Mathematical methods for science students pdf download

In recent offerings, students have written programs to simulate a model of housing segregation, determine the number of machines needed at a polling place, and analyze tweets from presidential debates. Instructor s : B. Sotomayor, B. This course meets the general education requirement in the mathematical sciences. Computer Science with Applications II. This course is the second in a three-quarter sequence that teaches computational thinking and skills to students in the sciences, mathematics, economics, etc.

Lectures cover topics in 1 data representation, 2 basics of relational databases, 3 shell scripting, 4 data analysis algorithms, such as clustering and decision trees, and 5 data structures, such as hash tables and heaps. Applications and datasets from a wide variety of fields serve both as examples in lectures and as the basis for programming assignments. In recent offerings, students have written a course search engine and a system to do speaker identification.

Students will program in Python and do a quarter-long programming project. Instructor s : A. Rogers, M. The course revolves around core ideas behind the management and computation of large volumes of data "Big Data". Topics include 1 Statistical methods for large data analysis, 2 Parallelism and concurrency, including models of parallelism and synchronization primitives, and 3 Distributed computing, including distributed architectures and the algorithms and techniques that enable these architectures to be fault-tolerant, reliable, and scalable.

Students will continue to use Python, and will also learn C and distributed computing tools and platforms, including Amazon AWS and Hadoop. This course includes a project where students will have to formulate hypotheses about a large dataset, develop statistical models to test those hypotheses, implement a prototype that performs an initial exploration of the data, and a final system to process the entire dataset.

Instructor s : M. Introduction to Data Engineering. Data-driven models are revolutionizing science and industry. Scalable systems are needed to collect, stream, process, and validate data at scale. This course is an introduction to "big" data engineering where students will receive hands-on experience building and deploying realistic data-intensive systems.

It will cover streaming, data cleaning, relational data modeling and SQL, and Machine Learning model training. A core theme of the course is "scale," and we will discuss the theory and the practice of programming with large external datasets that cannot fit in main memory on a single machine.

The course will consist of bi-weekly programming assignments, a midterm examination, and a final. This sequence, which is recommended for all students planning to take more advanced courses in computer science, introduces computer science mostly through the study of programming in functional Scheme and imperative C programming languages. Topics include program design, control and data abstraction, recursion and induction, higher-order programming, types and polymorphism, time and space analysis, memory management, and data structures including lists, trees, and graphs.

NOTE: Non-majors may use either course in this sequence to meet the general education requirement in the mathematical sciences; students who are majoring in Computer Science must use either CMSC or to meet requirements for the major. Introduction to Computer Science I.

Non-majors may use either course in this sequence to meet the general education requirement in the mathematical sciences; students who are majoring in Computer Science must use either CMSC or to meet requirements for the major. Introduction to Computer Science II. Feldman, A. Introduction to Computer Systems. This course covers the basics of computer systems from a programmer's perspective.

Extensive programming required. Instructor s : H. Gunawi Spring , H. Hoffmann Spring , M. Note s : Required of students who are majoring in Computer Science. Both courses in this sequence meet the general education requirement in the mathematical sciences; students who are majoring in Computer Science must use either CMSC or to meet requirements for the major. Honors Introduction to Computer Science I. Basic data structures, including lists, binary search trees, and tree balancing.

Basic mathematics for reasoning about programs, including induction, inductive definition, propositional logic, and proofs. This course emphasizes the C Programming Language, but not in isolation.

Instead, C is developed as a part of a larger programming toolkit that includes the shell specifically ksh , shell programming, and standard Unix utilities including awk. Nonshell scripting languages, in particular perl and python, are introduced, as well as interpreter!

We cover various standard data structures, both abstractly, and in terms of concrete implementations-primarily in C, but also from time to time in other contexts like scheme and ksh. The course uses a team programming approach. There is a mixture of individual programming assignments that focus on current lecture material, together with team programming assignments that can be tackled using any Unix technology. Team projects are assessed based on correctness, elegance, and quality of documentation.

We teach the "Unix way" of breaking a complex computational problem into smaller pieces, most or all of which can be solved using pre-existing, well-debugged, and documented components, and then composed in a variety of ways. Instructor s : F. Introduction to Creative Coding. Introduction to Human-Computer Interaction. An introduction to the field of Human-Computer Interaction HCI , with an emphasis in understanding, designing and programming user-facing software and hardware systems.

This class covers the core concepts of HCI: affordances, mental models, selection techniques pointing, touch, menus, text entry, widgets, etc , conducting user studies psychophysics, basic statistics, etc , rapid prototyping 3D printing, etc , and the fundamentals of 3D interfaces optics for VR, AR, etc. Creating technologies that are inclusive of people in marginalized communities involves more than having technically sophisticated algorithms, systems, and infrastructure.

It involves deeply understanding various community needs and using this understanding coupled with our knowledge of how people think and behave to design user-facing interfaces that can enhance and augment human capabilities. When dealing with under-served and marginalized communities, achieving these goals requires us to think through how different constraints such as costs, access to resources, and various cognitive and physical capabilities shape what socio-technical systems can best address a particular issue.

This course leverages human-computer interaction and the tools, techniques, and principles that guide research on people to introduce you to the concepts of inclusive technology design. You will learn about different underserved and marginalized communities such as children, the elderly, those needing assistive technology, and users in developing countries, and their particular needs. In addition, you will learn how to be mindful of working with populations that can easily be exploited and how to think creatively of inclusive technology solutions.

You will also put your skills into practice in a semester long group project involving the creation of an interactive system for one of the user populations we study. Introduction to Robotics. This course gives students a hands-on introduction to robot programming covering topics including sensing in real-world environments, path planning, localization, kinematics, and decision making under uncertainty.

This course will be centered around four to five main problem sets exploring some of these central concepts to robot programming. Each of these problem sets will involve students programming real, physical robots interacting with the real world during the academic year, students will program robots in simulation due to covid restrictions.

Through this hands-on robot programming, students will be able to 1 see the result of their programs come to life in a physical environment and 2 gain experience facing and overcoming the challenges of programming robots e.

Over time, technology has occupied an increasing role in education, with mixed results. Massive Open Online Courses MOOCs were created to bring education to those without access to universities, yet most of the students who succeed in them are those who are already successful in the current educational model.

This course focuses on one intersection of technology and learning: computer games. This course covers education theory, psychology e. Labs focus on developing expertise in technology, and readings supplement lecture discussions on the human components of education. Instructor s : D. Mathematical Foundations. This course is an introduction to formal tools and techniques which can be used to better understand linguistic phenomena.

A major goal of this course is to enable students to formalize and evaluate theoretical claims. Creative Machines and Innovative Instrumentation. An understanding of the techniques, tricks, and traps of building creative machines and innovative instrumentation is essential for a range of fields from the physical sciences to the arts.

In this hands-on, practical course, you will design and build functional devices as a means to learn the systematic processes of engineering and fundamentals of design and construction. The kinds of things you will learn may include mechanical design and machining, computer-aided design, rapid prototyping, circuitry, electrical measurement methods, and other techniques for resolving real-world design problems. In collaboration with others, you will complete a mini-project and a final project, which will involve the design and fabrication of a functional scientific instrument.

The course will be taught at an introductory level; no previous experience is expected. The iterative nature of the design process will require an appreciable amount of time outside of class for completing projects. The course is open to undergraduates in all majors subject to the pre-requisites , as well as Master's and Ph. Data Science for Computer Scientists.

This course covers computational methods for structuring and analyzing data to facilitate decision-making. We will cover algorithms for transforming and matching data; hypothesis testing and statistical validation; and bias and error in real-world datasets. A core theme of the course is "generalization"; ensuring that the insights gleaned from data are predictive of future phenomena.

The course will include bi-weekly programming assignments, a midterm examination, and a final. Introduction to Software Development. Besides providing an introduction to the software development process and the lifecycle of a software project, this course focuses on imparting a number of skills and industry best practices that are valuable in the development of large software projects, such as source control techniques and workflows, issue tracking, code reviews, testing, continuous integration, working with existing codebases, integrating APIs and frameworks, generating documentation, deployment, and logging and monitoring.

The course also emphasizes the importance of collaboration in real-world software development, including interpersonal collaboration and team management. The course will be organized primarily around the development of a class-wide software project, with students organized into teams.

Collaboration both within and across teams will be essential to the success of the project. The course discusses both the empirical aspects of software engineering and the underlying theory. Topics will include, among others, software specifications, software design, software architecture, software testing, software reliability, and software maintenance. Students will be expected to actively participate in team projects in this course.

For mathematics study, the most useful languages are French, German, and Russian. In recent years graduating math majors have obtained employment in a variety of jobs in business, industry, and governmental agencies and also have obtained teaching positions at the secondary school level such teaching positions normally require teaching certification.

Departments in a variety of fields which use mathematics, including the social and biological sciences as well as in engineering and the physical sciences, are interested in attracting math majors into their graduate programs. The math major requirements are flexible enough to allow preparation for various goals. Contact Information Receptionist math. Honors Program honors math. U niversity of W isconsin —Madison. On This Page.

MATH A sequence of two upper-level mathematics courses deemed acceptable by the Mathematics Honors advisor 7. State, explain, and apply the principal results, definitions, and theorems of a wide collection of mathematical areas including at least one area of advanced undergraduate mathematics.

Construct and evaluate mathematical proofs and arguments. Use mathematics to model and analyze phenomena in other disciplines. Departmental Expectations Historically, students who have successfully complete a three year undergraduate degree with a major in Mathematics have the following qualifications: a minimum of 29 advanced standing credits, which include completion of the following with either course credit or via placement examination: MATH and MATH Communication Part A units of foreign language Therefore the plan below assumes these requirements, but none other.

In this case, students should adjust their plan by reorganizing the remaining degree requirements using the following priorities: 1 Ethnic Studies and Communication Part B obligatory in the first year ; 2 Physical, Biological, and Social Science Breadth which may be prerequisites for more advanced electives ; 3 Humanities and Literature.

Remaining schedule space should be considered electives. Biomedical Engineering, B. Civil Engineering, B. Engineering Computer Engineering, B. Engineering Electrical Engineering Technology, A. Engineering Electrical Engineering Technology, B.

Engineering Electrical Engineering, B. Engineering, B. Engineering Mechanical Engineering, B. Surveying Engineering, B. Print Options. Program Description Computer Science is the study of computation, including its principles and foundations, its efficient implementation, its analysis, and its practical use in a wide range of different application areas. What is Computer Science? Entrance to Major This program currently has administrative enrollment controls. General Education Connecting career and curiosity, the General Education curriculum provides the opportunity for students to acquire transferable skills necessary to be successful in the future and to thrive while living in interconnected contexts.

Foundations grade of C or better is required. Cultures Requirement 6 credits are required and may satisfy other requirements United States Cultures: 3 credits International Cultures: 3 credits Writing Across the Curriculum 3 credits required from the college of graduation and likely prescribed as part of major requirements. Total Minimum Credits A minimum of degree credits must be earned for a baccalaureate degree. Quality of Work Candidates must complete the degree requirements for their major and earn at least a 2.

Limitations on Source and Time for Credit Acquisition The college dean or campus chancellor and program faculty may require up to 24 credits of course work in the major to be taken at the location or in the college or program where the degree is earned. Requirements for the Major To graduate, a student enrolled in the major must earn a grade of C or better in each course designated by the major as a C-required course, as specified by Senate Policy CMPEN CMPSC EE PHYS Integrated B.

Program Educational Objectives In particular, within a few years after graduation, graduates in computer science should be able to: Apply appropriate theory, practices, and tools to the specification, design, implementation, maintenance and evaluation of both large and small software systems.

Stay current through professional conferences, certificate programs, post-baccalaureate degree programs, or other professional educational activities. Nonroutine problems require productive thinking because the learner needs to invent a way to understand and solve the problem. For example, for most adults a nonroutine problem of the sort often found in newspaper or magazine puzzle columns is the following:.

Since there are 80 wheels in all, the eight additional wheels 80—72 must belong to 8 tricycles. Another reduction of the number of tricycles by 4 gives 28 bikes, 8 tricycles, and the 80 wheels needed.

A more sophisticated, algebraic approach would be to let b be the number of bikes and t the number of tricycles. The solution to this system of equations also yields 28 bikes and 8 tricycles. A student with strategic competence could not only come up with several approaches to a nonroutine problem such as this one but could also choose flexibly among reasoning, guess-and-check, algebraic, or other methods to suit the demands presented by the problem and the situation in which it was posed.

Flexibility of approach is the major cognitive requirement for solving nonroutine problems. It can be seen when a method is created or adjusted to fit the requirements of a novel situation, such as being able to use general principles about proportions to determine the best buy.

For example, when the choice is between a 4-ounce can of peanuts for 45 cents and a ounce can for 90 cents, most people use a ratio strategy: the larger can costs twice as much as the smaller can but contains more than twice as many ounces, so it is a better buy. When the choice is between a ounce jar of sauce for 79 cents and an ounce jar for 81 cents, most people use a difference strategy: the larger jar costs just 2 cents more but gets you 4 more ounces, so it is the better buy. When the choice is between a 3-ounce bag of sunflower seeds for 30 cents and a 4-ounce bag for 44 cents, the most common strategy is unit-cost: The smaller bag costs 10 cents per ounce, whereas the larger costs 11 cents per ounce, so the smaller one is the better buy.

There are mutually supportive relations between strategic competence and both conceptual understanding and procedural fluency, as the various approaches to the cycle shop problem illustrate. The development of strategies for solving nonroutine problems depends on understanding the quantities involved in the problems and their relationships as well as on fluency in solving routine problems.

Similarly, developing competence in solving nonroutine problems provides a context and motivation for learning to solve routine problems and for understanding concepts such as given, unknown, condition, and solution. There are mutually supportive relations between strategic competence and both conceptual understanding and procedural fluency,.

Strategic competence comes into play at every step in developing procedural fluency in computation. As students learn how to carry out an operation such as two-digit subtraction for example, 86—59 , they typically progress from conceptually transparent and effortful procedures to compact and more efficient ones as discussed in detail in chapter 6.

For example, an initial procedure for 86—59 might be to use bundles of sticks see Box 4—3. A compact procedure involves applying a written numerical algorithm that carries out the same steps without the bundles of sticks.

Part of developing strategic competence involves learning to replace by more concise and efficient procedures those cumbersome procedures that might at first have been helpful in understanding the operation.

Begin with 8 bundles of 10 sticks along with 6 individual sticks. Because you cannot take away 9 individual sticks, open one bundle, creating 7 bundles of 10 sticks and 16 individual sticks. Take away 5 of the bundles corresponding to subtracting 50 , and take away 9 individual sticks corresponding to subtracting 9.

The number of remaining sticks—2 bundles and 7 individual sticks, or 27—is the answer. Students develop procedural fluency as they use their strategic competence to choose among effective procedures. They also learn that solving challenging mathematics problems depends on the ability to carry out procedures readily and, conversely, that problem-solving experience helps them acquire new concepts and skills. Interestingly, very young children use a variety of strategies to solve problems and will tend to select strategies that are well suited to particular problems.

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. Such reasoning is correct and valid, stems from careful consideration of alternatives, and includes knowledge of how to justify the conclusions. In mathematics, adaptive reasoning is the glue that holds everything together, the lodestar that guides learning. One uses it to navigate through the many facts, procedures, concepts, and solution methods and to see that they all fit together in some way, that they make sense.

In mathematics, deductive reasoning is used to settle disputes and disagreements. Answers are right because they follow from some agreed-upon assumptions through series of logical steps. Students who disagree about a mathematical answer need not rely on checking with the teacher, collecting opinions from their classmates, or gathering data from outside the classroom. In principle, they need only check that their reasoning is valid.

Many conceptions of mathematical reasoning have been confined to formal proof and other forms of deductive reasoning. Our notion of adaptive reasoning is much broader, including not only informal explanation and justification but also intuitive and inductive reasoning based on pattern, analogy, and metaphor.

After working in pairs and. In the context of cutting short bows from a meter package of ribbon and using physical models to calculate that 12 divided by is 36, fifth graders can reason that 12 divided by cannot be 72 because that would mean getting more bows from a package when the individual bow is larger, which does not make sense.

Proofs both formal and informal must be logically complete, but a justification may be more telegraphic, merely suggesting the source of the reasoning. Justification and proof are a hallmark of formal mathematics, often seen as the province of older students.

However, as pointed out above, students can start learning to justify their mathematical ideas in the earliest grades in elementary school. Classroom norms can be established in which students are expected to justify their mathematical claims and make them clear to others. Students need to be able to justify and explain ideas in order to make their reasoning clear, hone their reasoning skills, and improve their conceptual understanding.

It is not sufficient to justify a procedure just once. As we discuss below, the development of proficiency occurs over an extended period of time. Students need to use new concepts and procedures for some time and to explain and justify them by relating them to concepts and procedures that they already understand. For example, it is not sufficient for students to do only practice problems on adding fractions after the procedure has been developed.

If students are to understand the algorithm, they also need experience in explaining and justifying it themselves with many different problems. Adaptive reasoning interacts with the other strands of proficiency, particularly during problem solving. Learners draw on their strategic competence to formulate and represent a problem, using heuristic approaches that may provide a solution strategy, but adaptive reasoning must take over when.

Conceptual understanding provides metaphors and representations that can serve as a source of adaptive reasoning, which, taking into account the limitations of the representations, learners use to determine whether a solution is justifiable and then to justify it. Often a solution strategy will require fluent use of procedures for calculation, measurement, or display, but adaptive reasoning should be used to determine whether the procedure is appropriate. And while carrying out a solution plan, learners use their strategic competence to monitor their progress toward a solution and to generate alternative plans if the current plan seems ineffective.

This approach both depends upon productive disposition and supports it. Productive disposition refers to 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. Developing a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance, and to experience the rewards of sense making in mathematics.

A productive disposition develops when the other strands do and helps each of them develop. For example, as students build strategic competence in solving nonroutine problems, their attitudes and beliefs about themselves as mathematics learners become more positive. The more mathematical concepts they understand, the more sensible mathematics becomes.

In contrast, when students are seldom given challenging mathematical problems to solve, they come to expect that memorizing rather than sense making paves the road to learning mathematics, 41 and they begin to lose confidence in themselves as learners. Similarly, when students see themselves as capable of learning mathematics and using it to solve problems, they become able to develop further their procedural fluency or their adaptive reasoning abilities.

Students who view their mathematical ability as fixed and test questions as measuring their ability rather than providing opportunities to learn are likely to avoid challenging problems and be easily dis-. Cross-cultural research studies have found that U. Most U. It is critical that they encounter good mathematics teaching in the early grades. Otherwise, those positive attitudes may turn sour as they come to see themselves as poor learners and mathematics as nonsensical, arbitrary, and impossible to learn except by rote memorization.

The teacher of mathematics plays a critical role in encouraging students to maintain positive attitudes toward mathematics.

Teachers and students inevitably negotiate among themselves the norms of conduct in the class, and when those norms allow students to be comfortable in doing mathematics and sharing their ideas with others, they see themselves as capable of understanding.

An earlier report from the National Research Council identified the cause of much poor performance in school mathematics in the United States:. The unrestricted power of peer pressure often makes good performance in mathematics socially unacceptable. This environment of negative expectation is strongest among minorities and women— those most at risk—during the high school years when students first exercise choice in curricular goals.

Avoiding such courses may eliminate the need to face up to peer pressure and other sources of discouragement, but it does so at the expense of precluding careers in science, technology, medicine, and other fields that require a high level of mathematical proficiency. Research with older students and adults suggests that a phenomenon termed stereotype threat might account for much of the observed differences in mathematics performance between ethnic groups and between male and female students.

So-called wise educational environments 50 can reduce the harmful effects of stereotype threat. These environments emphasize optimistic teacher-student relationships, give challenging work to all students, and stress the expandability of ability, among other factors. Students who have developed a productive disposition are confident in their knowledge and ability.

They see that mathematics is both reasonable and intelligible and believe that, with appropriate effort and experience, they can learn. Hence, our view of mathematical proficiency goes beyond being able to understand, compute, solve, and reason. It includes a disposition toward mathematics that is personal. Mathematically proficient people believe that mathematics should make sense, that they can figure it out, that they can solve mathematical problems by working hard on them, and that becoming mathematically proficient is worth the effort.

Now that we have looked at each strand separately, let us consider mathematical proficiency as a whole. As we indicated earlier and as the preceding discussion illustrates, the five strands are interconnected and must work together if students are to learn successfully. Learning is not an all-or-none phenomenon, and as it proceeds, each strand of mathematical proficiency should be developed in synchrony with the others. That development takes time. One of the most challenging tasks faced by teachers in pre-kindergarten to grade 8 is to see that children are making progress along every strand and not just one or two.

How the strands of mathematical proficiency interweave and support one another can be seen in the case of conceptual understanding and procedural fluency. Current research indicates that these two strands of proficiency con-. In turn, as a procedure becomes more automatic, the child is enabled to think about other aspects of a problem and to tackle new kinds of problems, which leads to new understanding. When using a procedure, a child may reflect on why the procedure works, which may in turn strengthen existing conceptual understanding.

Consider, for instance, the multiplication of multidigit whole numbers. Familiarity with this algorithm may make it hard for adults to see how much knowledge is needed for it. As students learn to execute a multidigit multiplication procedure such as this one, they should develop a deeper understanding of multiplication and its properties.

On the other hand, as they deepen their conceptual understanding, they should become more fluent in computation. A beginner who has simply memorized the algorithm without understanding much about how it works can be lost later when memory fails. Mathematical proficiency cannot be characterized as simply present or absent. Every important mathematical idea can be understood at many levels and in many ways. For example, even seemingly simple concepts such as even and odd require an integration of several ways of thinking: choosing alternate points on the number line, grouping items by twos, grouping items into two groups, and looking at only the last digit of the number.

When children are first learning about even and odd, they may know one or two of these interpretations. The research cited in chapter 5 shows that schoolchildren are never complete mathematical novices. They bring important mathematical concepts and skills with them to school as well as misconceptions that must be taken into account in planning instruction. Students should not be thought of as having proficiency when one or more strands are undeveloped.

Proficiency in mathematics is acquired over time. Each year they are in school, students ought to become increasingly proficient. For example, third graders should be more proficient with the addition of whole numbers than they were in the first grade.

Acquiring proficiency takes time in another sense. Students need enough time to engage in activities around a specific mathematical topic if they are to become proficient with it. When they are provided with only one or two examples to illustrate why a procedure works or what a concept means and then move on to practice in carrying out the procedure or identifying the concept, they may easily fail to learn.

To become proficient, they need to spend sustained periods of time doing mathematics—solving problems, reasoning, developing understanding, practicing skills—and building connections between their previous knowledge and new knowledge.

One question that warrants an immediate answer is whether students in U. The answer is important because it influences what might be recommended for the future. If students are failing to develop proficiency, the question of how to improve school mathematics takes on a different cast than if students are already developing high levels of proficiency.

NAEP includes a large and representative sample of U. We sketched some of that performance in chapter 2 , but now we look at it through the frame of mathematical proficiency. Although the items in the NAEP assessments were not constructed to measure directly the five strands of mathematical proficiency, they provide some useful information about these strands.

As in chapter 2 , the data reported here are from the main NAEP assessment except when we refer explicitly to the long-term trend assessment. In general, the performance of year-olds over the past 25 years tells the following story: Given traditional curricula and methods of instruction, students develop proficiency among the five strands in a very uneven way.

They are most proficient in aspects of procedural fluency and less proficient in conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. Many students show few connections among these strands. Examples from each strand illustrate the current situation. The same is true for rational numbers.

An overall picture of procedural fluency is provided by the NAEP long-term trend mathematics assessment, 58 which indicates that U. A closer look reveals that the picture of procedural fluency is one of high levels of proficiency in the easiest contexts.

These scale scores include all content areas: number, geometry, algebra, and so on. Students are less fluent in operating with rational numbers, both common and decimal fractions. Again, this level of performance has remained quite steady since the advent of NAEP.

One conclusion that can be drawn is that by age 13 many students have not fully developed procedural fluency. Although most can compute well with whole numbers in simple contexts, many still have difficulties computing with rational numbers. Results from NAEP dating back over 25 years have continually documented the fact that one of the greatest deficits in U. In the NAEP, students in the fourth, eighth, and twelfth grades did well on questions about basic whole number operations and concepts in numerical and simple applied contexts.

However, students, especially those in the fourth and eighth grades, had difficulty with more complex problem-solving situations.

Performance on word problems declines dramatically when additional features are included, such as more than one step or extraneous information. One kind of item asks students to reason about numbers and their properties and also assesses their conceptual understanding.

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.

They are unable to understand the basic concepts of Mathematics and their technique due to various reasons. The problems that occur in the process of learning mathematics are relatively less in case of other subjects. Hence, for common students mathematics becomes a tougher subject and consequently, they try to avoid it.

Problems related to mathematics occur not only in case of students but also in case of teachers. It is seen that most of the mathematics teachers are not aware of alternative simple methods of teaching mathematics and different skills of solving the same problem.

Hence, there is a need to study in detail the problems faced both by students and teachers in learning and teaching mathematics. So this study has been conducted with the sole objective of identifying various problems faced by students and teachers in learning mathematics and solicit suggestive Measure in favour of those.

For the study, 5 higher secondary schools and 5 colleges are selected randomly from study area. As a result, total no of institutions are ten. To obtain information, five student from each institutions are randomly selected and as a result total number of sample students were Again, to know the attitude of teacher, one teacher from each institution were selected randomly and as a result, number of sample teachers respondent were ten.

The data have been collected by personal interview method by the investigators themselves with the help of an interview schedule. A lot of information have been obtained through personal observation and interaction with the students and teachers. Problems faced by the teachers Moststudents Most students come cometowith classroom negative attitudetowards attitude towards mathematics.