Authoring Parson's problems (drag and drop) in STACK

Parson’s problems require students to assemble pre-written text into a correct order by dragging blocks into a tree structure.

This page provides a quick-start to authoring whole questions. There are a number of parts to authoring a Parson's problem:

1. Defining strings and creating drag and drop area using the Parson's block.
2. Linking the Parson's block to a STACK input (string input).
3. Assessing the student's answer and providing meaningful feedback automatically.

While drag and drop is certainly not new in automatic assessment, these problems have become popular in computer science, e.g. Parsons (2006).

Parson’s problems enable students to focus on the structure and meaning, by providing all the right words but just not initially in the right order. Learning mathematical proof is well-known to be very difficult for most students. The scaffolding provided by Parson's problems has been found to be very useful to students in discrete mathematics. This may well be due to the expert reversal effect. The expert reversal effect is a well-established phenomenon that what is useful for a beginner is quite different, perhaps the opposite, of what is useful for an expert. Even when independent proof writing is the final goal of instruction, it is very likely that Parson's problems have an important place in formative learning towards that goal.

Traditionally Parson’s problems require the student to create an ordered list. For mathematical proofs, we believe it is more natural to ask students to create a tree. Parson's problems in mathematics, especially proofs, do not always have a unique answer which is considered correct. Hence, there are challenges in deciding what a correct answer is in particular cases, and how to develop authoring tools to robustly create problems with reliable marking algorithms. For example, in a proof of $$A\mbox{ if and only if } B$$ We might expect/require two conscious and separate blocks

1. Assume $A$, $\cdots$, hence $B$.
2. Assume $B$, $\cdots$, hence $A$.

The order in which these two sub-proofs are presented is (normally) irrelevant. That is the if and only if proof construct allows its two sub-proofs to commute. This is precisely the same sense in which $A=B$ and $B=A$ are equivalent. There are blocks within the proof which can change order. Furthermore, since proofs are often nested blocks these sub-proofs may themselves have acceptable options for correctness. Proofs often contain local variables. Use of explicit block-structures clarify the local scope of variables, and the local scope of assumptions.

STACK provides "proof construction functions" with arguments. For example, an if and only if proof would be represented as proof_iff(A,B). Here A and B are either sub-proofs or strings to be shown to the student.

If the student has an opportunity to indicate more structure, then the assessment logic becomes considerably simpler, more reliable and transparent. Actually, we think there is significant educational merit in making this structure explicit and consciously separating proof-structure from the finer grained details. It is true that professional mathematicians often omit indicating explicit structure, they abbreviate and omit whole blocks ("the other cases are similar") but these are all further examples of expert reversal.

Notes

• Lists are a special case of a tree with one root (the list creation function) and an arbitrary number of nodes in order. Hence our design explicitly includes traditional Parson's problems as a special case.
• Teachers who do not want to scaffold explicit block structures (e.g. signal types of proof blocks) can choose to restrict students to (i) flat lists, or (ii) lists of lists.

Example 1: a minimal Parson's question

The following is a minimal Parson's question where there student is expected to create a list in one and only one order. It shows the proof that $\log_2(3)$ is irrational.

Question variables

Define the following question variables:

stack_include("contribl://prooflib.mac");
ta:proof("assume","defn_rat","defn_rat2","defn_log","defn_log2","alg","alg_int","contra","conc");

The optional library prooflib.mac contain many useful functions for dealing with student's answers which represent proofs.

The variable ta holds the teacher's answer which is a proof construction function proof. The arguments to this function are string keys, e.g. "alg" which refer to lines in the proof. The teacher expects these lines in this order.

Question text

The example question text below contains a Parson's block. Within the header of the Parson's block, ensure that input="ans1" is included, where ans1 is the identifier of the input. Place the following in the Question text field:

<p>Show that \(\log_2(3)\) is irrational. </p>
[[parsons input="ans1"]]
{
    "assume" : "Assume, for a contradiction, that \\(\\log_2(3)\\) is rational.",
    "defn_rat" : "Then \\(\\log_2(3) = \\frac{p}{q}>0\\) where ",
    "defn_rat2" : "\\(p\\) and \\(q\\neq 0\\) are positive integers.",
    "defn_log" : "Using the definition of logarithm:",
    "defn_log2" : "\\[ 3 = 2^{\\frac{p}{q}}\\]",
    "alg" : "\\[ 3^q = 2^p\\]",
    "alg_int" : "The left hand side is always odd and the right hand side is always even.",
    "contra" : "This is a contradiction.",
    "conc" : "Hence \\(\\log_2(3)\\) is irrational."
};
[[/parsons ]]
<p>[[input:ans1]] [[validation:ans1]]</p>

Notes:

1. The Parson's block requires a JSON object containing "key":"string" pairs. The string will be shown to the student. The student's answer will be returned in terms of the key tags. Numbers can be used as keys, but named keys above will be more specific. We strongly recommend using named keys.
2. The \ character in the string must be protected! Notice that \(...\) needs to be typed as \\(...\\).
3. The Parson's block has a wide range of options such as height and width which are documented elsewhere.

Input: ans1

1. The Input type field should be String.
2. The Model answer field should construct a JSON object from the teacher's answer ta using proof_parsons_key_json(ta, []). You can replace the empty list in the second argument with a proof_steps list if you want to display unused steps as well. (How to construct and use a proof_steps list will be documented below.)
3. Set the option "Student must verify" to "no".
4. Set the extra options to "hideanswer" to make sure the JSON representation of the teacher's answer is not shown to the student later as an answer.

Potential response tree: prt1

Define the feedback variables:

sa:proof_parsons_interpret(ans1);

The student's answer will be a JSON string, but we need to interpret which of the strings have been used and in what order. The proof_parsons_interpret function takes a JSON string and builds a proof representation object.

Then you can set up the potential response tree to be ATAlgEquiv(sa,ta) to confirm the student's answer is the same as the teacher's answer.

Example question 2: a proof with interchangeable block order

The following Parson's question is an if and only if proof, containing two blocks in order.

stack_include("contribl://prooflib.mac");

ta:proof_iff(proof("assodd","defn_odd","alg_odd","def_M_odd","conc_odd"), proof("contrapos","assnotodd","even","alg_even","def_M_even","conc_even"));

proof_steps: [
    ["assodd",     "Assume that \\(n\\) is odd."],
    ["defn_odd",   "Then there exists an \\(m\\in\\mathbb{Z}\\) such that \\(n=2m+1\\)."],
    ["alg_odd",    "\\[ n^2 = (2m+1)^2 = 2(2m^2+2m)+1.\\]"],
    ["def_M_odd",  "Define \\(M=2m^2+2m\\in\\mathbb{Z}\\) then \\(n^2=2M+1\\)."],
    ["conc_odd",   "Hence \\(n^2\\) is odd."],

    ["contrapos",  "We reformulate \"\\(n^2\\) is odd \\(\\Rightarrow \\) \\(n\\) is odd \" as the contrapositive."],
    ["assnotodd",  "Assume that \\(n\\) is not odd."],
    ["even",       "Then \\(n\\) is even, and so there exists an \\(m\\in\\mathbb{Z}\\) such that \\(n=2m\\)."],
    ["alg_even",   "\\[ n^2 = (2m)^2 = 2(2m^2).\\]"],
    ["def_M_even", "Define \\(M=2m^2\\in\\mathbb{Z}\\) then \\(n^2=2M\\)."],
    ["conc_even",  "Hence \\(n^2\\) is even."]
];

/* Generate the alternative proofs. */
tal:proof_alternatives(ta);
/* Create a set of flattened proofs. */
tas:setify(map(proof_flatten, tal));

The complete question text is

<p>Let \(n\in\mathbb{N}\).  Show that \(n\) is odd if and only if \(n^2\) is odd. </p>
[[parsons input="ans1"]]
{# stackjson_stringify(proof_steps) #}
[[/parsons ]]
<p>[[input:ans1]] [[validation:ans1]]</p>

Notice the function stackjson_stringify turns the variable proof_steps into a JSON object.

Notice in this example the teacher's proof is nested. This can be seen if we use numerical keys, not string keys and define

ta:proof_iff(proof(1,2,3,4,5),proof(6,7,8,9,10,11));

The two blocks can be in either order. Prooflib provides a function to automatically create both options. Notice the command tal:proof_alternatives(ta); in the question variables. The variable tal will be a list of both options for the if and only if proof. Note that proof_alternatives will recurse over all sub-proofs. Types of supported proof structure are documented within the prooflib file. Then we have to "flatten" each of these proofs to a set of list-based proofs: tas:setify(map(proof_flatten, tal));

There is one change in input from the above example:

1. The Model answer field should construct a JSON object from the teacher's answer ta using proof_parsons_key_json(ta, proof_steps).

In this example all steps are used, however if you add extra steps (distracters) then the model answer field has to separate these into used and unused lists, hence both the teacher's answer ta and the whole proof_steps list is needed.

As before, define the feedback variables to interpret the JSON as a proof:

sa:proof_parsons_interpret(ans1);

Set up the potential response tree to check if the student's proof sa is in the set of possible teacher's proofs. The simplest way is ATAlgEquiv(elementp(sa,tas), true) to confirm the student's answer is in the set of answers equivalent to teacher's answer.

To see this in action, try the following in the general feedback to display both proof options.

This is the proof, written with some structure
{@proof_display(tal[2], proof_steps)@}
Notice this proof has two sub-proofs, which can occur in any order.  Therefore we have two correct versions of this proof.
<table><tr>
<td><div class="proof">{@proof_display_para(tal[1], proof_steps)@}</div></td>
<td><div class="proof">{@proof_display_para(tal[2], proof_steps)@}</div></td>
</tr></table>
Can you see the differences between these proofs?

We have much more sophisticated general assessment tools for establishing the edit distance between the student's and teacher's proof and providing feedback on how to correct a partially correct proof. These are documented elsewhere.

Polish and tidy the question.

You should hide the inputs from students with CSS after testing, e.g. <p style="display:none">...</p>.

Note that all connection between the Parson's block and a string input is JSON format. Therefore input ans1 is a string, and we convert to and from JSON at various places in the process.

It is likely you will want to randomly permute the strings in the proof_steps list before the student sees them. This is documented in the Parson's block reference documentation.

"The teacher's answer is"....

The design of the interaction between the Parson's block and a STACK input is through JSON. This raw JSON will not be meaningful to students, hence the suggestion to hide the inputs from students with CSS after testing, e.g. <p style="display:none">...</p>.

We recommend the string input holding the JSON does not get shown to the student as a "correct answer". Set the extra options to "hideanswer" in the input to stop this input being displayed.

To display a correct proof as a "teacher's answer"

1. Create a new input ans2.
2. The Input type field should be String.
3. The Model answer field should display the correct proof constructed from a proof construction functions ta and a list of proof steps proof_steps. Set the model answer to proof_display(ta, proof_steps_prune(proof_steps)). You can choose any of the other display functions in the CAS libraries for representing text-based proofs. We choose to prune out the narrative here (with proof_steps_prune), which isn't really appropriate when saying "A correct answer would be....".
4. Set the option "Student must verify" to "no" and "Show the validation" to "no".
5. Hide this input with CSS <p style="display:none">...</p>.

This input is not used in any PRT.