Assessment is the process of gathering and interpreting evidence to make judgements

about student learning. It is the crucial link between learning outcomes, content and

teaching and learning activities. Assessment is used by learners and their teachers to

decide where the learners are at in their learning, where they need to go, and how best to

get there. The purpose of assessment is to improve learning, inform teaching, help students

achieve the highest standards they can and provide meaningful reports on students’

achievement. Assessment is the gathering of information about what students know

and can do in order to make decisions that will improve teaching and

learning. Assessment and evaluation are necessary and important

elements of the instructional cycle.

Evaluation is a judgement regarding the quality, value or worth of a

response, product or performance, based on established criteria and

curriculum standards. Evaluation gives students a clear indication of

how well they are performing based on the learner outcomes of the

curriculum. The payoff of effective evaluation is that students learn

how they can improve their performance. Assessment and evaluation

always go together.22

With information from assessment and evaluation, teachers can make

decisions about what to focus on in the curriculum and when to focus

on it. Assessment identifies who needs extra support, who needs

greater challenge, who needs extra practise and who is ready to move

on. The primary goal of assessment is to provide ongoing feedback to

teachers, students and parents, in order to enhance teaching and The Holly Lane Elementary

Staff believes the primary purpose of assessment is to gather

information and to promote student learning. Assessing students allows staff the opportunity to

identify areas of strengths and weaknesses in both individual students, as well as whole class

instruction. Assessments are a means to evaluate student progress in order to provide evidence

for reporting. Our staff believes assessments are used to drive instruction and adjust curriculum

as

needed. Assessments can be used to determine instructional groups as well. We believe a final

purpose for assessment is to provide both written and verbal feedback to students, parents,

teachers and administrators.

Assessment: from the Latin root assidere, to sit beside another. ago, Thomas Gilbert summed up

the principles of good feedback in his delightful and informative book Human Competence. In it,

he catalogued the requirements of any information system "designed to give maximum support to

performance." The requirements involved eight steps:

1. Identify the expected accomplishments.

2. State the requirements of each accomplishment. If there is any doubt

that people understand the reason why an accomplishment and its

requirements are important, explain this.

3. Describe how performance will be measured and why.

4. Set exemplary standards, preferably in measurement terms.

5. Identify exemplary performers and any available resources that people

can use to become exemplary performers.

6. Provide frequent and unequivocal feedback about how well each

person is performing. This confirmation should be expressed as a

comparison with an exemplary standard. Consequences of good and

poor performance should also be made clear.

7. Supply as much backup information as needed to help people

troubleshoot their own performance.

8. Relate various aspects of poor performance to specific remedial

actions.

Gilbert sardonically adds that "these steps are far too simple to be called a 'technology,' but it

may be that their simplicity helps explain why they are so rarely followed." He elaborates, "In

years of looking at schools and jobs, I have almost never seen an ideal [feedback] system.

Managers, teachers, employees, and students seldom have adequate information about how well

they are performing." A key question to ponder is: why are such "simple" steps "rarely followed"?

What views and practices in schools cause us to ignore or violate such commonsensical views

about performance?

One reason we rarely follow such simple steps is that there are fundamental misconceptions about assessment generally and feedback in particular among educators. As I have argued, far too many educators treat assessment as something one does after

teaching and learning are over instead of seeing assessment as central to learning. (If I were to say that learning requires Assessment and feedback are crucial for helping people learn. Assessment should mirror good instruction; happen continuously as part of instruction; and provide information about the levels of understanding that students are reaching. In order for learners to gain insight into their learning and their understanding, frequent feedback is critical: students need to monitor their learning and actively evaluate their strategies and their current levels of understanding. (How People Learn by Bransford, Brown, and Cocking 1999)Individuals acquire a skill much more rapidly if they receive feedback about the correctness of what they have done. One of the most important roles for assessment is the provision of timely and informative feedback to students during instruction and learning so that their practice of a skill and its subsequent acquisition will be effective and efficient. (Knowing What Students Know: The science and design of educational assessment by Pellegrino, Chudowsky, and Glaser 2001)

Assessment Is Needed for Effective Teaching

Two important conclusions about the best college teachers:

• How do they prepare to teach? They begin with questions about student learning objectives rather than about

what the teacher will do.• How do they check their progress and evaluate their efforts? They have some systematic program to assess

their own efforts and to make appropriate changes. They assess their students based on the primary learning objectives rather than on arbitrary standards.

(What the Best College Teachers Do by Bain 2004)

“People tend to learn most effectively (in ways that make a sustained, substantial, and positive influence on the way they think, act, or feel) when

• they are trying to solve problems (intellectual, physical, artistic, practical, abstract, etc.) or create something

new that they find intriguing, beautiful, and/or important;

• they are able to do so in a challenging yet supportive environment in which they can feel a sense of control

over their own education;

• they can work collaboratively with other learners to grapple with the problems;

• they believe that their work will be considered fairly and honestly; and

• they can try, fail, and receive feedback from expert learners in advance of and separate from any summative

judgment of their efforts.” – Ken Bain 2004

all of the Assessement methods

assessments we have experienced and give today fall into one of four basic categories of

methods:

1. Selected response

1. Selected response and short answer2. Extended written response3. Performance assessment4. Personal communicationAll four methods are legitimate options when their use correlates highly with the learning target and the intended use of the information.Performance AssessmentPerformance assessment is assessment based on observation and judgment; we look at a performance or product and make a judgment as to its quality. Examples include the following:• Complex such as playing a musical instrument, carrying out the steps in a scientific experiment, speaking a foreign language, reading aloud with fluency, repairing an engine, or working productively in a group. In these cases it the doing—the process—that is importan

is • Creating complex products such as a term paper, a lab report, or a work of art. In these cases what counts is not so much the process of creation (although that may be evaluated, too), but the level of quality of the product itself.As with extended written response assessments, performance assessments have two parts: a performance task or exercise and a scoring guide. Again, the scoring guide can award points for specific features of a performance or product that are present, or it can take the form of a rubric, in which levels of quality are described. For example, to assess the ability to do a simple process, such as threading a sewing machine, doing long

division, or safely operating a band saw, points might be awarded for each step done in the correct order. Or, for more complex processes or products, you might have a rubric for judging quality that has several dimensions, such as ideas, organization, voice, word

choice, sentence fluency and conventions in writing, or content, organization,

presentation,

and use of language in an oral presentation. Again, scores could be reported in

number or percent of points earned, or in terms of a rubric score. WHAT IS IT? Performance assessment, also known as alternative or authentic assessment, is a form of testing that requires students to perform a task rather than select an answer from a ready-made list. For example, a student may be asked to explain historical events, generate scientific hypotheses, solve math problems, converse in a foreign language, or conduct research on an assigned topic. Experienced raters--either teachers or other trained staff--then judge the quality of the student's work based on an agreed-upon set of criteria. This new form of assessment is most widely used to directly assess writing ability based on text produced by students under test instructions.

HOW DOES IT WORK? Following are some methods that have been used successfully to assess performance:

• Open-ended or extended response exercises are questions or other prompts that require students to explore a topic orally or in writing. Students might be asked to describe their observations from a science experiment, or present arguments an historic character would make concerning a particular proposition. For example, what would Abraham Lincoln argue about the causes of the Civil War?

• Extended tasks are assignments that require sustained attention in a single work area and are carried out over several hours or longer. Such tasks could include drafting, reviewing, and revising a poem; conducting and explaining the results of a science experiment on photosynthesis; or even painting a car in auto shop.

• Portfolios are selected collections of a variety of performance-based work. A portfolio might include a student's "best pieces" and the student's evaluation of the strengths and weaknesses of several pieces. The portfolio may also contain some "works in progress" that illustrate the improvements the student has made over time.

These methods, like all types of performance assessments, require that students actively develop their approaches to the task under defined conditions, knowing that their work will be evaluated according to agreed-upon standards. This requirement distinguishes performance assessment from other forms of testing.

WHY TRY IT? Because they require students to actively demonstrate what they know, performance assessments may be a more valid indicator of students' knowledge and abilities. There is a big difference between answering multiple choice questions on how to make an oral presentation and actually making an oral presentation.

More important, performance assessment can provide impetus for improving instruction, and increase students' understanding of what they need to know and be able to do. In preparing their students to work on a performance task, teachers describe what the task entails and the standards that will be used to evaluate performance. This requires a careful description of the elements of good performance, and allows students to judge their own work as they proce

For performance assessments to have a positive impact on instructional practices in the classroom,

teachers need to become familiar with the nature of the tasks, what content and thinking skills the tasks

assess, and what constitutes a high quality response. Teachers in the QUASAR schools participated in

workshops and collaborated informally with their colleagues to gain these types of experiences. They

focused on two important aspects of performance assessment:• Performance assessments allow students to show how they arrived at their solutions and providing

explanations for their answers, thereby providing rich information about students' thinking and reasoning.• Performance assessments reveal different levels of understanding of the mathematics content.

Therefore, evaluations of student responses should focus on the content of the response, not its length.

ed.

mathematics

It is no doubt that the world today is greatly indebted to the contributions made by Indian

mathematicians. One of the most important contribution made by them was the

introduction of decimal system as well as the invention of zero. Here are some the

famous Indian mathematicians dating back from Indus Valley civilization and Vedas to

Modern times.Aryabhatta worked on the place value system using letters to signify

numbers and stating qualities. He discovered the position of nine planets and stated that

these planets revolve around the sun. He also stated the correct number of days in a year

that is 365.

The most significant contribution of Brahmagupta was the introduction of zero(0) to the

mathematics which stood for “nothing”.ramanujan is one of the celebrated Indian

mathematicians. His important contributions to the field include Hardy-Ramanujan-

Littlewood circle method in number theory, Roger-Ramanujan’s identities in partition of

numbers, work on algebra of inequalities, elliptic functions, continued fractions, partial

sums and products of hypergeometric series. Prasanta Chandra Mahalanobis is the

founder ofIndian Statistical Institute as well as the National Sample Surveys for which he

gained international recognition. Calyampudi Radhakrishna Rao, popularly known as C R

Rao is a well known statistician, famous for his “theory of estimation”.

D. R. Kaprekar discovered several results in number theory, including a class of numbers and a constant named after him. Without any formal mathematical education he published extensively and was very well known in recreational mathematics cricle. Harish Chandra is famously known for infinite dimensional group representation theory.satyendranath bose Known for his collaboration with Albert Einstein. He is best known for his work on quantum mechanics in the early 1920s, providing the foundation for Bose–Einstein statistics and the theory of the Bose–Einstein condensate.Bhaskara Bhāskara was the one who declared that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. He is also famous for his book “Siddhanta Siromani”. Narendra Karmarkar is known for his Karmarkar’s algorithm. He is listed as an highly cited researcher by Institute for Scientific Information.

RAMANUJAN

• He was born on 22naof December 1887 in a small village of Tanjore

district, Madras.He failed in English in Intermediate, so his formal studies

were stopped but his self-study of mathematics continued.

• He sent a set of 120 theorems to Professor Hardy of Cambridge. As a result

he invited Ramanujan to England.

• Ramanujan showed that any big number can be written as sum of not

more than four prime numbers.

• He showed that how to divide the number into two or more squares or

cubes.

• when Mr Litlewood came to see Ramanujan in taxi number 1729,

Ramanujan said that 1729 is the smallest number which can be written in

the form of sum of cubes of two numbers in two ways,i.e. 1729 = 93 + 103 =

13 + 123since then the number 1729 is called Ramanujan’s number.

• In the third century B.C, Archimedes noted that the ratio of circumference

of a circle to its diameter is constant. The ratio is now called ‘pi ( Π )’ (the

16th letter in the Greek alphabet series)

• The largest numbers the Greeks and the Romans used were 106 whereas

Hindus used numbers as big as 1053 with specific names as early as 5000

B.C. during the Vedic period.

ARYABHATTA

• Aryabhatta was born in 476A.D in Kusumpur, India.

• He was the first person to say that Earth is spherical and it revolves

around the sun.

• He gave the formula (a + b)2 = a2 + b2 + 2ab

• He taught the method of solving the following problems:

14 + 24 + 34 + 44 + 54 + …………+ n4 = n(n+1) (2n+1) (3n2+3n-1)/30

BRAHMA GUPTA• Brahma Gupta was born in 598A.D in Pakistan.

• He gave four methods of multiplication.

• He gave the following formula, used in G.P series

a + ar + ar2 + ar3 +……….. + arn-1 = (arn-1) ÷ (r – 1)

• He gave the following formulae :

Area of a cyclic quadrilateral with side a, b, c, d= √(s -a)(s- b)(s -c)(s- d)

where 2s = a + b + c + d

Length of its diagonals =

SHAKUNTALA DEVI• She was born in 1939

• In 1980, she gave the product of two, thirteen digit numbers within 28

seconds, many countries have invited her to demonstrate her

extraordinary talent.

• In Dallas she competed with a computer to see who give the cube root of

188138517 faster, she won. At university of USA she was asked to give the

23rdroot

of916748676920039158098660927585380162483106680144308622407

12651642793465704086709659327920576748080679002278301635492

485238033574531693511190359657754734007568186883056208210161

29132845564895780158806771.She answered in 50seconds. The answer

is 546372891. It took a UNIVAC 1108 computer, full one minute (10

seconds more) to confirm that she was right after it was fed with 13000

instructions.Now she is known to be Human Computer.

BHASKARACHARYA

• He was born in a village of Mysore district.

• He was the first to give that any number divided by 0 gives infinity (00).

• He has written a lot about zero, surds, permutation and combination.

• He wrote, “The hundredth part of the circumference of a circle seems to be

straight. Our earth is a big sphere and that’s why it appears to be flat.”

• He gave the formulae like sin(A ± B) = sinA.cosB ± cosA.sinB

M A H AV I R AMahavira was a 9th-century Indian mathematician from Gulbarga who asserted

that the square root of a negative number did not exist. He gave the sum of a

series whose terms are squares of an arithmetical progression and empirical rules

for area and perimeter of an ellipse. He was patronised by the great Rashtrakuta

king Amoghavarsha. Mahavira was the author of Ganit Saar Sangraha. He

separated Astrology from Mathematics. He expounded on the same subjects on

which Aryabhata and Brahmagupta contended, but he expressed them more

clearly. He is highly respected among Indian Mathematicians, because of his

establishment of terminology for concepts such as equilateral, and isosceles

triangle; rhombus; circle and semicircle. Mahavira’s eminence spread in all South

India and his books proved inspirational to other Mathematicians in Southern

India.

P Y T H A G O R A S [Samos, 582 - 500 BC]

Like Thales, Pythagoras is rather known for mathematics than for philosophy.

Anyone who can recall math classes will remember the first lessons of plane

geometry that usually start with the Pythagorean theorem about right-angled

triangles: a²+b²=c². In spite of its name, the Pythagorean theorem was not

discovered by Pythagoras. The earliest known formulation of the theorem was

written down by the Indian mathematician BaudhÄyana in 800BC. The principle

was also known to the earlier Egyptian and the Babylonian master builders.

However, Pythagoras may have proved the theorem and popularised it in the

Greek world. With it, his name and his philosophy have survived the turbulences

of history.

Aryabhata I

Aryabhata (476-550) wrote the Aryabhatiya. He described the important

fundamental principles of mathematics in 332 shlokas. The treatise contained:

• Quadratic equations

• Trigonometry

• The value of π, correct to 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata’s

contributions include:

Trigonometry:

• Introduced the trigonometric functions.

• Defined the sine (jya) as the modern relationship between half an angle

and half a chord.

• Defined the cosine (kojya).

• Defined the versine (ukramajya).

• Defined the inverse sine (otkram jya).

• Gave methods of calculating their approximate numerical values.

• Contains the earliest tables of sine, cosine and versine values, in 3.75°

intervals from 0° to 90°, to 4 decimal places of accuracy.

• Contains the trigonometric formula sin (n + 1) x – sin nx = sin nx – sin (n

– 1) x – (1/225)sin nx.

• Spherical trigonometry .

Arithmetic:

• Continued fractions .

Algebra:

• Solutions of simultaneous quadratic equations.

• Whole number solutions of linear equations by a method equivalent to the

modern method.

• General solution of the indeterminate linear equation .

Mathematical astronomy:

• Proposed for the first time, a heliocentric solar system with the planets

spinning on their axes and following an elliptical orbit around the Sun.

• Accurate calculations for astronomical constants, such as the:

• Solar eclipse .

• Lunar eclipse .

• The formula for the sum of the cubes, which was an important step

in the development of integral calculus.[60]

Calculus:

• Infinitesimals :

• In the course of developing a precise mapping of the lunar eclipse,

Aryabhatta was obliged to introduce the concept of infinitesimals

(tatkalika gati) to designate the near instantaneous motion of the

moon.[61]

• Differential equations :

• He expressed the near instantaneous motion of the moon in the

form of a basic differential equation.[61]

• Exponential function :

• He used the exponential function e in his differential equation of

the near instantaneous motion of the moon.[61]

Varahamihira

Varahamihira (505-587) produced the Pancha Siddhanta (The Five

Astronomical Canons). He made important contributions to trigonometry,

including sine and cosine tables to 4 decimal places of accuracy and the following

formulas relating sine and cosinefunctions:

• sin2(x) + cos2(x) = 1

•

•

N I E L S H E N R I K A B E LNiels Henrik Abel (August 5, 1802 – April 6, 1829) was a

noted Norwegian mathematician [1] who proved the impossibility of solving

the quintic equation in radicals.

C A R L F R I E D R I C H G A U S SJohann Carl Friedrich Gauss (pronounced /ˈɡaÊŠs/; German: Gauß

listen(help·info), Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February

1855) was aGerman mathematician and scientist who contributed significantly to

many fields, including number theory, statistics, analysis, differential

geometry, geodesy, geophysics,electrostatics, astronomy and optics. Sometimes

known as the Princeps mathematicorum[1] (Latin, “the Prince of

Mathematicians” or “the foremost of mathematicians”) and “greatest

mathematician since antiquity”, Gauss had a remarkable influence in many fields

of mathematics and science and is ranked as one of history’s most influential

mathematicians.[2] He referred to mathematics as “the queen of sciences.”[3]

Gauss was a child prodigy. There are many anecdotes pertaining to his precocity

while a toddler, and he made his first ground-breaking mathematical discoveries

while still a teenager. He completed Disquisitiones Arithmeticae, his magnum

opus, in 1798 at the age of 21, though it would not be published until 1801. This

work was fundamental in consolidating number theory as a discipline and has

shaped the field to the present day.

L E O N H A R D E U L E RLeonhard Paul Euler (15 April 1707 – 18 September 1783) was a

pioneering Swiss mathematician and physicist who spent most of his life

in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər in

English and [ËˆÉ”Ê �lÉ�] in German; the common English

pronunciation /ˈjuË �lÉ™r/ EW-lÉ™r is incorrect.

Euler made important discoveries in fields as diverse as infinitesimal

calculus and graph theory. He also introduced much of the modern mathematical

terminology and notation, particularly for mathematical analysis, such as the

notion of a mathematical function. He is also renowned for his work

in mechanics, fluid dynamics, optics, and astronomy.

Euler is considered to be the preeminent mathematician of the 18th century and

one of the greatest of all time. He is also one of the most prolific; his collected

works fill 60–80quarto volumes. A statement attributed to Pierre-Simon

Laplace expresses Euler’s influence on mathematics: “Read Euler, read Euler, he

is the master [i.e., teacher] of us all.”

Euler was featured on the sixth series of the Swiss 10-franc banknote and on

numerous Swiss, German, and Russian postage stamps. The asteroid 2002

Euler was named in his honor. He is also commemorated by the Lutheran

Church on their Calendar of Saints on 24 May – he was a devout Christian (and

believer in biblical inerrancy) who wroteapologetics and argued forcefully against

the prominent atheists of his time.

D AV I D H I L B E R TDavid Hilbert (January 23, 1862 – February 14, 1943) was

a German mathematician, recognized as one of the most influential and universal

mathematicians of the 19th and early 20th centuries. He discovered and

developed a broad range of fundamental ideas in many areas, including invariant

theory and the axiomatization of geometry. He also formulated the theory

of Hilbert spaces one of the foundations of functional analysis.

Hilbert adopted and warmly defended Georg Cantor‘s set theory and transfinite

numbers. A famous example of his leadership in mathematics is his 1900

presentation of a collection of problems that set the course for much of the

mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and some

tools to the mathematics used in modern physics. He is also known as one of the

founders ofproof theory, mathematical logic and the distinction between

mathematics andmetamathematics.

B E R N H A R D R I E M A N NGeorg Friedrich Bernhard Riemann (help·info) (German

pronunciation: [ˈriË man] ; September 17, 1826 – July 20, 1866) was an

influentialGerman mathematician who made lasting contributions

to analysis and differential geometry, some of them enabling the later

development of general relativity.

E U C L I D

Euclid (Greek: Εá½�κλείδης — EukleídÄ“s),

fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician

and is often referred to as the “Father of Geometry.” He was active in Hellenistic

Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is the most

successful textbook and one of the most influential works in the history of

mathematics, serving as the main textbook for teaching mathematics (especially

geometry) from the time of its publication until the late 19th or early 20th

century. In it, the principles of what is now called Euclidean geometry were

deduced from a small set of axioms. Euclid also wrote works on perspective, conic

sections, spherical geometry, number theory and rigor.