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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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• Intro to multiplication
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• Prepare for the 2020 AP®︎ Statistics Exam
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• Derivatives: definition and basic rules
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• Applications of derivatives
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• Applications of integrals
• Differentiation: definition and basic derivative rules
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• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
• AP Calculus BC solved exams
• AP®︎ Calculus BC Standards mappings
• Integrals review
• Integration techniques
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• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

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What are teachers saying about Khan Academy’s interactive math worksheets?

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Free Math Worksheets

Printable math worksheets from k5 learning.

Our  free math worksheets  cover the full range of elementary school math skills from numbers and counting through fractions, decimals, word problems and more. All worksheets are printable files with answers on the 2nd page.

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Free Math Printable Worksheets with Answer Keys and Activities

Other free resources.

Feel free to download and enjoy these free worksheets on functions and relations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Long Division with Remainders
• Long Division with Remainders #2 (Zeros in the Quotient)
• Long Division with 2 Digit Divisors
• Whole Number by Unit Fraction
• Equation of Circle
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (Advanced)
• End of Unit, Review Sheet
• Distance Formula
• Simplify Rational Exponents (Algebra 2)
• Solve Equations with Rational Exponents (Algebra 2)
• Solve Equations with variables in Exponents (Algebra 2)
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (No logs)
• Compound Interest Worksheet (Logarithms required)
• Factor Trinomials Worksheet
• Factor by Grouping
• Domain and Range (Algebra 1)
• Functions vs Relations (Distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations with Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)
• Logarithmic Equations
• Properties of Logarithms Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Solve Quadratic Equations by Factoring
• Quadratic Formula Worksheets (3 different sheets)
• Quadratic Formula Worksheet (Real solutions)
• Quadratic Formula (Complex solutions)
• Quadratic Formula (Both real and complex solutions)
• Discriminant and Nature of the Roots
• Solve Quadratic Equations by Completing the Square
• Sum and Product of Roots
• Radical Equations
• Mixed Problems on Writing Equations of Lines
• Slope Intercept Form Worksheet
• Standard Form Worksheet
• Point Slope Worksheet
• Write Equation of Line from the Slope and 1 Point
• Write Equation of Line From Two Points
• Equation of Line Parallel to Another Line and Through a Point
• Equation of Line Perpendicular to Another Line and Through a Point
• Slope of a Line
• Perpendicular Bisector of Segment
• Write Equation of Line Mixed Review
• Word Problems
• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet
• Dividing Radicals
• Simplify Radicals Worksheet
• Adding Radicals
• Multiplying Radicals Worksheet
• Radicals Review (Mixed review worksheet on radicals and square roots)
• Rationalizing the Denominator (Algebra 2)
• Radical Equations (Algebra 2)
• Solve Systems of Equations Graphically
• Solve Systems of Equations by Elimination
• Solve by Substitution
• Solve Systems of Equations (Mixed Review)
• Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
• Real World Connections (Compare cell phone plans)
• Identifying Fractions

Trigonomnetry

• Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)
• Law of Sines
• Ambiguous Case of the Law of Sines
• Law of Cosines
• Vector Worksheet
• Sine, Cosine, Tangent, to Find Side Length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
• Mixed Review
• Unit Circle Worksheet
• Graphing Sine and Cosine Worksheet
• Sine Cosine Graphs with Vertical Translations
• Sine, Cosine, Tangent Graphs with Phase Shifts
• Sine, Cosine, Tangent Graphs with Change in Period, Amplitude and Phase Shifts (All Translations)
• Tangent Equation, Graph Worksheet
• Graphing Sine, Cosine, Tangent with Change in Period
• Cumulative, Summative Worksheet on Periodic Trig Functions - period, amplitude, phase shift, radians, degrees,unit circle
• Ratio and Proportion
• Similar Polygons
• Area of Triangle
• Interior Angles of Polygons
• Exterior Angles of Polygons

• Identifying Fractions Worksheet
• Associated Powerpoint
• Simplify Fractions Worksheet (Regular Difficulty)
• Associated PowerPoint
• Simplify Fractions Worksheet (Challenging Difficulty level for advanced learners)
• System of Linear Equations Worksheet
• System of Linear Equations - Real World Application
• Compositions of Reflections. Reflections Over Intersecting Lines as Rotations

All of these worksheets and activities are available for free so long as they are used solely for educational, noncommercial purposes and are not distributed outside of a specific teacher's classroom.

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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Math Assignment

 What is an annotated bibliography?

Assignments in math courses are usually a list of problems to be solved. Each question may be of various difficulties and types, which are marked and returned to the student to be used as a source of feedback. Math assignments are designed to provide opportunities for 'doing math' and to consolidate students’ understanding of the content. The questions often come from the most recent week of learned material, but some questions may require students to synthesize concepts from further back. This is because learning math is cumulative by nature; you continuously build upon what you've learned before. Because math assignments are typically used as a checkpoint for understanding, they tend to weigh less than other assessments in terms of grade, therefore, it's important to treat them as learning opportunities instead of a tool for maximizing course grades.

Click on the Timeline for a visual representation of the timeline. Click on the Checklist for a document containing the checklist items for a math assignment.

According to your start and end dates ( 2024-08-21 to 2024-09-05 ), you have 15 days to finish your assignment.

Step 1: Understand the questions Complete by Wed Aug 21, 2024

Read through the assignment once it’s released and understand the questions. Read the questions early to prime your brain; too many students start too late.

• For some courses, assignments come out before or as content is covered, so having the questions in your mind as you see the content can help prime you to actively learn the content and will also help you know where to look for relevant material in the lecture notes and/or videos. Active learning of content means you are asking yourself questions as you are learning, actively monitoring your understanding of the topic, and anticipating next steps.
• Read all the questions to estimate the range of tested material, make note of anything you don’t understand. Understanding the question correctly reduces the chance of getting stuck.

The hardest thing being with a mathematician is that they always have problems.

– Tendai Chitewere

Step 2: Gather materials and review Complete by Sat Aug 24, 2024

• Gather required materials (lecture notes and/or videos, textbook sections).
• Study the material (definitions, concepts, solved problems, etc.) until you are familiar with it. Identify connections across concepts/topics and practice additional problems. Regularly reviewing your material will also help you to find relevant information quickly if you get stuck on a question.

Step3: Solve questions and get help Complete by Tue Sep 03, 2024

Part a: attempt to solve each question.

• Rule of thumb for time allocation: one hard question may take significantly longer than all easy questions combined. Consider how long you’ve taken to solve all the easy questions. It may take at least twice as long to solve one hard question. Try all easy questions as soon as possible to gauge how long the rest could take.
• Try giving yourself a break for a day before attempting them again to allow time for your mind to continue working subconsciously.
• Still stuck? Check out Math problems: What to do when you're stuck for more strategies and tips.
• Keep track of unsolved questions and what stages you were stuck at. Don’t throw away any work in progress.Your instructor or TA can provide better help if you can narrow down on the issue.

The only way to learn mathematics is to do mathematics.

– Paul Halmos

Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

– Paul Halmos  

Part B: Get help on unsolved questions

• The sooner you ask for support, the more time you and the instructor/TA will have for follow-up conversations.
• Focus on asking questions to help you close the gap between your understanding, and the understanding you need to complete the question.  Questions such as “Can I get a hint on #4?” or “What is the answer?” are not typically helpful in increasing your problem-solving abilities.
• For questions you were challenged by, but were able to solve after receiving help from the instructor or TA, make sure that you fully understand how to come up with the solution in case a similar question is asked on a test.
• Many students find it helpful to study together. Make sure you understand what your instructor’s expectations are around group work (if you’re not sure, ask!). Be mindful about how much you post about an assignment solution on the course’s discussion board and if you’re in doubt, consider posting privately. Familiarize yourself with the academic integrity standards expected in your courses.

The goal of seeking help is to try to find out how to approach questions that you are stuck on. From this extra help you still may not see the complete picture of how to solve the problem. To see this may require that you go back to Part A , now with the instructor/TA’s suggestions in mind.

I have not failed. I've just found 10,000 ways that won't work.

– Thomas Alva Edison

Step 4: Check your solutions Complete by Thu Sep 05, 2024

• It’s important that you spend some time away from your solutions before you check them. For problems you did solve, try to figure out if you can check your own answer like you would have to on a test (i.e. don’t look up answers in a book or online).
• You could ask the instructor/TA for guidance, but avoid questions such as “Did I do this right?” Instead, ask yourself whether you are asking the right questions about your solution. 
• Remember not to focus only on the final answer, but also on how you communicated your answer. Ask yourself if a typical student in the class could follow along with what you’ve written. 
• Remember to cite any external sources that you used in your work, including work completed with others (if this was permitted).

Step 5: Review Marked Assignments Complete by Thu Sep 05, 2024

After receiving your marked assignment check over incorrect questions as well as those you got right. Review posted solutions and read them critically. Often, there are different approaches to a problem and you may learn about some of these new approaches through this review.

Introduction to Sets

Forget everything you know about numbers.

In fact, forget you even know what a number is.

This is where mathematics starts.

Instead of math with numbers, we will now think about math with "things".

What is a set? Well, simply put, it's a collection .

First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property.

For example, the items you wear: hat, shirt, jacket, pants, and so on.

I'm sure you could come up with at least a hundred.

This is known as a set .

So it is just things grouped together with a certain property in common.

There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

The curly brackets { } are sometimes called "set brackets" or "braces".

This is the notation for the two previous examples:

{socks, shoes, watches, shirts, ...} {index, middle, ring, pinky}

Notice how the first example has the "..." (three dots together).

The three dots ... are called an ellipsis, and mean "continue on".

So that means the first example continues on ... for infinity.

(OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)

• The first set {socks, shoes, watches, shirts, ...} we call an infinite set ,
• the second set {index, middle, ring, pinky} we call a finite set .

But sometimes the "..." can be used in the middle to save writing long lists:

Example: the set of letters:

{a, b, c, ..., x, y, z}

In this case it is a finite set (there are only 26 letters, right?)

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

And so on. We can come up with all different types of sets.

We can also define a set by its properties, such as {x|x>0} which means "the set of all x's, such that x is greater than 0", see Set-Builder Notation to learn more.

And we can have sets of numbers that have no common property, they are just defined that way. For example:

Are all sets that I just randomly banged on my keyboard to produce.

Why are Sets Important?

Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.

Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets .

Universal Set

Some More Notation

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?

Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!

Example: Are A and B equal where:

• A is the set whose members are the first four positive whole numbers
• B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal!

And the equals sign (=) is used to show equality, so we write:

Example: Are these sets equal?

• A is {1, 2, 3}
• B is {3, 1, 2}

Yes, they are equal!

They both contain exactly the members 1, 2 and 3.

It doesn't matter where each member appears, so long as it is there.

When we define a set, if we take pieces of that set, we can form what is called a subset .

Example: the set {1, 2, 3, 4, 5}

A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.

But {1, 6} is not a subset, since it has an element (6) which is not in the parent set.

In general:

A is a subset of B if and only if every element of A is in B.

So let's use this definition in some examples.

Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 is in A, and 1 is in B as well. So far so good.

3 is in A and 3 is also in B.

4 is in A, and 4 is in B.

That's all the elements of A, and every single one is in B, so we're done.

Yes, A is a subset of B

Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

Let's try a harder example.

Example: Let A be all multiples of 4 and B be all multiples of 2 . Is A a subset of B? And is B a subset of A?

Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.

The sets are:

• A = {..., −8, −4, 0, 4, 8, ...}
• B = {..., −8, −6, −4, −2, 0, 2, 4, 6, 8, ...}

By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:

A is a subset of B, but B is not a subset of A

Proper Subsets

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.

Let A be a set. Is every element of A in A ?

Well, umm, yes of course , right?

So that means that A is a subset of A . It is a subset of itself!

This doesn't seem very proper , does it? If we want our subsets to be proper we introduce (what else but) proper subsets :

A is a proper subset of B if and only if every element of A is also in B, and there exists at least one element in B that is not in A.

This little piece at the end is there to make sure that A is not a proper subset of itself: we say that B must have at least one extra element.

{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

Notice that when A is a proper subset of B then it is also a subset of B.

Even More Notation

When we say that A is a subset of B, we write A ⊆ B

Or we can say that A is not a subset of B by A ⊈ B

When we talk about proper subsets, we take out the line underneath and so it becomes A ⊂ B or if we want to say the opposite A ⊄ B

Empty (or Null) Set

This is probably the weirdest thing about sets.

As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!"

And right you are. It is a set with no elements .

This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.

It is represented by ∅

Or by {} (a set with no elements)

Some other examples of the empty set are the set of countries south of the south pole .

So what's so weird about the empty set? Well, that part comes next.

Empty Set and Subsets

So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A?

Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A . But what if we have no elements?

It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.

A good way to think about it is: we can't find any elements in the empty set that aren't in A , so it must be that all elements in the empty set are in A.

So the answer to the posed question is a resounding yes .

The empty set is a subset of every set, including the empty set itself.

No, not the order of the elements. In sets it does not matter what order the elements are in .

Example: {1,2,3,4} is the same set as {3,1,4,2}

When we say order in sets we mean the size of the set .

Another (better) name for this is cardinality .

A finite set has finite order (or cardinality). An infinite set has infinite order (or cardinality).

For finite sets the order (or cardinality) is the number of elements .

Example: {10, 20, 30, 40} has an order of 4.

For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.

Arg! Not more notation!

Nah, just kidding. No more notation.

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Stanford Online

Introduction to mathematical thinking.

HSTAR-Y0001

Stanford Graduate School of Education

The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today's world. This course helps to develop that crucial way of thinking.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by  The Lemann Foundation ), English

Course Syllabus

Instructor's welcome and introduction

•  Introductory material
•  Analysis of language – the logical combinators
•  Analysis of language – implication
•  Analysis of language – equivalence
•  Analysis of language – quantifiers
•  Working with quantifiers
•  Proofs
•  Proofs involving quantifiers
•  Elements of number theory
• Beginning real analysis

Recommended Background

High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.

A good way to assess if your  basic  school background is adequate (even if currently rusty) is to glance at the topics in the book  Adding It Up: Helping Children Learn Mathematics  (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).

Dr Keith Devlin, Co-founder and Executive Director H-STAR Institute

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Assignments on writing

Examples of short assignments, term papers, designing assignments that enable students to write well.

Writing well requires mastery of writing principles at a variety of different scales, from the sentence and paragraph scale (e.g., ordering information within sentences so content flows logically ) to the section and paper scale (e.g., larger-scale structure ). To simplify teaching, you can begin the term with shorter assignments to address the smaller-scale issues so you can more easily focus on the larger-scale issues when you assign longer assignments later in the term. At all scales, students best learn to communicate as mathematicians if the assignments are as authentic as possible: if the genre and rhetorical context are as similar as possible to those encountered by mathematicians.

Many of the following ideas are currently implemented in M.I.T.’s communication-intensive offerings of Real Analysis and Principles of Applied Mathematics .

• Require that at least one question on each problem set be typed up and written in the style of an expository paper (rather than the usually terse and sometimes scattered style of a homework solution).
• Assign short exposition tasks such as summarizing the proof of a theorem done in class or filling in the gaps in an explanation given briefly in class.
• To help students learn LaTeX or how to use equation editors, have an assignment requiring at least basic math formatting due early in the semester so students aren’t required to learn it as they’re researching and writing their term papers. Begin with simple math formatting exercises, building to more complex: e.g., see the assignments for M.I.T.’s Real Analysis recitations 1 (text with math) , 2 (table and figure) and 13 (slides containing a figure with LaTeXed labels) .
• Begin with communicating simple arguments, building to more complex (e.g., having students explain the heapsort algorithm and then revise the explanation based on feedback provides a rich opportunity for teaching about writing clear definitions, giving conceptual explanations as well as rigorous details, and presenting information in an order that is helpful to readers.) See the sequence of assignments from M.I.T.’s Principles of Applied Mathematics .
• Have students revise part of a concise textbook such as Rudin’s, Principles of Mathematical Analysis in the style of a more-thorough lecture note.
• Before an exam, have students formulate and submit to you a list of 2+ questions they have about the material. Students have a hard time formulating precise questions, yet this is an important communication and learning skill. Some students may feel they understand the course material, so permit questions that go beyond the scope of the course. You can use the questions to focus a review session. More detail about this assignment is given in this lesson plan from M.I.T.’s communication-intensive offering of Real Analysis.

The following books, articles, and websites contain short writing assignments.

• Stephen Maurer’s Undergraduate Guide to Writing Mathematics has an extensive appendix of writing exercises designed to target various aspects of writing mathematics.
• Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go , by A Crannell et al . [link goes to MAA review] This 119 page book from the MAA contains “writing projects suitable for use in a wide range of undergraduate mathematics courses, from a survey of mathematics to differential equations.” Each prompt is written in the form of an (often amusing) letter from someone who needs help with a “real-world” problem that requires math expertise. Students must solve the problem and write a letter of response. On his website, Tommy Ratliff (one of the co-authors) gives a brief account of using such projects in his calculus course.
• Annalisa Crannell’s Writing in Mathematics website has writing assignment for Calculus I, II, and III as well as links to colleagues’ websites that have further writing assignments.
• Quantitative Writing from Pedagogy in Action, the SERC Portal for Educators, has many examples of short and long writing assignments based on “ill-structured problems,” which are “open-ended, ambiguous, data-rich problems requiring the thinker to understand principles and concepts rather than simply applying formulae. Assignments ask students to produce a claim with supporting reasons and evidence rather than ‘the answer.'”
• The Nuts and Bolts of Proofs by Antonella Cupillari includes exercises for an introductory proof-writing course. Proof topics include calculus and linear algebra.
• Platt, M. L.. (1993). Short essay topics for calculus. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies 03.1 , 42-46.

Additional information about journal-writing assignments and other writing-to-learn assignments can be found on the page about using writing to help students learn math .

For each assignment, indicate your expectations about audience and length, so students know how much explanation to include. An appropriate audience is often other students in the class who are unfamiliar with the specific topic of the assignment, or other math majors not in the class.

Term papers enable students to pursue areas of their own interest and so can be among the most rewarding assignments for students. To help students succeed, give students guidance for choosing a sufficiently focused topic, for finding helpful sources, and for using sources appropriately. See this assignment for proposing a term paper topic , from M.I.T.’s Principles of Applied Mathematics –it includes guidance for how to choose a good paper topic.

One of the (interesting) challenges of assigning a term paper is generating a list of possible paper topics. Ideally, each topic should have well-defined scope and have at least two or three available resources accessible to students in the course. You may want to emphasize to the students that they are not expected to do original mathematics research. However, the paper must be their own — they cannot paraphrase and closely follow a published survey paper.

One of your institution’s librarians may be happy to collaborate with you to show students how to find useful sources.

To provide students with an authentic rhetorical context for their term papers, consider showing them samples of expository papers and suggesting that they write for a journal that publishes expository papers (e.g., The American Mathematical Monthly , Math Horizons , Mathematics Magazine , and The College Mathematics Journal .

Don’t assign a term paper unless a variety of topics exist at an appropriate level. For example, a term paper may not be appropriate for an introductory class in analysis.

Be aware that plagiarism may be an issue particularly in large classes on subjects for which a wealth of material is available online. In such classes, you may find it to be helpful to tightly specify the paper topics or to supply a specific slant to the papers (e.g., apply such-and-such method to an application of your choice). Vary the assignments from year to year. These precautions may be less important in small classes.

In some classes (e.g., applied mathematics classes), it may be necessary to carefully guide students to choose topics that contain sufficient mathematical content. For that reason, using caution when approving unfamiliar topics.

A poorly focused assignment will leave students confused about what is expected of them and is likely to result in poor writing. Students are likely to write their best if the assignment is interesting and if students are told (or are able to confidently identify for themselves) the following:

• educational objectives of the assignment
• audience knowledge and interest, and author’s relationship to the audience
• purpose of the text to be written (e.g., to convince, to entertain mathematically, to teach, to spark interest)
• content to be addressed
• details of the genre ( proof ? research paper? funding proposal?)
• how the writing will be graded
• an effective writing process (you can provide support by assigning intermediate due dates or revision )

The following resources explain these points and give further guidance for designing effective assignments:

• Bahls, P., Student Writing in the Quantitative Disciplines: A Guide for College Faculty , Jossy-Bass 2012, pp. 36-46, contains sections on structuring writing assignments (includes sample prompts), sequencing assignments throughout a course, and sequencing writing from course to course.

General resources (not specific to mathematics)

• How can I avoid getting lousy student writing?
• What makes a good writing assignment?
• The webpage Integrating Writing and Speaking Into Your Subject , provided by MIT’s Writing Across the Curriculum, has several subpages about writing assignments.
• Creating Writing Assignments , MIT’s Writing Center

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment (1993)

Chapter: 4 assessing to support mathematics learning, 4 assessing to support mathematics learning.

High-quality mathematics assessment must focus on the interaction of assessment with learning and teaching. This fundamental concept is embodied in the second educational principle of mathematics assessment.

T HE L EARNING P RINCIPLE

Assessment should enhance mathematics learning and support good instructional practice .

This principle has important implications for the nature of assessment. Primary among them is that assessment should be seen as an integral part of teaching and learning rather than as the culmination of the process. 1 As an integral part, assessment provides an opportunity for teachers and students alike to identify areas of understanding and misunderstanding. With this knowledge, students and teachers can build on the understanding and seek to transform misunderstanding into significant learning. Time spent on assessment will then contribute to the goal of improving the mathematics learning of all students.

The applicability of the learning principle to assessments created and used by teachers and others directly involved in classrooms is relatively straightforward. Less obvious is the applicability of the principle to assessments created and imposed by parties outside the classroom. Tradition has allowed and even encouraged some assessments to serve accountability or monitoring purposes without sufficient regard for their impact on student learning.

A portion of assessment in schools today is mandated by external authorities and is for the general purpose of accountability of the schools. In 1990, 46 states had mandated testing programs, as

compared with 20 in 1980. 2 Such assessments have usually been multiple-choice norm-referenced tests. Several researchers have studied these testing programs and judged them to be inconsistent with the current goals of mathematics education. 3 Making mandated assessments consonant with the content, learning, and equity principles will require much effort.

Studies have documented a further complication as teachers are caught between the conflicting demands of mandated testing programs and instructional practices they consider more appropriate. Some have resorted to "double-entry" lessons in which they supplement regular course instruction with efforts to teach the objectives required by the mandated test. 4 During a period of change there will undoubtedly be awkward and difficult examples of discontinuities between newer and older directions and procedures. Instructional practices may move ahead of assessment practices in some situations, whereas in other situations assessment practices could outpace instruction. Neither situation is desirable although both will almost surely occur. However, still worse than such periods of conflict would be to continue either old instructional forms or old assessment forms in the name of synchrony, thus stalling movement of either toward improving important mathematics learning.

From the perspective of the learning principle, the question of who mandated the assessment and for what purpose is not the primary issue. Instruction and assessment—from whatever source and for whatever purpose—must be integrated so that they support one another.

To satisfy the learning principle, assessment must change in ways consonant with the current changes in teaching, learning, and curriculum. In the past, student learning was often viewed as a passive process whereby students remembered what teachers told them to remember. Consistent with this view, assessment was often thought of as the end of learning. The student was assessed on something taught previously to see if he or she remembered it. Similarly, the mathematics curriculum was seen as a fragmented collection of information given meaning by the teacher.

This view led to assessment that reinforced memorization as a principal learning strategy. As a result, students had scant oppor-

tunity to bring their intuitive knowledge to bear on new concepts and tended to memorize rules rather than understand symbols and procedures. 5 This passive view of learning is not appropriate for the mathematics students need to master today. To develop mathematical competence, students must be involved in a dynamic process of thinking mathematically, creating and exploring methods of solution, solving problems, communicating their understanding—not simply remembering things. Assessment, therefore, must reflect and reinforce this view of the learning process.

This chapter examines three ways of making assessment compatible with the learning principle: ensuring that assessment directly supports student learning; ensuring that assessment is consonant with good instructional practice; and enabling teachers to become better facilitators of student learning.

A SSESSMENT IN S UPPORT OF L EARNING

Assessment can play a key role in exemplifying the new types of mathematics learning students must achieve. Assessments indicate to students what they should learn. They specify and give concrete meaning to valued learning goals. If students need to learn to perform mathematical operations, they should be assessed on mathematical operations. If they should learn to use those mathematical operations along with mathematical reasoning in solving mathematical problems, they must be assessed on using mathematical operations along with reasoning to solve mathematical problems. In this way the nature of the assessments themselves make the goals for mathematics learning real to students, teachers, parents, and the public.

Mathematics assessments can help both students and teachers improve the work the students are doing in mathematics. Students need to learn to monitor and evaluate their progress. When students are encouraged to assess their own learning, they become more aware of what they know, how they learn, and what resources they are using when they do mathematics. "Conscious knowledge about the resources available to them and the ability to engage in self-monitoring and self-regulation are important characteristics of self-assessment that successful learners use to promote ownership of learning and independence of thought." 6

In the emerging view of mathematics education, students make their own mathematics learning individually meaningful. Important mathematics is not limited to specific facts and skills students can be trained to remember but rather involves the intellectual structures and processes students develop as they engage in activities they have endowed with meaning.

The assessment challenge we face is to give up old assessment methods to determine what students know, which are based on behavioral theories of learning and develop authentic assessment procedures that reflect current epistemological beliefs both about what it means to know mathematics and how students come to know. 7

Current research indicates that acquired knowledge is not simply a collection of concepts and procedural skills filed in long-term memory. Rather the knowledge is structured by individuals in meaningful ways, which grow and change over time. 8

A close consideration of recent research on mathematical cognition suggests that in mathematics, as in reading, successful learners understand the task to be one of constructing meaning, of doing interpretive work rather than routine manipulations. In mathematics the problem of imposing meaning takes a special form: making sense of formal symbols and rules that are often taught as if they were arbitrary conventions rather than expressions of fundamental regularities and relationships among quantities and physical entities. 9

L EARNING F ROM A SSESSMENT

Modern learning theory and experience with new forms of assessment suggest several characteristics assessments should have if they are to serve effectively as learning activities. Of particular interest is the need to provide opportunities for students to construct their own mathematical knowledge and the need to determine where students are in their acquisition of mathematical understanding. 10 One focuses more on the content of mathematics, the other on the process of doing mathematics. In both, the assessment must elicit important mathematics.

Constructing Mathematical Knowledge Learning is a process of continually restructuring one's prior knowledge, not just adding to it. Good education provides opportunities for students to connect what is being learned to their prior knowledge. One knows

mathematics best if one has developed the structures and meanings of the content for oneself. 11 For assessment to support learning, it must enable students to construct new knowledge from what they know.

One way to provide opportunities for the construction of mathematical knowledge is through assessment tasks that resemble learning tasks 12 in that they promote strategies such as analyzing data, drawing contrasts, and making connections. It is not enough, however, to expand mathematics assessment to take in a broader spectrum of an individual student's competence. In real-world settings, knowledge is sometimes constructed in group settings rather than in individual exploration. Learning mathematics is frequently optimized in group settings, and assessment of that learning must reflect the value of group interaction. 13

Some mathematics teachers are using group work in instruction to model problem solving in the real world. They are looking for ways to assess what goes on in groups, trying to find out not only what mathematics has been learned, but also how the students have been working together. A critical issue is how to use assessments of group work in the grades they give to individual students. A recent study of a teacher who was using groups in class but not assessing the work done in groups found that her students apparently did not see such work as important. 14 Asked in interviews about mathematics courses in which they had done group work, the students did not mention this teacher's course. Group work, if it is to become an integral and valued part of mathematics instruction, must be assessed in some fashion. A challenge to developers is to construct some high-quality assessment tasks that can be conducted in groups and subsequently scored fairly.

Part of the construction of knowledge depends on the availability of appropriate tools, whether in instruction or assessment. Recent experimental National Assessment of Educational Progress (NAEP) tasks in science use physical materials for a miniexperiment students are asked to perform by themselves. Rulers, calculators, computers, and various manipulatives are examples from mathematics of some instructional tools that should be a part of assessment. If students have been using graphing calculators to explore trigonometric functions, giving them tests on which calculators are banned greatly limits the questions they can be asked and

consequently yields an incomplete picture of their learning. Similarly, asking students to find a function that best fits a set of data by using a computer program can reveal aspects of what they know about functions that cannot be assessed by other means. Using physical materials and technology appropriately and effectively in instruction is a critical part of learning today's mathematics and, therefore, must be part of today's assessment.

Reflecting Development of Competence As students progress through their schooling, it is obvious that the content of their assessments must change to reflect their growing mathematical sophistication. When students encounter new topics in mathematics, they often cannot see how the unfamiliar ideas are connected to anything they have seen before. They resort to primitive strategies of memorization, grasping at isolated and superficial aspects of the topic. As learning proceeds, they begin to see how the new ideas are connected to each other and to what they already know. They see regularities and uncover hidden relationships. Eventually, they learn to monitor their thinking and can choose different ways to tackle a problem or verify a solution. 15 This scenario is repeated throughout schooling as students encounter new mathematics. The example below contains a description of this growth in competence that is derived from research in cognition and that suggests the types of evidence that assessment should seek. 16

A full portrayal of competence in mathematics demands much more than measuring how well students can perform automated skills although that is part of the picture. Assessment should also examine whether students have managed to connect the concepts they have learned, how well they can recognize underlying principles and patterns amid superficial differences, their sense of when to use processes and strategies, their grasp and command of their own understanding, and whether

they can bring these skills and abilities together to produce smooth, proficient performance.

P ROVIDING F EEDBACK AND O PPORTUNITIES TO R EVISE W ORK

An example of how assessment results can be used to support learning comes from the Netherlands. 17 Eleventh-grade students were given regular 45-minute tests containing both short-answer and essay questions. One test for a unit on matrices contained questions about harvesting Christmas trees of various sizes in a forest. The students completed a growth matrix to portray how the sizes changed each year and were asked how the forest could be managed most profitably, given the costs of planting and cutting and the prices at which the trees were to be sold. They also had to answer the questions when the number of sizes changed from three to five and to analyze a situation in which the forester wanted to recapture the original distribution of sizes each year.

After the students handed in their solutions, the teacher scored them, noting the major errors. Given this information, the students retook the test. They had several weeks to work on it at home and were free to answer the questions however they chose, separately or in essays that combined the answers to several questions. The second chance gave students the opportunity not simply to redo the questions on which they were unsuccessful in the first stage but, more importantly, to give greater attention to the essay questions they had little time to address. Such two-stage testing essentially formalizes what many teachers of writing do in their courses, giving students an opportunity to revise their work (often more than once) after the teacher or other students have read it and offered suggestions. The extensive experience that writing teachers have been accumulating in teaching and assessing writing through extended projects can be of considerable assistance to mathematics teachers seeking to do similar work. 18

During the two-stage testing in the Netherlands, students reflected on their work, talked with others about it, and got information from the library. Many students who had not scored well under time pressure—including many of the females—did much better under the more open conditions. The teachers could grade the students on both the objective scores from the first stage and

the subjective scores from the second. The students welcomed the opportunity to show what they knew. As one put it

Usually when a timed written test is returned to us, we just look at our grade and see whether it fits the number of mistakes we made. In the two-stage test, we learn from doing the task. We have to study the first stage carefully in order to do well on the second one. 19

In the Netherlands, such two-stage tasks are not currently part of the national examination given at the end of secondary school, but some teachers use them in their own assessments as part of the final grade each year. In the last year of secondary school, the teacher's assessment is merged with the score on the national examination to yield a grade for each student that is used for graduation, university admission, and job qualification.

L EARNING FROM THE S CORING OF A SSESSMENTS

Assessment tasks that call for complex responses require scoring rubrics. Such rubrics describe what is most important about a response, what distinguishes a stronger response from a weaker one, and often what characteristics distinguish a beginning learner from one with more advanced understanding and performance. Such information, when shared between teacher and student, has critically important implications for the learning process.

Teachers can appropriately communicate the features of scoring rubrics to students as part of the learning process to illustrate the types of performance students are striving for. Students often express mystification about what they did inadequately or what type of change would make their work stronger. Teachers can use rubrics and sample work marked according to the rubric to communicate the goals of improved mathematical explication. When applied to actual student work, rubrics illustrate the next level of learning toward which a student may move. For example, a teacher may use a scoring rubric on a student's work and then give the student an opportunity to improve the work. In such a case, the student may use the rubric directly as a guide in the improvement process.

The example on the following page illustrates how a scoring rubric can be incorporated into the student material in an assess-

ment. 20 The benefits to instruction and learning could be twofold. The student not only can develop a clearer sense of quality mathematics on the task at hand but can develop more facility at self-assessment. It is hoped that students can, over time, develop an inner sense of excellent performance so that they can correct their own work before submitting it to the teacher.

The rubrics can be used to inform the student about the scoring criteria before he or she works on a task. The rubric can also be used to structure a classroom discussion, possibly even asking the students to grade some (perhaps fictional) answers to the questions. In this way, the students can see some examples of how responses are evaluated. Such discussions would be a purely instructional use of an assessment device before the formal administration of the assessment.

S TIMULATING M OTIVATION , I NTEREST, AND A TTENTION

Because assessment has the potential to affect the learning process substantially, it is important that students do their best when being assessed. Students' motivation to perform well on assessments has usually been tied to the stakes involved. Knowing that an assessment has a direct bearing on a semester grade or on placement in the next class—that is, high personal stakes—has encouraged many students to display their best work. Conversely, assessments to judge the effectiveness of an educational program where results are often not reported on an individual basis carry low stakes for the student and may not inspire students to excel. These extrinsic sources of motivation, although real, are not always consonant with the principle that assessment should support good instructional practice and enhance mathematics learning. Intrinsic sources of motivation, such as interest in the task, offer a more fruitful approach.

Students develop an interest in mathematical tasks that they understand, see as relevant to their own concerns, and can manage. Recent studies of students' emotional responses to mathematics suggest that both their positive and their negative responses diminish as tasks become familiar and increase when the tasks are novel. 21 Because facility at problem solving includes facility with unfamiliar tasks, the regular use of nonroutine problems must become a part of instruction and assessment.

In some school districts, educational leaders are experimenting with nonroutine assessment tasks that have instructional value in themselves and that seem to have considerable interest for the students. Such a problem was successfully tried out with fifth-grade students in the San Diego City School District in 1990 and has

subsequently been used by other districts across the country to assess instruction in the fifth, sixth, and seventh grades. The task is to help the owner of an orange grove decide how many trees to plant on each acre of new land to maximize the harvest. 22 The yield of each tree and the number of trees per acre in the existing grove are explained and illustrated. An agronomist consultant explains that increasing the number of trees per acre decreases the yield of each tree and gives data the students can use. The students construct a chart and see that the total yield per acre forms a quadratic pattern. They investigate the properties of the function and answer a variety of questions, including questions about extreme cases.

The assessment can serve to introduce a unit on quadratic functions in which the students explore other task situations. For example, one group of sixth-grade students interviewed an elementary school principal who said that when cafeteria lunch prices went up, fewer students bought their lunches in the cafeteria. The students used a quadratic function to model the data, orally reported to their classmates, and wrote a report for their portfolios.

Sixth-grade students can be successful in investigating and solving interesting, relevant problems that lead to quadratic and other types of functions. They need only be given the opportunity. Do they enjoy and learn from these kinds of assessment activities and their instructional extensions? Below are some of their comments.

It is worth noting that the level of creativity allowable in a response is not necessarily tied to the student's level of enjoyment of the task. In particular, students do not necessarily value assessment tasks in which they have to produce responses over tasks in which they have to choose among alternatives. A survey in Israel of junior high students' attitudes toward different types of tests showed that although they thought essay tests reflected their knowledge of subject matter better than multiple-choice tests did, they preferred the multiple-choice tests. 23 The multiple-choice tests were perceived as being easier and simpler; the students felt more comfortable taking them.

A SSESSMENT IN S UPPORT OF I NSTRUCTION

If mathematics assessment is to help students develop their powers of reasoning, problem solving, communicating, and connecting mathematics to situations in which it can be used, both mathematics assessment and mathematics instruction will need to change in tandem. Mathematics instruction will need to better use assessment activities than is common today.

Too often a sharp line is drawn between assessment and instruction. Teachers teach, then instruction stops and assessment occurs. Results of the assessment may not be available in a timely or useful way to students and teachers. The learning principle implies that "even when certain tasks are used as part of a formal, external assessment, there should be some kind of instructional follow-up. As a routine part of classroom discourse, interesting problems should be revisited, extended, and generalized, whatever their original sources." 24

When the line between assessment and instruction is blurred, students can engage in mathematical tasks that not only are meaningful and contribute to learning, but also yield information the student, the teacher, and perhaps others can use. In fact, an oftstated goal of reform efforts in mathematics education is that visitors to classrooms will be unable to distinguish instructional activities from assessment activities.

I NTEGRATING I NSTRUCTION AND A SSESSMENT

The new Pacesetter™ mathematics project illustrates how instruction and assessment can be fully integrated by design. 25 Pacesetter is an advanced high school mathematics course being developed by the College Board. The course, which emphasizes mathematical modeling and is meant as a capstone to the mathematics studied in high school, integrates assessment activities with instruction. Teachers help the students undertake case studies of applications of mathematics to problems in fields, such as industrial design, inventions, economics, and demographics. In one activity, for example, students are provided with data on the population of several countries at different times and asked to develop mathematical models to make various predictions. Students answer questions about the models they have devised and tackle more extended tasks that are written up for a portfolio. The activity allows the students to apply their knowledge of linear, quadratic, and exponential functions to real data. Notes for the teacher's guidance help direct attention to opportunities for discussion and the interpretations of the data that students might make under various assumptions.

Portfolios are sometimes used as the method of assessment; a sample of a student's mathematical work is gathered to be graded by the teacher or an outside evaluator.

This form of assessment involves assembling a portfolio containing samples of students' work that have been chosen by the students themselves, perhaps with the help of their teacher, on the basis of certain focused criteria. Among other things, a mathematics portfolio might contain samples of analyses of mathematical problems or investigations, responses to open-ended problems, examples of work chosen to reflect growth in knowledge over time, or self-reports of problem-solving processes learned and employed. In addition to providing good records of individual student work, portfolios might also be useful in providing formative evaluation information for program development. Before they can be used as components of large-scale assessment efforts, however, consistent methods for evaluating portfolios will need to be developed. 26

Of course the quality of student work in a portfolio depends largely on the quality of assignments that were given as well as on

the level of instruction. At a minimum, teachers play a pivotal role in helping students decide what to put into the portfolio and informing them about the evaluation criteria.

The state of Vermont, for example, has been devising a program in which the mathematics portfolios of fourth- and eighth-grade students are assessed; 27 other states and districts are experimenting with similar programs. Some problems have been reported in the portfolio assessment process in Vermont. 28 The program appears to hold sufficient merit, however, to justify efforts under way to determine how information from portfolios can be communicated outside the classroom in authoritative and credible ways. 29

The trend worldwide is to use student work expeditiously on instructional activities directly as assessment. An example from England and Wales is below. 30

Assessment can be integrated with classroom discourse and activity in a variety of other ways as well: through observation, questioning, written discourse, projects, open-ended problems, classroom tests, homework, and other assignments. 31 Teachers need to be alert to techniques they can use to assess their students' mathematical understanding in all settings.

The most effective ways to identify students' methods are to watch students solve problems, to listen to them explain how the problems were solved, or to read their written explanations. Students should regularly be asked to explain their solution to a problem. Each individual cannot be asked each day, but over time the teacher can get a reading on each student's understanding and proficiency. The teacher needs to keep some

record of students' responses. Sunburst/Wings for Learning, 32 for example, recently produced the Learner Profile ™, a hand-held optical scanner with a list of assessment codes that can be defined by the teacher. Useful in informal assessments, a teacher can scan comments about the progress of individual students while walking around the classroom.

Elaborate schemes are not necessary, but some system is needed. A few carefully selected tasks can give a reasonably accurate picture of a student's ability to solve a range of tasks. 33 An example of a task constructed for this purpose appears above. 34

U SING A SSESSMENT R ESULTS FOR I NSTRUCTION

The most typical form of assessment results have for decades been based in rankings of performance, particularly in mandated assessment. Performances have been scored most

typically by counting the number of questions answered correctly and comparing scores for one individual to that for another by virtue of their relative percentile rank. So-called norm referenced scores have concerned educators for many years. Although various criticisms on norm referencing have been advanced, the central educational concern is that such information is not sufficiently helpful to improve instruction and learning and may, in fact, have counterproductive educational implications. In the classroom setting, teachers and students need to know what students understand well, what they understand less well, and what the next learning steps need to be. The relative rankings of students tested may have uses outside the classroom context, but within that context, the need is for forms of results helpful to the teaching and learning process.

To plan their instruction, for example, teachers should know about each student's current understanding of what will be taught. Thus, assessment programs must inform teachers and students about what the students have learned, how they learn, and how they think about mathematics. For that information to be useful to teachers, it will have to include an analysis of specific strengths and weaknesses of the student's understanding and not just scores out of context.

To be effective in instruction, assessment results need to be timely. 35 Students' learning is not promoted by computer printouts sent to teachers once classes have ended for the year and the students have gone, nor by teachers who take an inordinate amount of time to grade assessments. In particular, new ways must be found to give teachers and students alike more immediate knowledge of the students' performance on assessments mandated by outside authorities so that those assessments—as well as the teacher's own assessments—can be used to improve learning. Even when the central purpose of an assessment is to determine the accomplishments of a school, state, or nation, the assessment should provide reports about their performance to the students and teachers involved. School time is precious. When students are not informed of their errors and misconceptions, let alone helped to correct them, the assessment may have both reinforced misunderstandings and wasted valuable instructional time.

When the form of assessment is unfamiliar, teachers have a particular responsibility to their students to tell them in advance

how their responses will be evaluated and what criteria will be used. Students need to see examples of work a priori that does or does not meet the criteria. Teachers should discuss sample responses with their students. When the California Assessment Program first tried out some open-ended questions with 12-grade students in its 1987-1988 Survey of Academic Skills, from half to three-fourths of the students offered either an inadequate response or none at all. The Mathematics Assessment Advisory Committee concluded that the students lacked experience expressing mathematical ideas in writing. 36 Rather than reject the assessment, they concluded that more discussion with students was needed before the administration of the assessment to describe what was expected of them. On the two-stage tests in the Netherlands, there were many fewer problems in scoring the essays when the students knew beforehand what the teacher expected from them. 37 The teacher and students had negotiated a kind of contract that allowed the students to concentrate on the mathematics in the assessment without being distracted by uncertainties about scoring.

A SSESSMENT IN S UPPORT OF T EACHERS

The new visions of mathematics education requires teachers to use strategies in which they function as learning coach and facilitator. Teachers will require support in several ways to adopt these new roles. First, they will need to become better diagnosticians. For this, they will need "… simple, valid procedures that enable [them] to access and use relevant information in making instructional decisions"; "assessment systems [that] take into account the conceptualizations of learning, teaching, and the curriculum that are held by teachers"; and systems that "enable teachers to share assessment data with students and to involve students in making instructional decisions." 38 Materials should be provided with the assessments developed by others that will enable teachers to use assessment tasks productively in their instruction. Help should be given to teachers on using assessment results to encourage students to reflect on their work and the teachers to reflect on theirs.

Teachers will require assistance in using assessments consonant with today's vision of mathematics instruction. The Classroom Assessment in Mathematics (CAM) Network, for example, is an electronic network of middle school teachers in seven urban centers

who are designing assessment tasks and sharing them with one another. 39 They are experimenting with a variety of new techniques and revising tasks to fit their teaching situation. They see that they face some common problems regarding making the new tasks accessible to their students. Collaborations among teachers, whether through networks or other means, can assist mathematics teachers who want to change their assessment practice. These collaborations can start locally or be developed through and sponsored by professional organizations. Publications are beginning to appear that can help teachers assess mathematics learning more thoroughly and productively. 40

There are indications that using assessments in professional development can help teachers improve instruction. As one example, Gerald Kulm and his colleagues recently reported a study of the effects of improved assessment on classroom teaching: 41

We found that when teachers used alternative approaches to assessment, they also changed their teaching. Teachers increased their use of strategies that have been found by research to promote students' higher-order thinking. They did activities that enhanced meaning and understanding, developed student autonomy and independence, and helped students learn problem-solving strategies. 42

This improvement in assessment, however, came through a substantial intervention: the teachers' enrollment in a three-credit graduate course. However, preliminary reports from a number of professional development projects such as CAM suggest that improved teaching practice may also result from more limited interventions.

Scoring rubrics can also be a powerful tool for professional development. In a small agricultural county in Florida, 30 teachers have been meeting on alternate weekends, attempting to improve their assessment practice. 43 The county has a large population of migrant workers, and the students are primarily of Mexican-American descent. The teachers, who teach mathematics at levels from second-grade arithmetic to calculus, are attempting to spend less time preparing the students to pass multiple-choice standardized tests. Working with a consultant, they have tried a variety of new tasks and procedures. They have developed a much greater respect for how assessments may not always tap learning. They found, for

example, that language was the biggest barrier. For students who were just learning English requests such as "discuss" or "explain" often yield little information. The teacher may need, instead, to ask a sequence of questions: "What did you do first?" "Why did you do that?'' "What did you do next?" "Why?" and so on. Working with various tasks, along with the corresponding scoring rubrics, the teachers developed a new appreciation for the quality of their students' mathematical thinking.

Advanced Placement teachers have reported on the value of the training in assessment they get from the sessions conducted by the College Board for scoring Advanced Placement Tests. 44 These tests include open-ended responses that must be scored by judges. Teachers have found that the training for the scoring and the scoring itself are useful for their subsequent teaching of the courses because they focus attention on the most important features and lead to more direct instruction on crucial areas of performance that were perhaps ignored in the past.

Assessment tasks and rubrics can be devices that teachers use to communicate with parents and the larger community to obtain their support for changes in mathematics education. Abridged versions of the rubrics—accompanied by a range of student responses—might accomplish this purpose best. Particularly when fairly complex tasks have been used, the wider audience will benefit more from a few samples of actual student work than they will from detailed descriptions and analyses of anticipated student responses.

Teachers are also playing an active role in creating and using assessment results. In an increasing number of localities, assessments incorporate the teacher as a central component in evaluating results. Teachers are being recognized as rich sources of information about what students know and can do, especially when they have been helped to learn ways to evaluate student performance. Many students' anxiety about mathematics interferes with their test performance; a teacher can assess students informally and unobtrusively during regular instruction. Teachers know, in ways that test constructors in distant offices cannot, whether students have had an opportunity to learn the mathematics being assessed and whether they are taking an assessment seriously. A teacher can talk with students during or after an assessment, to find out how they inter-

preted the mathematics and what strategies they pursued. Developers of external assessment systems should explore ways of taking the information teachers can provide into account as part of the system.

In summary, the learning principle aims to ensure that assessments are constructed and used to help students learn more and better mathematics. The consensus among mathematics educators is that assessments can fulfill this expectation to the extent that tasks provide students opportunities to extend their knowledge, are consonant with good instruction, and provide teachers with an additional tool that can help them to become better facilitators of student learning. These are new requirements for assessment. Some will argue that they are burdensome, particularly the requirement that assessments function as learning tasks. Recent experience—described below and elsewhere in this chapter—indicates this need not be so, even when an assessment must serve an accountability function.

The Pittsburgh schools, for example, recently piloted an auditing process through which portfolios developed for instructional uses provided "publicly acceptable accountability information." 45 Audit teams composing teachers, university-based researchers, content experts, and representatives of the business community evaluated samples of portfolios and sent a letter to the Board of Education that certified, among other things, that the portfolio process was well defined and well implemented and that it aimed at success for all learners, challenged teachers to do a more effective job of supporting student learning, and increased overall system accountability.

There is reason to believe, therefore, that the learning principle can be honored to a satisfactory degree for both internal and external assessments.

To achieve national goals for education, we must measure the things that really count. Measuring What Counts establishes crucial research- based connections between standards and assessment.

Arguing for a better balance between educational and measurement concerns in the development and use of mathematics assessment, this book sets forth three principles—related to content, learning, and equity—that can form the basis for new assessments that support emerging national standards in mathematics education.

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Linear programming class xii chapter 12, math assignment class x ch-3 | linear equations in two variables.

  CHAPTER 3    CLASS   X

PAIR OF LINEAR EQUATIONS IN TWO VARIABLE

Extra questions of chapter 3 class 10 : Pair of Linear Equations in Two Variable with answer and  hints . Useful math assignment for the students of class 10

Basic points of  Chapter 3 Pair of Linear Equations in two variable .

ASSIGNMENT BASED ON CH-3 CLASS 10

a)   2x - 3y = 1 ;  x + 2y = 2

Ans: Unique solution, intersecting lines, consistent

Ans: No solution, parallel lines, inconsistent

Ans: Many solution, coincident lines, consistent

a)    2x + y - 5 = 0;  6x + 3y + p = 0       Ans [p = -15]

a)   8x + 5y = 9;   kx + 10y = 15   Ans: k =16

Answer:   a)   k = 16 ,    b) k = -1,     c) k = 2,     d) k = 土 6

a)  (a + b)x  - 2by = 5a + 2b + 1;    3x - y = 14             Ans [a = 5, b =  1]

b)     2x - 3y = 7;  (a + b)x  +  (a + b)y = 4a + b           Ans [-5, -1]

c) (2a - 1)x + 3y – 5 = 0;   3x + (b - 1)y – 2 = 0          Ans [a = 17/4,   b= 11/5]

d)   kx + 3y = 2k + 1;  2(k + 1)x + 9y = 7k + 1            Ans [k = 2]

e)   2x + 3y = 7;   (k – 1)x + (k + 2)y = 3k                  Ans [k = 7]

a) x + y = 3;   2x + 5y = 12                        Ans [1, 2]

b)  2x + 3y + 5 = 0;   3x - 2y - 12 = 0        Ans [2, -3]

a) x + y = 5;      2x - y = - 2                     

[Sol. (1, 4) & Area= 12]

  [Area= 7.5]

a)     x + y = 7    ;   3x - 2y = 1                  Ans [3, 4]

Ans [45 and 30]

Ans [Rs1800 & Rs 1400]

Ans  [Rs 6.50 and Rs 1.50]

Ans  [93]

Ans [47 or 74]

Ans   [64]  

Ans  [12/25]

Ans  [7/18]

Ans  [42, 10]  

Sum of two numbers is 105 and their difference is 45. Find the numbers.

Answer: Required Numbers are: 75, 30

Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old are Rashmi and Nazma now ?

Let present age of Rashmi and Nazma be x years and y years respectively

Therefore, x - 3 = 3(y - 3)

Or x - 3y + 6 = 0

And x + 10 = 2(y + 10)

Or x – 2y – 10 = 0

Solving equations to get x = 42, y = 16

Present age of Rashmi is 42 years and that of Nazma is 16 Years

Let number of correct answers be x and

number of incorrect answers be y

3x - y = 40

4x - 2y = 50

Solving, we get x = 15, y = 5

Total number of questions = 20

Ans  [10 km/h and  2 km/h]

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