is one that asks students to justify and explain their solutions.
One such item Box 4—5 required that students use subtraction and division to justify claims about the population growth in two towns. The results were only slightly better at grade Students apparently have trouble justifying their answers even in relatively simple cases. Research related to productive disposition has not examined many aspects of the strand as we have defined it. Such research has focused on attitudes.
Box 4—5 Population Growth in Two Towns. In the populations of Town A and Town B were 5, and 6,, respectively. The populations of Town A and Town B were 8, and 9,, respectively. Brian claims that from to the populations of the two towns grew by the same amount. Use mathematics to explain how Brian might have justified his claim. Darlene claims that from to the population of Town A grew more.
Use mathematics to explain how Darlene might have justified her claim. Cited in Wearne and Kouba, , p. Used by permission of National Council of Teachers of Mathematics. In general, U. The complex relationship between attitudes and achievement is well illustrated in recent international studies.
Although within most countries, positive attitudes toward mathematics are associated with high achievement, eighth graders in some East Asian countries, whose average achievement in mathematics is among the highest in the world, have tended to have, on average, among the most negative attitudes toward mathematics. In Asian countries, perhaps because of cultural traditions encouraging humility or because of the challenging curriculum they face, eighth graders tend to perceive themselves as not very good at mathematics.
In the United States, in contrast, eighth graders tend to believe that mathematics is not especially difficult for them and that they are good at it. The data do not indicate, however, whether the students thought they could make sense out of the mathematics themselves or depended on others for explanations. Despite the finding that many students associate mathematics with memorization, students at all grade levels appear to view mathematics as useful.
Although students appear to think mathematics is useful for everyday problems or important to society in general, it is not clear that they think it is important for them as individuals to know a lot of mathematics.
Although our discussion of mathematical proficiency in this report is focused on the domain of number, the five strands apply equally well to other domains of mathematics such as geometry, measurement, probability, and statistics.
Regardless of the domain of mathematics, conceptual understanding refers to an integrated and functional grasp of the mathematical ideas. These may be ideas about shape and space, measure, pattern, function, uncertainty, or change. When applied to other domains of mathematics, procedural fluency refers to skill in performing flexibly, accurately, and efficiently such procedures as constructing shapes, measuring space, computing probabilities, and describing data.
It also refers to knowing when and how to use. The five strands apply equally well to other domains of mathematics such as geometry, measurement, probability, and statistics.
Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them whether the problems arise in the context of number, algebra, geometry, measurement, probability, or statistics. Similarly, the capacity to think logically about the relationships among concepts and situations and to reason adaptively applies to every domain of mathematics, not just number, as does the notion of a productive disposition.
The tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics applies equally to all domains of mathematics. We believe that proficiency in any domain of mathematics means the development of the five strands, that the strands of proficiency are interwoven, and that they develop over time.
Further, the strands are interwoven across domains of mathematics in such a way that conceptual understanding in one domain, say geometry, supports conceptual understanding in another, say number. Becoming mathematically proficient is necessary and appropriate for all students. People sometimes assume that only the brightest students who are the most attuned to school can achieve mathematical proficiency.
Those students are the ones who have traditionally tended to achieve no matter what kind of instruction they have encountered. But perhaps surprisingly, it is students who have historically been less successful in school who have the most potential to benefit from instruction designed to achieve proficiency.
Historically, the prevailing ethos in mathematics and mathematics education in the United States has been that mathematics is a discipline for a select group of learners. The continuing failure of some groups to master mathematics—including disproportionate numbers of minorities and poor students—has served to confirm that assumption.
More recently, mathematics educators have highlighted the universal aspects of mathematics and have insisted on mathematics for all students, but with little attention to the differential access that some students have to high-quality mathematics teaching. One concern has been that too few girls, relative to boys, are developing mathematical proficiency and continuing their study of mathematics. That situation appears to be improving, although perhaps not uniformly across. The and NAEP assessments indicated that the few gender differences in mathematics performance that did appear favored male students at grade 12 but not before.
These differences were only partly explained by the historical tendency of male students to take more high school mathematics courses than female students do, since that gap had largely closed by With regard to differences among racial and ethnic groups, the situation is rather different.
The strong connection between economic advantage, school funding, and achievement in the United States has meant that groups of students whose mathematics achievement is low have tended to be disproportionately African American, Hispanic, Native American, students acquiring English, or students located in urban or rural school districts.
Over the same period, African American and Hispanic students recorded increases at grades 4 and 12, but not at grade 8. The mathematics achievement gaps between average scores for these subgroups did not decrease in Students identified as being of middle and high socioeconomic status SES enter school with higher achievement levels in mathematics than low-SES students, and students reporting higher levels of parental education tend to have higher average scores on NAEP assessments.
At all three grades, in contrast, students eligible for free or reduced-price lunch programs score lower than those not eligible. Goals for mathematics instruction like those outlined in our discussion of mathematical proficiency need to be set in full recognition of the differential access students have to high-quality mathematics teaching and the differential performance they show.
Those goals should never be set low, however, in the mistaken belief that some students do not need or cannot achieve proficiency. In this day of rapidly changing technologies, no one can anticipate all the skills that students will need over their lifetimes or the problems they will encounter. Proficiency in mathematics is therefore an important foundation for further instruction in mathematics as well as for further education in fields that require mathematical competence.
Schools need to prepare students to acquire new skills and knowledge and to adapt their knowledge to solve new problems. The currency of value in the job market today is more than computational competence. It is the ability to apply knowledge to solve problems. They need to be able to learn new concepts and skills. They need to be able to apply mathematical reasoning to problems.
They need to view mathematics as a useful tool that must constantly be sharpened. In short, they need to be mathematically proficient. Students who have learned only procedural skills and have little understanding of mathematics will have limited access to advanced schooling, better jobs, and other opportunities. If any group of students is deprived of the opportunity to learn with understanding, they are condemned to second-class status in society, or worse.
Many people in the United States consider procedural fluency to be the heart of the elementary school mathematics curriculum. They remember school mathematics as being devoted primarily to learning and practicing computational procedures. In this report, we present a much broader view of elementary and middle school mathematics.
We also raise the standard for success in learning mathematics and being able to use it. In a significant and fortuitous twist, raising the standard by requiring development across all five strands of mathematical proficiency makes the development of any one strand more feasible. Because the strands interact and boost each other, students who have opportunities to develop all strands of proficiency are more likely to become truly competent with each. We conclude that during the past 25 years mathematics instruction in U.
It has developed some procedural fluency, but it clearly has not helped students develop the other strands very far, nor has it helped them connect the strands. Consequently, all strands have suffered. We consider not just performance levels but also the nature of the learning process itself.
We describe what students are capable of, what the big obstacles are for them, and what knowledge and intuition they have that might be helpful in designing effective learning experiences. This information, we believe, reveals how to improve current efforts to help students become mathematically proficient. See Hiebert and Carpenter, , for a discussion of the ways that cognitive science informs mathematics education on the nature of conceptual understanding.
For views about learning in general, see Bransford, Brown, and Cocking, ; Donovan, Bransford, and Pellegrino, For discussion of learning in early childhood, see Bowman, Donovan, and Burns, For a broader perspective on classrooms that promote understanding, see Fennema and Romberg, See, for example, Hiebert and Carpenter, , pp.
For work in psychology, see Baddeley, ; Bruner, , pp. See, for example, Nunes, a, b; Saxe, It should be emphasized that, as discussed above, conceptual understanding requires that knowledge be connected.
Bransford, Brown, and Cocking, , pp. For each of the five levels in the stack of blocks, there are two options: red or green. Similarly, for each of the five toppings on the hamburger, there are two options: include the topping or exclude it. The connection might be made explicit as follows: Let each level in the stack of blocks denote a particular topping e. English, a, p. See English, b, for an extended discussion of these ideas. Alexander, White, and Daugherty, , propose these three conditions for reasoning in young children.
There is reason to believe that the conditions apply more generally. The term disposition should not be taken to imply a biological or inherited trait. See, for example, Stevenson and Stigler, Other researchers claim that Asian children are significantly more oriented toward ability than their U.
For evidence that U. Cobb, Yackel, and Wood, , For a more general discussion of classroom norms, see Cobb and Bauersfeld, ; and Fennema and Romberg, Steele, ; and Steele and Aronson, , show the effect of stereotype threat in regard to subsets of the GRE Graduate Record Examination verbal exam, and it seems this phenomenon may carry across disciplines. A recent synthesis by Rittle-Johnson and Siegler, , on the relationship between conceptual and procedural knowledge in mathematics concludes that they are highly correlated and that the order of development depends upon the mathematical content and upon the students and their instructional experiences, particularly for multidigit arithmetic.
See Leder, , and Fennema, , for summaries of the research. In NAEP, gender differences may have increased slightly at grade 4 in the past decade, although they are still quite small; see Ansell and Doerr, For a review of the literature on race, ethnicity, social class, and language in mathematics, see Secada, Alexander, P. Analogical reasoning and early mathematics learning. English Ed. Mahwah, NJ: Erlbaum. Alibali, M. How children change their minds: Strategy change can be gradual or abrupt.
Developmental Psychology , 35 , — Ansell, E. NAEP findings regarding gender: Achievement, affect, and instructional experiences. Kenney Eds. Backer, A. Every child can succeed: Reading for school improvement. Baddeley, A. The psychology of memory. New York: Basic Books. Ball, D. Making believe: The collective construction of public mathematical knowledge in the elementary classroom.
Phillips Ed. Chicago: University of Chicago Press. Beaton, A. Bempechat, J. Review of Educational Research , 69 3 , — Bowman, B. Eager to learn: Educating our preschoolers. Bransford, J. How people learn: Brain, mind, experience, and school. Brownell, W. Psychological considerations in the learning and the teaching of arithmetic. Reeve Ed. AT classic: Meaning and skill—maintaining the balance. Arithmetic Teacher , 34 8 , 18— Original work published Bruner, J.
The process of education. New York: Vintage Books. Campbell, J. Carpenter, T. Results from the second mathematics assessment of the National Assessment of Educational Progress. Journal for Research in Mathematics Education 29 , 3— Teaching and learning mathematics with understanding.
Romberg Eds. Mahway, NJ: Erlbaum. Developing conceptions of algebraic reasoning in the primary grades. Cobb, P. The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale, NJ: Erlbaum. Adams Eds. New York: Springer-Verlag. The teaching experiment classroom. Bauersfeld Eds. Connecting students to a changing world: A technology strategy for improving mathematics and science education: A statement.
New York: Author. Davis, R. How students think: The role of representations. Donovan, M. How people learn: Bridging research and practice. Druckman, D. Dweck, C. Motivational processes affecting learning. American Psychologist , 41 , — English, L. Analogies, metaphors, and images: Vehicles for mathematical reasoning. Mathematical reasoning: Analogies, metaphors, and images. Fennema, E. Mathematics, gender and research.
Hanna Eds. Lund, Sweden: Lund University Press. Mathematics classroom that promote understanding. Fuson, K. Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction 7 , — Your prime time is your money. Avoid Interruptions — Do not answer your phone during certain time blocks in your plan.
Do not allow email pop-ups while working at your computer. More on interruptions in the next Chapter. Schedule Reactive Activity — Set appointments during the day when you will read and respond to email, when you will answer telephones, when you will take interruptions! Have A Plan B — This might be one the greatest time savers you will ever use.
No matter how well you plan, things happen that you will have no control over, events or occurrences that suddenly give you unplanned time. Your appointment might not show, you could be kept waiting for an appointment, or an employee could be late. Keep productive activities available as a plan B to substitute at any given time. Keep Meeting Times — Start meetings promptly and always give the ending time the same precise attention. Follow meeting plans. Be Realistic — Be as accurate as possible about the amount of time you allot for activities.
Break Yourself In — While most SME owners are optimistic with great expectations, if you are not used to following a schedule, start slow. Leave yourself those times during the day when you need to think or get away.
Make it a Team Effort and Inform Others — Make sure employees, clients, vendors, co-workers, and other associates know you are going to be following a new schedule. You might be surprised how many people will help you protect your plan. Create Expectations — Train others about what they should expect from you. Let them know a sense of urgency exists in business. Show up early, start meetings and appointments on time, get to the point, and conclude and leave on time.
Respect Others — Showing up late or keeping people waiting is disrespectful. Time is the very substance of life. Keeping people waiting for appointments and meetings, does not show you are important. It shows you are impolite. If you want people to respect your time, start by respecting theirs. Be Generous - Give Others Specific Time — Give people regular and specific times when you are available for impromptu meetings or conversations. Be strict but give them time to adjust to your schedule.
This does not mean you should surrender to any negative time wasters. Blame it on the Clock! Set goals and keep track of whether or not you are accomplishing them. Use organizational tools to help you complete your goals.
Determine what tasks must be done today, what your long-term deadlines are, and what things need to be accomplished in the future. Reevaluate this list as necessary. Always leave time in your schedule for the unexpected.
Twenty to thirty hours of your week should be devoted to specific tasks. However, attempt to leave around ten hours free for the unexpected.
You never know when you will need more time on a project than you expect or when a personal issue may arise that must be addressed. In order to choose tasks you must be aware of as many chores and projects as possible. In order to do this, every planning session must have a list. You Must Have a List! List all your tasks, then rate them, and list them again in order. Then you can schedule them. Every task you do should first have to pass this benchmark.
You have goals, priorities, and objectives. Does every task contribute to your big picture? Estimate how much time each task will take, then imagine what you would do with the time if the task were cancelled. While not always possible, everything you do should contribute to your objectives. Why is the Task Urgent? While urgency should be a mindset of business, urgency should also be questioned — ruthlessly. Is the urgency only appeasing someone else? What has caused the urgency? Many urgent situations have been caused by mistakes.
Determining the cause of urgency can eliminate or postpone a task and lead to prevention measures of interruptions and mistakes. Customers might be making demands that are unnecessary. Check with all parties involved. The Delegation Qualifier Are you the only person that can handle the task? Sometimes you might be, but many times someone else can perform for you. Delegate everything possible to free up your schedule. Are You Utilizing Technology Could an in person appointment be a phone call?
Conference calling can rule out travel and save an enormous amount of time. Could you email instead of calling? Email can be done on your terms when you want. You will have time to articulate better compared to the live telephone conversation. Time can be wasted and sales lost by leaving phone messages for people. Email eliminates phone tag. Can the Task Be Dissected? Are there portions of the work that can be delegated, eliminated, or postponed? There are many jobs throughout the day that are actually not worth the time to do.
Applying the dollar figure when considering cancelling a task is another measure of the task value. Time — How much time will it take? Effect — Completed versus Cancelled Effectiveness — What is the most effective way to perform the task? Contribution to Your Objectives Replacement — What could be done with the time instead? One great trick for prioritizing is to give every task a deadline. Give tasks a deadline. Having a clear deadline makes tasks easier to prioritize.
Prioritizing while planning is easy. You have time to think. Prioritizing while working is a bit more challenging. Prioritizing Interruptions While most people are familiar with prioritizing tasks, few people prioritize their interruptions. Hence, few people have defined the types of events that interrupt them.
Define Interruption Types In order to take control of your time, you must minimize interruptions. While having a job definition for yourself is a great start, most owners have not defined or classified these problems.
They just catch every ball tossed at them. The tail is often wagging the dog. Make a list of every kind of interruption you have experienced in the last three months.
Define to them what an emergency is by making a list of specific examples. Be very clear as the word emergency is open to broad definition. Some people classify an emergency only as fire or pending death. Others consider a paper jam in the copier a crisis. Then schedule appointments daily or weekly with each person, based on how often they presently communicate with you. If your manager is accustom to talking with you throughout the day, dedicate a time just for talking. If each person knows they have this appointed time, and a precise understanding of an emergency, they will be less likely to interrupt you.
While service to customers is an utmost concern, customers come to expect the type of service you provide. Their expectations have often been set by your business. They might be used to having free access to you by phone at any time, or having you respond instantly or within a certain amount of time. When planning your week, if you set aside a time each day or as needed, to respond to telephone calls and emails, let your clients or customers know. Define for yourself what client emergencies are and which ones you will respond to.
Family You can do the same thing with family members. If your spouse has a habit of communicating during your OUT, discuss emergency situations and set appointments at usual intervals to talk.
Devices Most devices that interrupt us are actually human interruptions. The good news here is we are in control of these devices. They can all be temporarily shut off. Processes Select the interruptions from your list that are process oriented. Now divide then into preventable and unpreventable.
Many process interruptions can be prevented. Running short on inventory, running out of supplies, computer viruses, etc. With a bit of planning and implementing new measures, you should be able to eliminate most of these interruptions types. Some process interruptions are not preventable.
Electrical outages, equipment breakdowns, cash flow shortages, etc. Most of us spend countless number of hours on repeating tasks we have already done, reacting to situations that should never have arisen, and ranting about how much we have to do and what little time we have to do them in.
In order to figure out where all our time is going we have to become aware of our actions. Keep a journal of what you do with the time you spend at work for at least one week. This can be extremely tedious and seem to take up even more of your precious time, but in order to improve time management we have to become aware of where our time goes. Look for time stealers. Time stealers are anything that reduces your effectiveness in the workplace.
Where does your time go? The first step in sorting out your time management problems is to identify where your time goes — keep a log of what you do and when you do it.
Complete that for a week or two. After you have compiled your log, take a look at it and identify the most frequent time stealers that reduce your effectiveness in the workplace. You can reclaim your time! Here are 10 techniques and strategies that you can use to manage your time more effectively. Make a decision on where you want to focus your energy and then look at everything else you have been doing and decide what to do with it.
Could you outsource some of your tasks and operation to a third party outsider? Can you automate any tasks like getting email on your mobile phone? This will keep you focused on what is important. Continually revisit your strategy to make sure you keep on track and away from all of the distractions. Keep analyzing where you are spending your time on a monthly basis and make adjustments to what you should be doing and how you are doing it.
Now that you know what your particular time stealers are, how do you arrest them? In most cases, people have to come to terms with the fact that they cannot do everything but that they can do what is most important. Planning and goal setting is of the utmost importance. Most small businesses overbook projects to keep the income hopper full. Break your projects into parts and at times delegate these parts to colleagues. When you delegate a task, however, make sure you communicate clearly what results you desire, what the deadline is, and all the necessary instructions that the person will need to complete the project.
Make sure that you verify that the person understands of what is expected of them. As each part of the project is accomplished, recognize your successes and build on them. Revise them monthly. Stay aware of them daily. Keep a list where you will see them daily. Keep information in trusted system where you can organize and classify it. You should have clear to do list for every workday. If they are harmful or useless, get rid of them.
If anything must be postponed — reschedule immediately. Be optimistic and positive about your life. Meetings In a recent survey, sixty two per cent of business owners and personnel listed meetings as the number one source of wasted time. Send everyone a copy before they attend. This is where anyone who wants—IBMers, partners, clients, product owners, and others—can come together to collaborate, ask questions, share knowledge, and support each other in their everyday work efforts.
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