ce05916 May 14, 2019
7 contributors

Users who have contributed to this file

429 lines (253 sloc) 26.4 KB

Author quick start 1: a minimal working question

The author quick start guide shows you how to write STACK questions. Part 1 gets a minimal question working.

Consider students who are learning to symbolically integrate expressions of the form ( r(px+q)^n ). Below is a typical set of practice exercises.

(Reproduced with permission from Advanced Mathematics for AS and A level 2 (A-level mathematics), Haese Mathematics (2018) 978-1-925489-32-3)

At the end of this guide you will be able to:

• Create a new STACK question, ensuring mathematical notation is displayed correctly using (\LaTeX) notation.
• Catch, and provide feedback on, common errors by building a Potential Response Tree. These include forgetting to include the constant of integration or accidentally differentiating instead of integrating.
• Create random versions, and ensure the marking algorithms, feedback and worked solutions reflect the particular version.
• Preview and test STACK questions.

Before you begin

We assume you are familiar with the following:

1. Adding questions to a Moodle quiz, via the question bank.
2. Simple (\LaTeX) formatting for mathematics. Some basic examples are provided at the end of the CASText documentation. The full document preparation system is not needed or supported.
3. Using a strict syntax for representing mathematical expressions, e.g. 1/2*sin(n*pi). Knowing about the computer algebra system (CAS) Maxima would be useful in the longer term, but this introduction is self-contained.

Creating a minimal STACK question

Navigate to the question bank and create a new question.

There are lots of fields, but only a few are compulsory.

1. The "question name".
2. The "question text", which is shown to the student.
3. The teacher's "model answer", (inside "Input: ans1" on a default question).
4. A test of "correctness".

By default a new question automatically has one input, and one algorithm to test correctness of the answer.

Question name

You must give the question a name. Use something meaningful so that you can easily identify it later. For example question1.

Question text

Next we need to write the question text itself. Copy the following into the Question text box:

<textarea> Find $$\int 5(3x-2)^{-3} \mathrm{d}x$$. [[input:ans1]] [[validation:ans1]] </textarea>

Notes:

• Moodle has a wide choice for text editors, so the screenshots in this quick start guide might look slightly different to your version of Moodle. Also the cut and paste may, or may not, include some of the formatting.
• The text contains LaTeX mathematics environments. Do not use mathematics environments $..$ and $$..$$. Instead you must use $$..$$ and $..$ for inline and displayed mathematics respectively. (There is an automatic bulk converter if you have a lot of legacy materials.)
• Internally the student's answer will be assigned to a variable ans1.
• The tag [[input:ans1]] denotes the position of the box into which the student puts their answer.
• The tag [[validation:ans1]] will be replaced by any feedback related to the validity of the input ans1. E.g. syntax errors caused by missing brackets.
• The tags could be positioned anywhere in the question text: more on this later.

Input: ans1

Scroll down: there will be an inputs section of the editing form. Click on the heading Input: ans1 to reveal the relevant settings.

For a minimal question we must specify the model answer field.

-5/(6*(3*x-2)^2)+c

Notes

1. The student's response is stored in the answer variable ans1.
2. The model answer must be a syntactically valid expression in CAS (Maxima) syntax, not LaTeX. E.g. -5/(6*(3*x-2)^2)+c not \frac{-5}{6(3x-2)^2}+c.
3. Inputs can have a variety of types selected by the Input type drop-down menu. The Algebraic input is default, and what we need here.
4. A question can have many inputs for multiple parts. These are discussed later in this guide.

Assessing correctness of a response - the Potential Response Tree (PRT)

Next we have to decide if the student's answer is correct.

To grade the student's response we need to determine its mathematical properties using an algorithm known as a potential response tree.

By default, a new question contains one potential response tree called prt1. Feedback generated by the tree replaces the tag [[feedback:prt1]] at the appropriate time. By default the tag [[feedback:prt1]] is placed in the "Specific feedback" field so that Moodle can control the timing of the feedback via the Moodle quiz settings. For a minimal question you don't need to edit or move this tag.

Configuring a potential response node

A potential response tree is a non-empty acyclic directed graph of potential response nodes. By default we have one potential response node.

1. SAns is compared to TAns with the answer test, possibly with an option.
2. If true then we execute the true branch.
3. If false then we execute the false branch.

Each branch can then

• Assign/update the score.
• Assign formative feedback to the student.
• Leave an answer note for statistical reporting purposes.
• Nominate the next potential response node, or end the process [stop].

Let us configure the first node to determine if the student has integrated correctly.

1. Specify the variable ans1 in the SAns setting.
2. Specify the correct answer in the TAns setting: -5/(6*(3*x-2)^2)+c
3. Confirm we have AlgEquiv in the Answer test drop-down menu (this is the default).

The node should now be configured as follows:

Saving the question

Now scroll to the bottom of the page and press the [Save changes and continue editing] button. If the question fails to save check carefully for any errors, correct them and save again.

We now have a minimal question.

To recap we have

1. The "question name".
2. The "question text", which is shown to the student.
4. A test of "correctness".

Next we should try out our question, by pressing the Preview link at the bottom of the page:

Previewing the question

To speed up the testing process, scroll down the preview window and under Attempt options, make sure you have "How questions behave" set to "Adaptive Mode". If necessary "Start again with these options". This will allow you to check your answers without having to Submit and Start again repeatedly.

With the preview open, try typing in

-5/6*(3*x-2)^-2 + c

into the answer box. The system first establishes the syntactical validity of this answer.

Press the [Check] button.

The system executes the potential response tree and establishes whether your answer is equivalent to the model answer -5/6*(3*x-2)^-2+c.

Try -5/(54*x^2-72*x+24)+c. The system should also accept this as correct.

Next type in -5/6*(3*x-2)^-2 + C. Now, if we compare the teacher's -5/6*(3*x-2)^-2+c with -5/6*(3*x-2)^-2 + C using algebraic equivalence (recall we specified AlgEquiv in the potential response tree). This will not be accepted as equivalent. The reason is that c and C are different. A reasonable teacher will probably not care which letter is used for the constant of integration.

We will need to edit the question now to use a different answer test. Close the preview window and return to the page "Editing a STACK question". Find your potential response tree settings and click on the drop-down menu where we selected AlgEquiv and select Int from the list. Type x into the Test options setting. Now press the [Save changes and continue editing] button and once more click the preview button. We have just selected a special answer test for dealing with integration questions.

Next, try getting the question wrong. If your server does not have "instant validation" switched on (an administrator/installation option) you will need to submit each answer twice. Notice all your responses are stored in an attempts table.

STACK establishes specific mathematical properties. To demonstrate that it's the mathematical properties of the student's response that is being established type

-5/6*(3*x-2)^-2 + K

into the answer box. Since this includes a constant of integration this is considered as correct.

Built into the Int answer test is a check to ensure the response includes a constant of integration. Now type

-5/6*(3*x-2)^-2

We also wanted to check that the student hadn't differentiated by mistake. Fortunately this is also handled by the Int answer test. Finally, type

-45*(3*x-2)^-4

See that built-in feedback is provided to the student - a warning that they have forgotten the constant of integration.

If you don't want students to see the automatic feedback select the Quiet option in the potential response node.

Using question variables

The model answer will normally be referred to more than once, so it is usually easiest to assign the model answer to a "question variable" using the optional question variables field.

Add the following to the question variables

p: 5*(3*x-2)^-3;
ta: int(p,x)+c;

STACK uses Maxima's syntax for assignment, which is unusual. In particular the colon : is used to assign a value to a variable. So to assign the value of 5 to n we use the syntax n:5.

Notice we are using the CAS to determine the model answer by calling the int() function to find the anti-derivative. When the CAS determines an anti-derivative it does not include a constant of integration so we have to add it ourselves.

Now we need to update the rest of the question to use the variables. Replace the question text with

<textarea readonly="readonly" rows="3" cols="100"> Find $$\int{@p@}\mathrm{d}x$$ [[input:ans1]] [[validation:ans1]] </textarea>

Notice that now we have defined a local variable p, and used the value of this in the Question text. The difference is between mathematics enclosed between $$..$$ symbols and {@..@} symbols. All the text-based fields in the question, including feedback, are CAS text. This is HTML into which mathematics can be inserted. LaTeX is placed between $$..$$s, and CAS expressions (including your variables) between matching {@..@} symbols. The CAS expressions are evaluated in the context of the question variables and displayed as (\LaTeX).

Since we have used {@p@} here, the user will not see a (p) on the screen when the question is instantiated, but the displayed value of p.

In the input ans1 replace the model answer with ta.

In the potential response tree, node 1, replace the expression TAns with ta.

These changes just propagate the new variables through out the question. We should test the question again, but this can be done in an automatic way.

Student validation

Notice in the above there is a two-step process for the student to enter their answer.

First is "validation", and normally servers have "instant validation" enabled. If the expression is valid STACK shows the student "Your last answer was interpreted as follows:" and displays their expression. An invalid response creates an error message.

The second stage executes when a valid expression is entered, and this evaluates the potential response tree to assess the student's answer.

This two-stage process is a unique and essential feature of STACK. There are lots of options for validation to help the student. For example, in the above all example expressions have a strict syntax. Here we used expressions like -5/6*(3*x-2)^-2+c which has all the * symbols to denote multiplication. STACK has lots of options, and you could choose to let students type in expressions like -5/6(3x-2)^-2+c and accept implied multiplication. Documentation on these options is given in the inputs section.

Question tests

Testing questions is time consuming and tedious, but important to ensure questions work. To help with this process STACK enables teachers to define "question tests". The principle is the same as "unit testing" in software engineering.

Scroll to the bottom of the page and press the [Save changes and continue editing] button. Press the Preview link.

From the question preview window, click on Question tests & deployed versions link in the top right of the page.

Click Add a test case to add a test to your question. Fill in the following information

ans1 = ta
score = 1
penalty = 0

The system will automatically evaluate ta to create the value taken to be the student's answer ans1 and then assess the answer using this information. It will match up the actual outcomes with those you specified. This automates the testing process.

You can add as many tests as you think is needed, and it is usually a sensible idea to add one for each case you anticipate. Add in another test case for

ans1 = ev(int(p,x),simp)
score = 0
penalty = 0.1

Here we create a test case without a constant of integration. Note that test cases are not "simplified", so we need to use Maxima's command to ev(int(p,x),simp to "evaluate int(p,x) with simplification`. For future reference, information in simplification is provided elsewhere. In this case STACK should fail to give students any marks, indicating the test passes! This can be verified by running the test case.

Now try

ans1 = int(p,x)+c
score = 1
penalty = 0

Notice the system evaluates the integral, but does not fully simplify the algebraic result. You may not want students to use the CAS int command in this question! There is no particular difference between the int function and the * function, and so we allow student to input answers with calculus operations in them. Indeed, when a student needs to type in a differential equation they will need to use the diff command! You can render this answer "invalid" by entering int into the "forbidden words" option of the Input: ans1 part of the question editing form.

This example nicely illustrates the way validity can be used to help students. An answer int(p,x)+c is a correct response to the question, but it is invalid. In this example we want them to perform integration, not to have the CAS do it!

Quality control is essential, and more information is given in the page on testing.

Back to the mathematics!

So far we have authored question (i) from the text book, shown in the image above: ( \int \frac{5}{(3x - 2)^3} \mathrm{d}x )

Below is a student's written response, which demonstrates two common slips:

Notice that the student has:

• Forgotten to include the constant of integration.
• Differentiated the outer function, instead of integrating.

These are things which students are likely to do with any integration question. Indeed, through force of habit students have been known to differentiate by mistake and still add a constant of integration! Also, there are mistakes students have made which are much more specific to this particular question:

• Forgetting to use substitution and hence not dividing by (p), and effectively integrating ( \int r(px+q)^n \mathrm{d}x \rightarrow \frac{r}{n+1}(px+q)^{n+1}+c ).
• Having difficulties in increasing a negative number (in this case (-3) by one). In our example ( \int \frac{5}{(3x - 2)^3} \mathrm{d}x \rightarrow \frac{5}{3}\frac{1}{(3x - 2)^4}+c).

The whole point of STACK is that the CAS enables teachers to check for these kinds of errors and provide students with meaningful feedback which helps them improve their performance, without using multiple choice options which give the game away!

When checking a student's answer with STACK a teacher needs to ask themselves "What are the mathematical properties which makes a student's answer correct/incorrect?" In our case these questions include:

• Is the student's answer a symbolic anti-derivative of the integrand?
• Does the student have a constant of integration in an appropriate form?

The built-in int answer test answers these questions, so a teacher does not have to write code to do so for every similar question.

Next, a teacher needs to ask "What might a student do incorrectly, and what will this give them as an answer?" This second question is more difficult. The answer might come through experience or from asking upfront diagnostic questions (again using STACK). It is often sensible to review students' responses to STACK questions after a year and build in better feedback in the light of experience with students.

Enhancing the feedback

There are two further common mistakes for students to make when finding the anti-derivative of simple functions-of-functions:

1. Accidentally finding the derivative of the outer function (multiplying by the power and taking one off the power - i.e. following the wrong process). In this case effectively integrating ( \int r(px+q)^n \mathrm{d}x \rightarrow r , n(px+q)^{n-1}+c ).
2. Expanding brackets when they didn't need to - not remembering to leave their final answer in factored form.

Let us continue to enhance feedback by checking that the student has differentiated the outer function by mistake. We do this by adding another potential response node.

Close the preview, scroll down to the Potential Response Tree and click [Add another node] button at the bottom of the list of nodes:

From the false branch of Node 1, change the "Next" field so it is set to [Node 2]. If the first test is false, we will then perform the test in Node 2.

Update the form so that Node 2 has

SAns = diff(ans1, x)
TAns = 5*(-3)*(-4)*3*(3*x-2)^-5

Notice that we are using Maxima to differentiate the student's answer, which helps to remove a constant of integration. We then compare that result, algebraically, to an expression we would expect had the student responded in the way they have. If the student has made this mistake they will end up with (r, n(px+q)^{n-1}+c) so differentiating this we have the expression (r, n, p, (n-1)(px+q)^{n-2}) from which we get 5*(-3)*(-4)*3*(3*x-2)^-5 in this example. We might as well have the CAS calculate the value.

This gives us the test, but what about the outcomes?

1. On the true branch set the score=0
2. On the true branch set the feedback to It looks like you have subtracted one off the power of the outer function instead of adding one to the power!

Notice here that STACK also adds an "intelligent note to self" in the answer note field:

This is useful for statistical grouping of similar outcomes when the feedback depends on randomly generated questions, and different responses. You have something definite to group over. This is discussed in reporting.

Press the [Save changes and continue editing] button and preview the question.

To test this feedback type the following response into the answer box:

-15*(3*x-2)^-4+c

We are using Maxima to differentiate the student's response, so it does not matter whether or not the student includes a constant of integration in their answer. You can verify this by typing the above response but missing off the constant.

The Form of a Response: not leaving an answer in factored form

Because we are using the mathematical properties of a student's response to judge its accuracy, we can even check to ensure that the student has responded in the correct form. For example, we expect a student answering this correctly to respond with (-\frac{5}{6}(3x-2)^{-2} + c ) but they might equally well respond with ( -\frac{5}{54x^{2}-72x+24} + c). This is, of course, mathematically correct but not in the factored form convention demands. The student is correct but we still should guide them towards not expanding brackets when they don't need to.

We need to go back and [Add another node] to the Potential Response Tree. A third node is added.

To use this potential response, edit Node 1, and now change the true branch to make the Next node point to the new Node 3. If we enter Node 3, we know the student has the correct answer and just need to establish if it is factored or not and provide the appropriate feedback. To establish this we need to use the FacForm answer test.

Update the form so that Node 3 has

SAns = strip_int_const(ans1,x)
TAns = strip_int_const(ta,x)
Test options = x
Quiet = Yes.

STACK provides a function strip_int_const to remove any constant of integration which a student may have used. We have both SAns and TAns the same here as the FacForm test also checks for algebraic equivalence, which is not relevant here. The FacForm answer test provides automatic feedback which would be inappropriate here, hence we choose the quiet option.

We also need to assign outcomes in Node 3.

1. On the true branch set the score=1, and mod to =
2. On the false branch set the score=1, and mod to =
3. On the false branch set the feedback to Your answer is not factored. Well done for getting the correct answer but remember that there is no need to expand out the brackets.

Having developed our integration question to the point where we can provide some quite detailed guidance to students (based on the mathematical properties of their answer) we can now consider using this particular question as the basis for a whole set of random questions.

You will need to review the test cases you created above. These will now "fail" because the new node means that the last answer note no longer matches up with the actual outcomes. If you have lots of tests, you will need to update them all. However, updating the test outcomes and confirming they are working, is easier than testing and re-testing by hand...

Before moving on you might consider saving the current question as a new question so you don't lose your work.

Random questions

To generate random questions we again make use of the question variables field.

Earlier in this guide we introduced the idea of question variables. Let's take a look again at the question variables we declared:

p:5*(3*x-2)^-3
ta: int(p,x)+c

We defined two local variables p and ta, and used these values in other places such as the question text, input, potential response tree and question tests.

Notice also that in the variable ta used to define the model answer contains a CAS command to integrate the value of p with respect to x. It wasn't necessary for the CAS to work out the answer to our original question (we could have specified it ourselves) but it is certainly necessary in a random question.

We are now in a position to generate a random question. To do this modify the question variables to be

a1 : 1+rand(6);
a2 : 1+rand(6);
n : -(2+rand(4));
p : a1*(x-a2)^n;
ta: int(p, x)+c;

In this new question we are asking the student to find the anti-derivative of a question with a definite form (\frac{a_1}{(x-a_2)^n}). a1, a2 and n are all variables, which are assigned random numbers. These are then used to define the variable p, used in the question itself. We also have the CAS integrate the expression p and store the result in the variable ta.

Remember that when generating random questions in STACK then we talk about random numbers when we really mean pseudo-random numbers. To keep track of which random numbers are generated for each user, there is a special rand command in STACK, which you should use instead of Maxima's random command. The rand command is a general "random thing" generator, see the page on random generation for full details. rand can be used to generate random numbers and also to make selections from a list.

Question note

Now that our question contains random numbers we need to record the actual question version seen by a particular student. As soon as we use the rand function STACK forces us to add a Question note. Fill the question note in as

$\int {@p@}\mathrm{d}x = {@ta@}.$

Two question versions are considered to be the same if and only if the question note is the same. It is the teacher's responsibility to create sensible notes.

Handling random variables in the Potential Response Tree

We also need to ensure the test answers, TAns, in each node of the potential response tree are updated accordingly. If the student has differentiated the outer function by mistake then the derivative of their response will be of the form a1*n*(n-1)*(x-a2)^(n-2).

We will need to update TAns of node 2 of the potential response tree to add in this value. It is sensible to create another question variable

taw1:a1*n*(n-1)*(x-a2)^(n-2)

So this possible outcome can be used in the potential response tree, question tests and so on by referring to the variable taw1.

Edit your trial question, save and preview it to get new random versions of the question.

Deploying random versions

Before a student sees the questions it is sensible to deploy random versions. See deploying for more information on this process.

Next steps

STACK's question type is very flexible.

• You can add a worked solution in the General feedback.
• You can change the behaviour of the question with the options.
• You can add plots to all the CASText fields with the plot command.
• You can add support for multiple languages.
• You might like to look at Moodle's quiz settings, creating a simple quiz. This is, strictly speaking, a completely Moodle issue and there is every reason to combine STACK questions with other Moodle question types. Some very brief notes are included in the quiz quick start guide.

The next part of the authoring quick start guide looks at multi-part mathematical questions.

You can’t perform that action at this time.