'Problem solving' has been a buzzphrase in mathematics education for around 15 years (Don Coleman says it is much longer than that). In 1989, the National Council of Teachers of Mathematics (NCTM) published a set of national standards for teachers to go by when teaching mathematics to students from Kindergarten to 12th grade. The standards emphasized that 'problem solving' aspect of mathematics, and this idea has caught on.
Most universities and colleges now have problem solving courses as part of the curriculum for aspiring teachers, and there have been several very good books written to use in these courses. So why should we write another book on problem solving? Well, as the title suggests, our approach to problem solving is to emphasize the teacher role in problem solving. We will concentrate on the activities a teacher engages in when making up problems to teach the student to solve problems, with special emphasis on the preparation of homework sets for posting on the Web Homework System, hosted on mathclass.com, a University of Kentucky Mathematical Sciences server .
The text is made available in multiple formats. The most familiar form is the portable document format or pdf file . A pdf file can be read in an Acrobat reader, which you can download free from http://www.adobe.com/products/acrobat/readstep2.html. This file can be read on a computer screen, printed out and read like a book, or printed out in relevant pieces. It is paginated, has a table of contents at the front showing where topics are located by page number, text divided into chapters and sections, and an index at the back for looking up terms and other keywords and phrases, again by page number.
The text is also made available as an html (hypertext markup language) document. You are familiar with this from the internet. Most web pages you look at when you surf the internet for information are written in html. The text in this format is not paginated, but it does have a table of contents and index, which are active in the sense that the entries are hyperlinked to their references. So if you run across a word or phrase in the text which you don't know or can't figure out from the context, you can go to the index and look it up. If you find it in the index, you just click the link, and you are taken to a place in the text where the term is defined either explicitly or by its context. One can read a well made html document more efficiently than a hardcopy version of the same text. If you want to make notes, you can print out relevant portion of the text and make notes on that.
The third format the text is available in is as a book of Maple worksheets . This is the source document for the text, that is, any changes to the text are made in the Maple worksheet and then exported to pdf and html. Perhaps this will be the most useful of all the formats as you become familar with it. Maple is a mathematical word processor and high powered graphing calculator. There are other programs, such as Mathematica and Matlab, but we have chosen Maple. You must have ready access to Maple to make much use of this text.
The book of worksheets format has a table of contents/index worksheet which acts a hub for your reading; that is, you open this worksheet first, then click on the links in either the table of contents or the index to go to where you want to read and work. While there you can execute cells or make notes right in the worksheet. When you are finished, you can click on the back arrow at the top of the worksheet to go back to the hub. If you have made changes, you are given the opportunity to save them.
Making up homework problems is an activity that every math teacher does. Often, it is simply a matter of choosing some problems from a textbook, or a list of problems already made up. But it should include some problems that you have had a hand in constructing, possibly alone or working with a group of teachers. When you make up quizzes and tests for your students, you are engaging in problem creation. It is work, and takes effort. But it is worthwhile and interesting work, especially if you have some interesting ways to make up problems.
Polya's four steps to solving a problem
George Polya (1887-1985), a Hungarian mathematician, wrote "How to solve it." for high school students in 1957. Here is his four step method.
Understand the problem:
Read the problem over carefully and ask yourself: Do I know the meaning of all the words? What is being asked for? What is given in the problem? Is the given information sufficient (for the solution to be unique)? Is there some inconsistent or superfluous information which is given? By way of checking your understanding, try restating the problem in a different way.
Design a plan for solving the problem:
In essence, decide how you are going to work on the problem. This involves making some choices about what strategies to use. Some possible strategies are:
Draw a picture or diagram
-- making a picture which relates the information given to what is asked for can often lead to a solution.
Make a list
-- this is a strategy which is especially useful in problems where you need to count the members of a set.
Solve smaller versions of the problem and look for a pattern
-- almost any problem can be made simpler in some way. By working out simpler versions, you can often see patterns which help solve the original problem.
Decompose the problem
-- Many problems can be broken into a series of smaller problems. This strategy can turn a problem which on first glance seems intractable into something more doable.
Use variables and write an equation
-- the method of algebra. Very useful in a lot of problems. See the section below for more details and examples.
Carry out the plan
: Spend a reasonable amount of time trying to solve the problem using your plan. If you are not successful, go back to step 2. If you run out of strategies, go back to step 1. If you still don't have any luck, talk the problem over with a classmate.
Look back
: After you have a proposed solution, check your solution out. Is it reasonable? Is it unique? Can you see an easier way to solve the problem? Can you generalize the problem?
A method for setting up and solving problems with algebra
A huge number of problems that you or I make up can be solved by setting up and solving a system of equations. Here is a method for setting up and solving such a problem.
Problem: Given that the area of a rectangle is 40 square feet, and its height is 10 feet, find its width.
Solution: Call its width x. That is the unknown. Call the area A and the length b. These are the parameters.
We know the area of a rectangle is its height times its width, that is,
. Since we only have one unknown, we can stop and solve the equation for x in terms of A and b:
feet. For the original problem. x = 40/10 = 4 feet.
Problem: Given that 1 hp pump can fill a tank in 30 minutes and a 1.5 hp pump can fill it in 20 minutes, how long would it take to fill the tank when both pumps are working?
Solution: Let t be the time in minutes it takes for both of them to fill the tank together. So in 1 minute, working together they fill 1/t of the tank. The first motor fills 1/30 of the tank in 1 minute, and the second motor fills 1/20 of the tank. So together
1/t = 1/30 + 1/20 of the tank. Solve for t to get t = 600/50 = 12 minutes. Note: If the first motor fills in a minutes and the second in b minutes, then the equation is 1/t = 1/a + 1/b, so t = 1/(1/a+1/b) = a*b/(a+b).
Basically, what we are doing here adding a fifth step to Polya's method: Pose the problem .
Listed below are some ways that you can use to provide inspriration, motivation, or focus while you are trying to think up problems.
Take a problem you can already solve
This is a way people do research in mathematics. Start with an interesting problem you can solve and try to think up related problems. There are some standard ways to do this. We describe some of them briefly here, and in more detail later.
1. You can insert a parameter into the problem. That is, replace one of the given numbers in the problem with a letter, which we regard as the name of an unspecified number. Then see if you can solve the 'parameterized' problem. Of course, the solution will be stated in terms of the parameter, so it may be an algebraic expression.
Example. Jack has 5 apples and Jill has 3 pears. How many pieces of fruit do they have together? Answer: 5 + 3 or 8 pieces of fruit.
Parameterized problem: Jack has a apples and Jill has 3 pears. How many pieces of fruit do they have together? Answer: a + 3 pieces of fruit.
Solving a parameterized version of a problem exposes the method of solution and leads naturally to other investigations.
2. If you have parameterized a problem, you can investigate the domains of the parameters, that is you can ask 'for what values of the parameters does the answer make sense?'. In the above example, we can see that the parameter a should be a non-negative integer.
3. We can pose some of a problem's inverse problems. For example, an inverse problem to the fruit problem above is: If Jack has 6 apples and Jill has pears so that together they have 15 pieces of fruit, then how many pears does Jill have?
4. What are some other interesting questions you can ask with the same givens as in the problem?
Virtually any problem will be stated in some context or setting, and so without fail, there will be other questions you can ask in that context.
Example. A famous problem from plane geometry is to show that if three points in the plane are noncollinear, then they are on a circle. What are some other questions you might ask in this setting?
Question. Must any 4 noncollinear but coplanar points lie on a circle?
Solution: This one is easy. The first three points determine a circle. Now the 4th point is probably not going to lie on that circle, and in any case we could pick another 4th point that does not lie on it. So, the answer is, no, 4 random points need not lie on a circle.
Question. Choose 3 noncollinear points at random in the plane. What is the probability that they lie in a semicircle? QED
Solution:
This one is not so easy. Call the points
,
, and
. Call C the circle thru the points. Call
the semicircle of C going counterclockwise from
for i = 1,2, and 3. Now all three points lie in a semicircle if and only if they lie in exactly one of the
's. Since the points were chosen at random, the probability that
and
both lie on
is
=
. Since there are 3 mutually exclusive possibilities, we get a probability of
that 3 randomly chosen points in the plane lie in a semicircle. QED
If the problem is one you haven't had luck in solving, then by asking and solving related questions, sometimes you can sneak up on the solution to the original problem.
Take a setting, picture or theme
The setting for a problem is there before the problem was posed, so you don't have to start with a problem, you can take an interesting setting and look for problems in that setting.
Quadrille paper.
There is no end to the interesting problems you can pose using a sheet of quadrille paper. For example,
Problem. Make a 2 by 4 rectangular grid, and insert the digits 1 thru 8 in the cells so that no two adjacent cells contain concurrent numbers. Note: Two cells that touch on an edge or at a corner are adjacent.
Pyramids
Let B be a region in the xy-plane and let v be a point above the plane. The pyramid P with base B and vertex v is the union of the set of segments connecting v with a point in B. For each edge E of B, the union of the segments connecting v with a point e of E is called the side of P based on E. The edges of P consist of the segments which are edges of one or more sides of B. The vertices of P are defined similarly. There are a ton of questions you could ask about pyramids. For example.
Problem. How many edges will a pyramid P have? What is it's volume? How would you define the interior of a pyramid?
Inscribed figures
If P and Q are convex polygons, and each vertex of P lies on a side of Q, then P is said to be inscribed in Q. Give some examples of pairs of polygons P and Q with P inscribed in Q.
This is an experiment I have tried. Rip pages at random out of an old magazine and distribute them to your class, one to each student. Give them 15 minutes to come up with two interesting problems motivated by what they have read in their page. You may be pleasantly surprised by what you get, and in any case it is a good exercise for the students.
Have a set of 'standards'
We can make up interesting homework problems which exercise and test a mastery of any set of standards.
Standards are in. In 1989, the National Council of Teachers of Mathematics (NCTM) issued its first set of national standards for grades K-12. These were such a big hit that they were greatly enlarged and updated in 2000. The NCTM web page at http://www.nctm.org/ has all kinds of materials related to their standards in K-5, Middle, and High school (which is called 9-11 for some reason). The province of Ontario has an extremely detailed set of standards that their students are to be taught to and measured against.
http://www.edu.gov.on.ca/eng/document/curricul/curr97ma/curr97m.html . And, closer to home, we have the Kentucky Dept. of Education core content assessment standards: look for Core Content in this link
http://www.kde.state.ky.us/KDE/Instructional+Resources/default.htm
One of the main goals of this section is to discuss how to develop WHS homework which can be used by teachers to measure (and document that they do so) their students against standards of various kinds.
Course Standards
In our teaching, we speak of 'course standards', by which we mean a list of things we want our students to know, or be able to do, or be able to use at the end of the course. These standards are usually listed in the course syllabus which we give to our students at the beginning of the course.
Course standards can be organized along several lines. Probably the simplest is to organize them temporally according to the order in which they are first addressed in the course. Then the standards can be assigned ordinal numbers to distinguish them from one another.
As an example of a set of course standards, we could cite the ones that are used in Ma 123 Elementary Calculus at the University of Kentucky. There are 40 standards arranged in logical (and thus chronological) order. These are broken into 4 groups for examination purposes. So the student knows in advance which standards will be tested at each exam.
Here are the first 5 standards written in Maple strings for use with WHS homework.
| > | ma123[1]:="1. Student has an understanding of the relationship between the geometry of lines in the plane and analytic expressions (i.e. equations) describing them. In particular, student can calculate equations of lines from point or point-slope, and information about intersection, parallel and orthogonal relationships. Conversely they must be able to deduce the corresponding geometric information from the analytic.": ma123[2]:="2. Student knows the quadratic formula and is able to calculate the point(s) of intersection of two quadratics, a quadratic and a line, line and circle, etc.": ma123[3]:="3. Student is able to correctly and precisely define/describe the derivative of a function at a point in both geometric and analytic terms. They should understand that the analytic form represents the limiting value of the slopes of secant lines.": ma123[4]:="4. Student can calculate the derivative of a simple function (linear , quadratic, f(x) = a/(x-b) ) at a given point as a limit (i.e. using the analytic definition)": ma123[5]:="5. Student knows the standard forms of the equation for a circle with a given radius about a given point and should be able to find the tangent line to a circle at a point.": |
A start on a set of standards for a pre-calculus course
It is an interesting and important exercise to make up course standards for the courses we teach.
The student will be able to:
1. use the associative, commutative, and distributive properties of addition and mulitplication to decide if two expressions for numbers obtained by combining other numbers (possibly unspecified) using those operations represent the same number;
2. state the definitions of subtraction and division, and say what the basic properties of these operations are;
3. do item 1 with the operations of subtraction and division thrown in;
4. .....
Project: A sample of what is to come
Let's make a start on a set of standards for a course we teach (Algebra I?) Break up into teams of 3 or 4. Choose a name for your team. One member of the team will put in the team answers to the question below, either into a Maple worksheet or a Word document. This member will post the homework referred to in exercise 7 below, and toggle downloading.
| > |
Introduction to homework preparation for WHS
WHS tags: A brief description with examples
The source. Login to WHS, go to Teacher Resources, and click on Authoring Tools. At the bottom of the page, click on Author Documents and print it out. This gives a succinct description of all the WHS tags, including the answer format tags. Below are several examples of problems which have been composed by hand using the tags. This can be done in any text editing program such as word 2000, edit, etc.
Homework Assignments from Maple Worksheets
Input files for WHS are most conveniently made using the computer algebra system Maple (current version 9.5). This means that the author can use the computational power of Maple to check results and draw plots for the assignment as well as the high level worksheet interface to format mathematical formulas. One intersperses these supporting computations with the input to be read by WHS. The Maple .mws file is then written out as HTML and .gif image files; these are made into a .zip file which is uploaded to WHS and converted into the .xml file which is actually read by WHS.
In order for the worksheet to be usable by WHS, it must have the format which we are about to describe. Note that the current implementation of WHS should accept any worksheet input in the specified format. It does, in fact, accept certain no longer supported formats in the interest of backwards compatibility.
The lines of text in the spreadsheet can be divided into two classes:
Control lines are instructions to WHS. They specify such things as the beginning of a question, the location of an input widget such as a text box, and the beginning of a part of the worksheet that should be ignored by WHS. Control lines have a control entity, strings of a particular form which always contains an underscore character. The precise format of these entities are described below. But, it is recommended that worksheet authors should simply avoid use of the underscore character entirely except in control lines. Other lines are simply referred to as normal text lines .
Input Ignored by WHS
Each control line together with all the normal text lines immediately following it and appearing before the next control line make up a section of the input file. The normal text lines appearing before the first control line are also said to form a section.
The following sections are ignored by WHS:
All sections prior to the first control line containing H_, T_ or Q_ are ignored.
All sections with a control line containing SKIP_ and all immediately following sections with control lines not containing T_ or Q_ are ignored by WHS.
If the first section not ignored by WHS has a control line containing H_[x] where x is a positive integer or H_, then the text of the section is the global header. It is output at the beginning of the Homework assignment. Typically, it is used for a title. If the [x] is specified, then x is the multiplicity , which defaults to the value 1. If the multiplicity is greater than 1, then out of each group of x questions, only one randomly selected one is included in the displayed assignment. Each time the assignment is displayed, different selections of problems are obtained.
In more detail, when an assignment is requested, one can specify a numerical version number. There are several possibilities:
If the version is -1, one gets the Common version. If the multiplicity is m, then this consists of questions whose numbers are 1 mod m.
If one requests the Personal version, then one gets the version whose version number is equal to the account id of the user.
One can specify a Specific version by giving the version number.
One can specify a Specific version 0 or simply leave the version number field blank. This causes a random version to be selected.
Versions other than the Common version are made up by pseudo random selection of one problem from the last m - 1 of the problems in each group of m contiguous problems.
The parameters between the square brackets are semicolon separated. The first is the multiplicity as defined above. The second argument will not be addressed here; simply leave it blank. By putting a third parameter called base , you can cause the randomization to start after question number base, and for the Question XXX not to be displayed. This was a hack for KEMTP. Experimental feature: if you set this parameter to -1 and if the multiplicity is greater than 1, then the problems selected for a version will appear in the same order as in the Maple or Word document; if you need to have the original functionality of base, then replace it with -1 - base, e.g. base = 1 gives a new value base of -2 if you want no scrambling.
By putting a fourth parameter called header , one can vary the logo image at the top of the page. The value of the parameter is a filename; the file itself must be located in the html subdirectory of the main wqs directory of the server.
Finally, by putting a fifth parameter called hwFlags , one can affect how the homework xml is generated. If the value of hwFlags is 1, then video links on the client machine will be to files in the root directory. If you do not want this, then use 0 or no value for this parameter.
Header examples :
1. A homework with a 56 question database and a header H_[5;;6;0;1] will have 16 questions in each retrieved homework. The first 6 will be unnumbered and will be the first 6 questions in the database. The remaining 10 questions will be numbered and will be the first question in each group of 5 questions in the database counting from the 7th question in the database, if the version is -1. For other version numbers, the questions are chosen from the last 4 in each group of 5 counting from the 7th. Any videos that are associated with the homework are accessible from the top of any drive they are stored on.
2. If the header in the above setting is changed to H_[5;;-7;0;1], the only thing that changes is that the problems occur in the same order they are listed in the database.
The Leader Tag and Question tag
The remaining sections not ignored by WHS can be grouped into questions. Each question consists of :
An optional leader section. The control line contains T_ . The normal text is used as a lead-in to the question. In extreme cases, it can be an entire lecture of material before the first question.
A mandatory question section whose control line contains QM_ or QN_
A series of answer sections. There should be at least one in a QM_ question. The control line is identified by the string Ax_ where x is one of the upper case letters C, F, H, L, R, and S. The letter specifies particular answer formats. In addition, for QN_ questions, one can use X.
Question Control Line Formats
Questions whose answers are not to be graded by WHS are QN_ questions ; those whose answers are to be graded by WHS are
QM_ questions
. Students are given the opportunity to submit comments about QM_ questions only if they submit a proposed answer; these comments are sent via e-mail to their instructors.
If the character immediately following the underscore is an upper case N, then the correct answer is not displayed when the student submits an incorrect answer to the question. Otherwise, the correct answer is displayed when an incorrect answer is submitted. The correct answer is always displayed when a correct answer is submitted.
Correct answer specification occurs immediately after the underscore character (or the letter N following the underscore character. It should occur for both QM_ and QN_ questions, even though the answers are ignored in the case of QN_ questions. The specification is enclosed in square brackets and consists of a sequence of at least two quantities. The quantities are semicolon separated. The first quantity is the error bound; its use will be described below. The remaining quantities are the correct answers for each of the input fields in the answer section. The format of these will also be described below.
Note that checking of answers is done with an .asp script interpreted by a VBasic interpreter. This has the unfortunate consequence that expressions are evaluated using vbasic's non-standard operator precedence in which unary minus is of higher precedence than exponentiation. For example, -3^2 is interpreted as 9 and not -9. It is recommended that one reduce the likelihood of misinterpreted results either parenthesizing expressions or using numerical answers whenever possible.
The correct answer specification of QM_ questions can be followed by an additional square bracket enclosed semicolon separated list of non-empty description strings. These strings should be case insensitive and contain neither square bracket characters nor single or double quote characters or semicolons. The intention is that these strings will be the names of standard skills being tested by the question. The list of the strings together with the question numbers where they are listed is stored in the homework descriptor. The homework results pages show numbers of problems correctly answered for each description string.
The various answer types correspond to different ways to input or to check answers. In order for a question to be marked correct, all the input answers must be correct.
Header Answers: These have AH_ in the control line. A single parameter with value 0, 1, 2, or 3 can be specified by adding [x] after the AH_. The normal text lines are output and, if the parameter is specified and has value 1 or 3, then it is followed by a line break, i.e. <br/>. If the value is 2 or 3, then it is preceded by a line break. There is no (correct) answer for a header answer section.
Header answers can also be used to put in a video link. In this case the first parameter is as before, the second parameter is the name of the video file, and the third parameter is a label. The body of the header will be displayed followed on the right with a pull down menu and a button with the label specified as the third parameter. The pulldown menu allows the user to choose a disk drive or more typically a cd rom drive; in this case the video file is assumed to be located in a subdirectory of the root directory; the name of this subdirectory should be the homework identifier. The default value of the drive letter is So, a typical example would be
AH_[0;myvideo.wmv;Show Video]
For a video about this this or a similar problem:
simply a blank; this causes the file to be retrieved from the WHS video server. Note: if the Homework header has a fifth argument of 1, then the video is assumed to be in the top directory of the drive letter chosen (so, if the video bill.wmv is stored on a CD which is inserted into the L drive, then WHS will look in L://video.wmv for the video.
These have AC_[x] in their control line where x is positive integer indicating the length in characters of the input text box. The input text box is followed by the normal text lines of the section. The student should input a single number expressed in 'calculator syntax'. This means an expression with explicit asterisks for multiplication and can include pi for 3.14159..., e for 2.718..., and elementary function calls such as ln(...), exp(...), sin(...), etc. The correct answer should have the same format. The submitted answer is said to be correct if its value differs in absolute value from the value of the correct answer by no more than the error bound.
Example:
QM_[0;4]
What is 1 + 3?
AC_[5]
SKIP_
These have the same format as Constant Answers except that you use AW_[x] instead of AC_[x]. The answer is a sharp character separated list of strings. The submitted answer is said to be correct if at least one of these strings is the same string as the answer. Both answer and submitted string are trimmed of leading and trailing whitespace. The comparison is case sensitive.
Example.
QM_[0;triangle#Triangle#TRIANGLE]
If A, B, and C are noncollinear points, then the union of the three line segments AB, BC, and CA is called a
AW_[8]
.
SKIP_
Constant answers can also be input as entries in tabular arrays. To do so, use
AT_
[....] control lines where the semicolon separated parameters are:
The size S of the textboxes, in characters.
The number R of rows in the table
The number C of columns in the table
The actual entries in the table -- there should be R times C of these entries. Any entry which is empty is replaced by a textbox of length S.
The correct answers are specified as several entries in the list of correct answers in the QM_[...] control line for the question. The answers occur in row major order.
Example:
QM_[0;12;15]
If f(x) = x + 3, then complete the table of function values.
AT_[4;3;2;x;f(x);9;;12;]
SKIP_
These have AF_[x;t;n;a;b] in their control line. The parameters are semicolon separated. The integer parameter x is the size in characters of the input text box. The t parameter is the variable of the function used in both the correct answer and in the input answer. When the answer is checked, n random values of this variable are chosen from the half closed interval [a,b); the functional values of the correct and submitted answers are calculated, and the absolute values of the difference of these values must all be no larger than the error bound in order for the answer to be considered correct.
Example:
QM_[0;x+12]
AH_[0]
If f(x) is a linear function and f(9)=12, and f(12)=15, then f(x) =
AF_[10;x;2;0;1]
SKIP_
These are like function answers but allow for multiple variables. These answers have AE_[x;m;x1;...;xm;n;lo1;hi1;...;lom;him] in their control line. The parameters are semicolon separated. The integer parameter x is the size in characters of the input text box. The parameter m is the number of variables, and the variable names expected in the submitted answer are x1, x2, ..., xm. When the answer is checked, n random m-tuples will be generated where the jth coordinate lies in the range of [loj, hij) and is used for the variaable xj. The functional values of the correct and submitted answers are calculated, and the absolute values of the difference of these values must all be no larger than the error bound in order for the answer to be considered correct.
Example:
QM_[0;x+2*y]
AH_[0]
If f is a linear function of two variables, with f(0,0)=1 and f(1,0)=2 then f(x,y) =
AE_[10;2;x;y;2;0;1;0;1]
SKIP_
These have exactly the same format as Function answers except that they use AI_ in place of AF_. When integral answers are checked, the functions are checked for equality up to a constant summand. The test is done with n - 1 randomly selected points in (a,b).
Example:
QM_[0;x^2+2*x]
AH_[0]
Give an antiderivative for 2*x + 2. Use calculator notation.
Answer =
AI_[10;x;2;0;1]
SKIP_
These have
AS_
[val1;val2;...;valk] in their control line. The input is a pull down menu with val1, val2, ..., valk as the menu choices plus the possibility of a blank answer. The possible answers are semicolon separated strings. The blank answer is originally displayed; an answer is selected if the display is any of the other values. The correct answer must be one of the strings val1, val2, ..., valk. The normal text lines are displayed after the pull down menu. The strings val1, val2, ..., valk should not contain either semicolons or sharp characters.
Example:
QM_[0;8]
AH_[0]
If f(x) = x + 3, then the value of f at x = 5 is
AS_[8;2;6;7]
SKIP_
Horizontal Line of Radio Button Answers:
These have AL_ [val1;val2;...;valk] in their control line. The possible answers are the semicolon separated strings val1, val2, ..., valk. The correct answer must be one of the strings val1, val2, ..., valk. The radio buttons are displayed in a centered single row table with the corresponding answer string to the right of the radio button. The strings val1, val2, etc. should not contain either semicolons or sharp characters.
Example:
QM_[0;13]
If f(x) = 2*x + 3, then the value of f at x = 5 is
AL_[13;8;10;11]
SKIP_
Horizontal Line of Checkbox Answers:
these have AB_ [val1;val2;...;valk] in their control line. The interface generates a one line centered table of checkboxes labeled by val1, val2, ..., valk. The correct answer string should have k semicolon separated answers which are either vali or the empty string depending on whether or not the i-th checkbox should be checked in order for the result to be counted right. Because checkboxes are two-state, one cannot distinguish between the answer not being given and not having any of the boxes checked. So, the author should always word the question in such a way that the student should have at least one box checked in order for the question to be answered -- e.g. one of the boxes could have the value 'none' and the others could have numbers for various possibilities which might be true.
Example:
QM_[0;;2;;;10/5]
If f(x) = x + 3, then the value of f at x = 5 is
AB_[8;2;6;7;10/5]
SKIP_
Vertical Row of Radio Button Answers:
This is used when the value of the button needs a longer description. Each radio button is described by its own answer section, whose control line is of the form
AR_[x]
where x is a small integer used to identify the radio button group. A group consists of contiguous AR sections with the same group identifier -- they act in unison, in that only one of the group may be selected at once. The entire group of radio buttons is indented as a <dir> group. After each radio button, the section's normal text lines are inserted. The correct answer is an integer corresponding to the number of the radio button in its group (counting from 1). When the submitted answer is displayed on the feedback page, it is specified in the same way. Because some authors prefer to list the correct answer first in the list, the first answer is re-inserted in a random location in the group.
Example:
QM_[0;2]
Which is the correct definition of parallel lines in space?
AR_[1]
Two lines whose intersection is empty.
AR_[1]
Two coplanar lines whose intersection is empty.
AR_[1]
Neither of the others
SKIP_
These are used only with QN_ type questions. The
AX_[x]
where x is a small positive integer causes a textarea element to appear; it has x rows and 40 columns. Typically, you will want to proceed it with a line break, which can be accomplished with an AH_[3] section before it. Since they only occur in QN_ questions, there is no grading of AX_ answers.
Example:
QN_[0;to be hand-graded]
Discuss the universe and give 3 examples.
AX_[3]
SKIP_
QN questions can be hand-graded.
To do so, the instructor goes to Access Records for the class and then clicks on Assignment submission. For the particular assignment, one can click on Display and then click for each student to display submission of the student. There is a Right/Wrong box for each problem to be hand graded.
Creating a homework using maple.
How do we create a homework? The main tool is the word zipit from the MCtools package. It has a help page: access it by typeing
| > | mctools(zipit); |
Now the steps to creating the homework are:
1 If necessary, Get the most recent MCtools section from
https://www.msc.uky.edu/carl/communicating_math/MCtools_page.htm
and copy the section into the top of your homework. Or, if you are starting fresh, just use the worksheet that MCtools comes in as your homework worksheet. It has a Kentucky standards section, a tagit examples section, and a start on a homework that you can modify. Note: If you start with a fresh worksheet, you should merge the styles in the MCtools worksheet you downloaded into your worksheet. Click on the Format menu at the top of the worksheet, then click on Merge existing ..., then browse for the downloaded MCtools worksheet (save it somework if you haven't), select it, click on open, then click on done.
2. Open the MCtools section and execute it, then close the section (This is something you do each time you start a Maple session.
3. Make a title for the homework. This involves putting some text in the title section.
4. Create a folder to save the homework in and save it there. You can do this from Maple from the File Menu.
5. Create the correct zipit line for this homework.
Modify the input line starting with zipit at the top of the worksheet so that the first argument is the homework worksheet name and the second argument is the correct path to your homework folder. For example, if your homework worksheet is called hw1.mws and the path is C://temp/hw1, then your zipit line would look like this
zipit("hw1","C://temp/hw1");
Don't execute this just yet (wait until you have created the problems and exported to html).
6. Make the homework problems . Put problems in their own sections so that you can eliminate them from the homework easily by just closing off the section. All work pertaining to the problem can be put in this section. Put in just one or two problems to get started, and post the homework just to see everything is working. Before you get the homework like you want it, you will go through the editing cycle ( Save Export Zipit Install and Preview ) several times.
7. Save the worksheet again and export it to html. To save it again, just click on File, then click on Save. The changes you have made will be saved. To export to html, click on File, then click on Export As, then click on HTML. Accept all suggestions Maple makes.
8. Now execute the zipit line . Put your cursor in the input cell and press enter. A zipfile named hw1whs.zip (if your homework was called hw1) is created and placed just above the homework folder hw1. Important: Note that the zipfile's name hw1whs is not the same as the worksheet's name hw1. The reason is that zipit creates the html file hw1whs.html in the folder which is the file to be used by WHS to create the homework. Unless you tell it otherwise, zipit will always create a zipfile whose name is the name of your worksheet
Note: If you get an error message saying you don't have the zip program on your system, you can get the (free) info-zip from the MCtoolss Page at
or from the Info-zip Home page. Put it in C://Winnt or C://Windows. There is an info-zip unzip program, but in XP you can extract the files in a zipfile with the window explorer, and in windows 2000 you can use 7-zip or winzip to unzip a zipfile.
9. You are ready to install the new homework zipfile on WHS
Using the export to html with Mathml option in Maple
With the MCtools of June 2004 or later, you can choose to export your files to html with Mathml . WHS can handle this format now. Its principle advantage is that formatted mathematics is exported in a scalable form. Its principle drawback is that current browsers do not display it properly unless a plugin is installed first. IE6 requires Mathplayer (free from http://www.dessci.com/en/products/mathplayer/). So homeworks installed this way should have a notice in the homework header telling the students that they may need to install mathplayer before they can view the homework properly.
1. Log into your WHS account, using Internet Explorer.
2. Highlight the Teacher Resources menu item and click on the Authoring Tools Button.
3 Go to the Homework Installation table and press the Browse button. Use the window that opens up to locate the homework zipfile hw1whs.zip on your computer (Remember, it is just above the directory hw1whs) The icon for the zipfile has a little zipper on it to distinguish it as a zipfile.
4. Highlight the zipfile and click it to make it appear in the Open line in the browse window.
5. Press the Open Button in the browse window. The browse window disappears, and the path to the zipfile appears in the upload line of the bottom table.
6. Press the Install and test homework button, and wait. After a few seconds, the zipfile is uploaded, installed, and the homework is retrieved for you to preview.
Once you have a homework installed, you will find that you need to make some changes. You might want to correct an error in the homework, or add a problem, or delete a problem.
If you are working on the computer from which you made the last installation of the homework, all you have to do is open the source worksheet (unless it is already open, because you just installed the homework), make the changes, then go through the edit cycle : S ave, E xport, Z ipit, I nstall The first 3 steps take place in the Maple worksheet, and the last 2 take place in Internet Explorer. Keep both applications open as you edit. That way, you can go more easily through the cycle. Occasionally, you will have to re-authenticate your login to mathclass, if you spend more than a few minutes making changes in the source worksheet.
If you are working on a different computer (but one that has Maple and Internet Explorer), then you can login to mathclass, go to Teacher Resources , then click on Authoring Tools, then find your homework in the Homework Testing Table and click on Download. If you get the message saying you don't have permission to download, go to the Homework Installation Table and find your homework on the line with Toggle Downloading, which you should then click. Now go back and download your homework zipfile and open it up with Winzip or 7zip and click on the source worksheet in there. Maple will open the worksheet. Create a folder to put the worksheet in and go through the edit cycle. Note: you will need to modify the zipit line to reflect where your worksheet is on the local machine.
Here are some algorithms for common tasks that WHS authors are confronted with. If you discover something that is missing or wrong, please report it to the authors.
What to do if you are having trouble gettting a homework to install
Sometimes, when you try to install a homework, you cannot get it to install at all. What to do?
Go through the steps to install it again. If you still get the error message, try this:
1 Close Maple
2 From the Start Button, go to My Computer and locate the homework zipfile. Delete it. Then go down into the homework directory and double click to go down into it. Then select the html files in the directory and delete them. (don't delete the source worksheet.) Also delete the images folder you see in the directory.
3 . Now start Maple up again and open the source worksheet.
4 . Open the MCtools section, execute it, and close it.
5 . export the worksheet to html
6 . execute the zipit line.
7. Log into WHS and try to install the homework. If you still cannot get it to install, email the zipfile to me and I'll try to install it.
If you already have the directory containing the source worksheet on your computer,
1 . Open up the source worksheet for the homework.
2. Open the MCtools section at the top, execute it, and close it.
3. Make the modifications.
4 Save the worksheet.
5 Export the worksheet to html
6. Execute the zipit line.
7 . Upload the homework zipfile to WHS (Note: Don't exit Maple here.)
8 . Install and preview the homework.
9. If there is some additional change to make, go to the source worksheet which you left open and proceed from step 3 above.
This is the editing cycle, which you will perform several times before the homework is just like you want it.
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How to download and unzip a homework zipfile from WHS.
You can download homework zipfiles that you have previously installed on WHS. Also, you can download a homework that some other author has installed, provided that author gives permission.
1 F irst open your browser (Internet Explorer) and log into WHS (if you haven't already done so) .
2 Download the homework zipfile:
3 Now Unzip the homework file:
Nota Bene: Usually, all you want when you download a homework zipfile is the Maple worksheet that it contains. It is the source, and with it you can regenerate the html export. So, often rather than saving the homework zipfile as described above, you can open it up (with 7-zip if you are using Windows 2000) and just click on the Maple worksheet in there to open it in Maple. Then you can save the worksheet
How to delete a homework that you have installed.
If you have a homework that you want to remove, just do this:
1 From the Homework Installation table at the bottom of the Authoring Tools page, select the homework you want to remove.
2 . Click the remove this homework button.
3 . You will be asked to verify that you wish to remove the homework. After checking that you are removing the correct homework, click ok.
How to create a homework list and add homeworks to it.
A homework list contains a sequence of homeworks. Every class (see below on how to create a class) must have a homework list.
1 . Click Authoring Tools , and go to the middle of the Homework List Management table.
2 . Type in a name for the List and then click the Create Homework List button. This creates an empty homework list.
3 . Now at the bottom of the table click the toggle Publication flag button. This will allow you and others to use your homework list in a class.
4 . You can add your homeworks or homeworks from other authors to the list.
A class in WHS is the vehicle by which you make your homework list available to your students.
1
. Login to mathclass
2
. Go to
Teacher Resources
->
Manage Classes
: A page will appear with
Class management
at the top.
3 In the Create a Class table, choose a name for your class:
4 . Select a school. If your school is not in the list, you can put the class in any school
5 . Don't bother with the homepage part. You can reconfigure the class and install a homepage later.
6. Select your homework list (or any homework list will do for the moment.).
7 . Ignore the box to put a version. That is a relic from past days.
8. Now click the Create Class button. You have created your class. Now students can request registration in your class.
How to approve a students request for registration in your class
1. Login to mathclass
2. Go to Teacher Resources -> Access Records : A page will appear with 2 columns of buttons. Click the button in the 2nd row 2nd column that says Registration and Passwords .
3. You will see a button called Approve for Registration . Highlight the students name and click the button. The student will appear on class roll after you refresh the browser.
Note: You can remove a student from your roll at any time. You can also change a students password at any time, unless the student has teacher standing in WHS.
1. Login to mathclass
2. Go to Teacher Resources -> Access Records : A page will appear with 2 columns of buttons.
3 To check comments click the button Student Comments. This collects comments which students make in the feed back sheet. You also receive email copies, to ensure that you get the comment.
4. To check performance on homeworks, click the button Assignment Submissions button (to get collated class results, including comma separated files of individual responses on each homework) or the button Homework Standards Download (to get a comma separated file of all the standards measurements on each student.), or finally and most often the button Summaries by student (to get a summary of each students homework results on each homework and calculate selected averages.)
4. To check student uploads (not frequently used yet), click the Student uploads button.
How to use the Class Downloads button
If you want to provide a document, worksheet, etc to the students in your class, you can use the Class Downloads Button. Go to Teacher Resources -> Access Records : The button in the fourth row second column is labelled Class Downloads. Click on that: a browse window opens. Write in a short description of the file for you students to identify it by. Browse for the file you want to upload to the class and click open with it highlighted. The window disappears and the filename appears above the upload file button. Click on that, and the file is uploaded to the class. You can verify this by going back to the File Uploads Page and refreshing the browser. If you make a mistake and upload the wrong file, you can delete the file with the delete button.
Your students (the ones you have approved for registration) can download the file by going to the bottom of the Announcements page. They will see a Class downloads button. Clicking that will take them to a page listing the files you have uploaded. Note: If you are a student in your own class, you can use this same place to upload the files.
This provides a secure method of providing files for your students. It is more convenient and less public than putting a link on your webpage.
How to install and maintain a webpage for a class.
Each class comes with a place to put a webpage. It provides a place you can put your syllabus and other information relavant to the class. The students and anyone else with the URL can access it without logging into mathclass, so you can put announcements there as well as on the announcement section of Access Records.
To start:
1. Use Word 2000 to create a web page. Start with something simple.
2. Create a folder, say c://temp/webpage and save your webpage in this folder. Name it default.doc
3. Also save default.doc as type webpage in the same folder. This will create a file default.htm and a subfolder of webpage, default_files, which contains any images you have put into the webpage.
4. Make the zipfile. If you are in Windows XP, compress the folder from Start --> My Computer by locating the folder, rightclicking its icon and choosing Send to Compressed file. This will make a zipfile with the same name as the folder, which is ok. The name of a webpage zipfile is not important, just the name of the webpage, which should be default.htm. If you are in Windows 2000, use 7-zip or Winzip to zip the folder. Alternatively, you can open a command window, cd to c://temp and issue the command zip -r webpage webpage. This assumes you have zip in your path. If you don't, you can get it from www.info-zip.org
To install the webpage in your class:
1 . Login to mathclass
2 . Go to Teacher Resources -> Manage Classes : A page will appear with Class Management at the top
3 . Choose the class you want to Configure: A page will appear with Configure Classes at the top.
4. Scroll down to the Upload webpage zip archive portion and click Browse
5. A Browse window opens up. Search for the zipfile webpage.zip in C:\temp
6. Click Open . The browse window will close and the name webpage.zip will appear in the Upload line.
7.
Then click
Upload Zipfile
You will get a message saying the file installed and the webpage can be found at
the url
www://mathclass.org/Homepages/xxxx
where xxxx is the identifier for your class.
8. Copy that website url into Homepage line in the Configure Class table at the top. Then when you check announcements in the class, you can go to the class webpage from there. Of course, you can also go directly to the class web page.
To maintain the webpage in your class:
If you want to add some links or assignments to the class webpage, you need to modify it and zip it up again.
1 . You can download your webpage zipfile and go through 7 steps at the top.
2 . Or if you have access to the files you used to make the webpage, just go through steps 2-7 at the top.
After you have made a new webpage zipfile, go through the 8 step procedure to install the webpage in your class.
How to find out your Magic number
If you want to have hidden sections in your homework that are stored in the homework zipfile, then you will need to supply zipit with your accId, which we are calling your magic number. Here's how to get yours. Login to WHS, then bring up the html souce in your browser. On Internet Explorer 6, the source is accessed via the View Menu. You'll see the word source about halfway down. Click that and you will get a page of gobbletygook. About halfway down the page, you will see the line var accId = ###
where ### is some number. That is your Magic number . When you have a homework that has a hidden section that you have pasted into a WHS problem or in the Header section, or one that you have created using addsectionhint, make sure that you include the option Magic=### (where ### is your magic number) in the zipit line. So for example, if you have a homework c://temp/stuff with hidden sections and you want to zip it up, then you would use zipit("stuff","c://temp/stuff",Magic=###);
How to construct WHS homeworks with Word 2000
WHS homeworks can be constructed with Word2000. If you are used to using tagit in MCtools to format the problems, you will need to review the question and answer format tags because these have to be inserted by hand in a Word homework. Also, you will need to zip the files by hand.
Note: It is important to save the file as a filtered web page. This creates an htm file which WHS can process into an xml file.
The steps in creating and installing a homework using Word2000 (in WindowsXP, or Windows2000) are:
1
Open a new word document.
2
. Type in one or two of the problems, beginning with a header tag.
3
. From the 'save as' menu, create a directory with a suitable name (say wordsample) and save the homework as a filtered web page in that directory, giving the file the same name as you gave the directory. Later if you want to make changes, you can open this with Word and use it as a source document.
4
. Now, if you are in XP (as you are in CB 313 or Mathskeller), go to My Computer from the Start Menu, then browse for the directory wordsample that you have saved wordsample in and right-click it. A window will open up. Choose Send to Compressed Zip folder. A zipfile wordsample.zip will appear (it will look like the wordsample directory, except it will have a zipper on it.)
If you are in Windows 2000 or Windows 98, you can use Winzip or 7-Zip. These are programs which you can install that you can use to zip and unzip folders. 7-zip is free to install on your home computer. It can be downloaded from http://www.7-zip.org/download.html Winzip has an evaluation version which you can download at http://www.winzip.com/ddchomea.htm
5
. Log into WHS and install the wordsample.zip zipfile as usual.
6
Now go back to Word2000 and insert the rest of the problems in your homework, then resave (as a filtered web page), rezip and reinstall, until you have it like you want it.
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