Merge branch 'projects-authomath'

mkdocs.yml

@@ -30,7 +30,10 @@ nav:
         - 'Annual conferences': 'Conference.md'
         - 'Training events in 2020': '2020meetings.md'
         - 'Meeting reports': 'MeetingReports/index.md'
-    - Projects: Projects/
+    - Projects:
+        - 'Overview': 'Projects/index.md'
+        - 'AuthOMath': 'Projects/AuthOMath/index.md'
+        - 'IDIAM': 'Projects/IDIAM/index.md'
     - Case Studies: CaseStudies/
     - Demo: 'https://stack-demo.maths.ed.ac.uk/demo/'
     - Documentation: 'https://docs.stack-assessment.org/en/'

website_files/Projects/AuthOMath/ActivatingFeedback.md

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+# Combining STACK and GeoGebra for better feedback
+
+### Guido Pinkernell, Heidelberg University of Education, Germany
+
+STACK offers the possibility of Combining STACK and GeoGebra for better feedback. When placed in feedback, interactive applets allow for what could be called "activating feedback", i.e. adaptive learning material that  students to work out the necessary knowledge themselves.
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/-yERZtbRtNk" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+The object of this task is the well-known translation of an algebraic expression into its geometric representation.
+
+<div class="float-none img-screenshot">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/interactivefeedback02.png" alt="Answered AuthOMath Task">
+<figcaption class="figure-caption">Figure: An answered AuthOMath task</figcaption>
+</figure></div>
+
+Here, GeoGebra provides an interactive applet in both STACK's task and feedback area. The latter comes in three steps, each after some delay:
+
+* First, it allows the learner to compare her or his wrong solution with the correct, thus giving an experienced user immediate hints about his error, presumably made inadvertently.
+* For those needing more help, it provides an interactive version of the situation together with questions that guide the student through working out himself how the algebraic and geometrical representations of this function relate.
+* Finally, it gives access to a worked solution, thus serving those learners which rely on a stepwise instruction to solve questions like this.
+
+<div class="float-none img-screenshot">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/interactivefeedback03.png" alt="First Feedback Step"/>
+<figcaption class="figure-caption">Figure: The first feedback step in an answered AuthOMath task</figcaption>
+</figure></div>
+
+<div class="float-none img-screenshot">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/interactivefeedback05.png" alt="Second Feedback Step">
+<figcaption class="figure-caption">Figure: The second feedback step in an answered AuthOMath task</figcaption>
+</figure></div>
+
+<div class="float-none img-screenshot">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/interactivefeedback06.png" alt="Third Feedback Step">
+<figcaption class="figure-caption">Figure: The third feedback step in an answered AuthOMath task</figcaption>
+</figure></div>

website_files/Projects/AuthOMath/Example-generation-research.md

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+# Example generation and educational research
+
+### George Kinnear, The University of Edinburgh, UK
+
+STACK offers the possibility of undertaking educational research.
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/Q_m-j5rZF6o" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+

website_files/Projects/AuthOMath/Fractions-part-whole.md

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+# Fractions as Part of One Whole 
+
+### Gunter Ehret, Heidelberg University of Education, Germany
+
+STACK offers the possibility to give adaptive feedback. STACK tasks could therefore be an enrichment for maths lessons and a relief for teachers, especially in schools where self-directed learning plays an important role.
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/KO5hbh7iRWM" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+This example task was created by teacher-education students in a didactics seminar at the Heidelberg University of Education.  In this task children test their understanding fractions as part of one whole.  Different types of feedback can be shown, depending on the input of the students.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Fractions-part-whole2.png" alt="A correct response">
+<figcaption class="figure-caption">Figure: A question in which children test their understanding fractions as part of one whole.</figcaption>
+</figure></div>
+
+If the sliders are set so that the dynamic illustration correctly represents the required fraction, there is a detailed confirmation of correctness in mathematically correct language, as well as a request to repeat the task five times in a row, with the required fraction changing each time.
+
+If the input is incorrect, this is fed back and the student can either repeat the task directly or wait 15 seconds for the sample solution.
+
+The programme also identifies if an equivalent fraction is entered; the feedback can be different here, in the case presented in the video it refers to a learner who has not yet formally learned to expand and reduce fractions.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Fractions-part-whole1.png" alt="An incorrect response, with feedback">
+<figcaption class="figure-caption">Figure: Illustrating feedback to an incorrect response.</figcaption>
+</figure></div>

website_files/Projects/AuthOMath/Random-generation.md

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+# Randomly generating mathematical questions
+
+### Konstantina Zerva, The University of Edinburgh, UK
+
+STACK offers the possibility to randomly generate questions and give feedback, which is relevant to each specific random variable. The task presented in the video is a very simple one, and it was deliberately chosen to be simple, because the main focus here is the randomisation and the things that question authors need to consider when creating random variables in a question. 
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/HKeEqr7ep8g" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+The task asks students to drag the point \(A\) so that the slope of the line is the same as the given value; in this case \(-7/2\). The values of the slope are randomised. 
+
+When the student load the question they all see the same graph. The initial slope of the line is set to \(1\) in the GeoGebra app and the point \(A\) is placed at \((2,2)\). The students can move the point \(A\) along the line and drag it around so that the line changes slopes.
+
+In this example the slop needs to be \(-7/2\). The slope is defined as \[\frac{\text{Change in Y}}{\text{Change in X}}\] so the students can drag the point \(A\) and place it at \(Y=-7\) and \(X=2\)  (point \((2,-7)\) ). Then click the check button and see if their answer is correct or not. Another possible solution here \(Y=7\) and \(X=-2\). 
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Correct_slope.png" alt="Correct slope">
+<figcaption class="figure-caption">Figure: Defining the correct slope.</figcaption>
+</figure></div>
+
+A common mistake that students do is believe that the slope is defined as \[\frac{\text{Change in X}}{\text{Change in Y}}\] so if they put \(A\) at the point \((-7,2)\) they'll receive specific feedback about this mistake. 
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Wrong_slope.png" alt="Wrong slope">
+<figcaption class="figure-caption">Figure: A common misconception.</figcaption>
+</figure></div>
+
+Let's look at the background of the question and how to deal with the randomisation. All the slopes need to be ratios of two integers (e.g. \(3/2\)) and also they shouldn't simplify to an integer (avoid cases like \(4/2=2\)). 
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Code.png" alt="An example of Maxima code">
+<figcaption class="figure-caption">Figure: An example of Maxima code for creating the variables for this question.</figcaption>
+</figure></div>
+
+The variable \(rd\) defines the denominator of the slope. We pick the denominator to be an even number, in this case a power of \(2\). We could also predefine a list of even numbers and randomly pick from the list. 
+The variable \(rn\) defines the numerator of the slope. For the numerator we pick the values from a predefined list and these values are all odd numbers. 
+We define the slope as \(\dfrac{rn}{rd}\) and in our case all the slopes will be fractions. 

website_files/Projects/AuthOMath/Sample-factor-quadratic.md

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+# Relating factored equations to the area of a rectangle
+
+### Cecilia Russo, Universidad de Cantabria, Spain
+
+STACK offers the possibility of immediate formative feedback. It's great to have everyone in a class of 100 students know they would be able to get feedback even if you are away. The teacher's presence is vital, but it is impossible to be there all the time, and AuthOmath is the perfect teaching assistant combining STACK and GeoGebra.
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/62PLll_kM_A" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+This example is a task sequence to help the student relate factored equations to the area of a rectangle. The first task asks the student to factorise a  square of binomial.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/JKU-pic1.png" alt="A blank question">
+<figcaption class="figure-caption">Figure: A question in which student should write a quadratic in factored form.</figcaption>
+</figure></div>
+
+If students write an incorrect answer, they are get the following feedback:
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/JKU-pic2.png" alt="Feedback to an incorrect attempt">
+<figcaption class="figure-caption">Figure: Feedback to a student's attempt.</figcaption>
+</figure></div>
+
+Now, a student can drag point \(A\) to observe the area of square \(ABCD\) and try to find a relation between the expression, the area and the length of the sides. When they drag point \(A\), they get a square composed of \(4\) rectangles. The area of each rectangle is shown in the applet.  The idea is for the student to add these areas, and obtain the expression needed as the answer to the task. STACK and GeoGebra help the student during the feedback.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/JKU-pic3.png" alt="Feedback to an incorrect attempt">
+<figcaption class="figure-caption">Figure: Students drag the applet.</figcaption>
+</figure></div>
+
+Then, the second part of the sequence is related to \((x+a)(x+b)\) expressions. Where  "\(a\)" and "\(b\)" are different natural numbers.  In this second part we don't have a square.
+
+Again, if students write a wrong answer, they get feedback with a GeoGebra applet. This applet lets the student try out the relation between the length of the sides of the rectangle \(ABCD\) and the area of it.  In that case, they can drag points \(A\) and \(C\) to get a rectangle composed of four rectangles. The area of each one is shown.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/JKU-pic5.png" alt="Feedback to an incorrect attempt">
+<figcaption class="figure-caption">Figure: Students drag the applet, in this case it is a rectangle.</figcaption>
+</figure></div>
+
+This feedback allows the students to interact with a geometric polynomial representation. At the same time, they can find out a strategy to find a factorized expression.  In this way, the geometry and the algebra reinforce each other.
+

website_files/Projects/AuthOMath/Sample-linear-algebra.md

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+# Illustrate eigenvectors by positioning vectors
+
+### Chris Sangwin, The University of Edinburgh, UK
+
+The original purpose of STACK was to accept answers which are algebraic expressions, moving away from reliance on multiple choice questions.  Through AuthOMath we have included GeoGebra diagrams within questions, allowing a student's answer to include the configuration of a diagram.  This question illustrates how dynamic diagrams can be used to support more advanced topics in university mathematics.
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/TlnwKcVhqEg" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+In the problem below students need to relate the eigenvectors of a matrix to the transformation represented by the matrix.  Eigenvectors is a technical term for vectors which are scaled by a transformation, but remain in the same direction (or reverse direction). Understanding the effect of transformations through calculating eigenvectors and the corresponding eigenvalues is an important topic in vector spaces.  Students need to find a matrix and perform some routine calculations.  In addition, four points in the diagram are ready to be dragged by the user to define the endpoints of two eigenvectors.
+
+<div class="float-right img-tall">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/LinearAlgebra-1.png" alt="A question in which students indicate the position of the eigenvectors">
+<figcaption class="figure-caption">Figure: A question in which students indicate the position of the eigenvectors.</figcaption>
+</figure></div>
+
+Students get very adept at calculating eigenvectors through a mechanical procedure, but their geometric understanding can remain fragile. Excessive calculation can reinforce this problem.
+
+<div class="float-left img-tall">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/LinearAlgebra-2.png" alt="A student's partially correct attempt at a question in which students indicate the position of the eigenvectors">
+<figcaption class="figure-caption">Figure: A student's attempt to indicate the position of the eigenvectors.</figcaption>
+</figure></div>
+
+In the second figure a student has attempted to illustrate the positions of the eigenvectors.  The vector \(\mathbf{u}\) is correct, but \(\mathbf{v}\) is not an eigenvector.  The GeoGebra diagram returns the endpoints of the vectors, the computer algebra system establishes the relevant properties (is \(\mathbf{v}\) an eigenvector?) and then generates the feedback shown in the yellow feedback box.
+
+GeoGebra is used to provide a visual representation of the vectors.  In principal this problem could be extended to a wider range of 2D transformations, and other situations where students can demonstrate their understanding of linear algebra by positioning vectors.
+
+

website_files/Projects/AuthOMath/Sample-linear-equations.md

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+# Providing formative feedback to linear equations
+
+### María Sanz Ruiz, Universidad de Cantabria, Spain
+
+
+The AuthOMath Project combines GeoGebra and STACK. It allows us to present multiple representations of the same mathematical object. STACK is especially useful to create different approaches to feedback.
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/-tvUJudb6XA" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+This task is about solving linear equations step by step. Whenever the students make a mistake, STACK is able to identify it and provide specific feedback to it.  Below is an empty question ready for a student.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Linear-equation-1.png" alt="A blank question">
+<figcaption class="figure-caption">Figure: A question in which student should write their answer.</figcaption>
+</figure></div>
+
+The image below shows feedback generated automatically by the system.  Notice some lines are correct, but the student has made one mistake.
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Linear-equation-2.png" alt="A blank question">
+<figcaption class="figure-caption">Figure: A question in which student should write their answer.</figcaption>
+</figure></div>
+
+These formative hints can be delivered in many ways. Some students are more receptive to symbolic feedback, while others appreciate written comments. It is also possible to combine both.
+
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Linear-equation-3.png" alt="A blank question">
+<figcaption class="figure-caption">Figure: A question in which student should write their answer.</figcaption>
+</figure></div>
+
+
+This is a useful way to distinguish the characteristics of the different feedbacks and determining which are more helpful and effective in fostering mathematical abilities and learning.
+

website_files/Projects/AuthOMath/Tangram-task.md

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+# Tangram tasks for pre-service teachers
+
+### José Manuel Diego Mantecón, Universidad de Cantabria, Spain 
+
+The AuthoMath Project allows us to use GeoGebra applets on STACK. This offers 
+a wide range of tasks that we can create and provide feedback to. Students 
+have the chance to work in an interactive environment and use different 
+tecniques to solve the same task. 
+
+<center>
+<iframe class="embed-responsive-item" width="560" height="315" src="https://www.youtube.com/embed/vMCC9aSmYaI" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+</center>
+
+There are many ways to compute the area and the perimeter of a figure. In 
+this task, we go through a few of them using Tangram pieces that our 
+pre-service teachers can move around and rotate. They can use this applet to 
+make the calculations using different units of measurement. 
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Tangram1.png" alt="Using Tangram pieces">
+<figcaption class="figure-caption">Figure: Illustrating using Tangram pieces within a response.</figcaption>
+</figure></div>
+
+
+In this task we ask them to compute and compare the area and the perimeter of 
+two figures, but also to reflect on what they are doing as a way to teach 
+these concepts to primary school students. Mathematical concepts go hand in 
+hand with didactical notions. 
+
+<div class="float-none img-middle">
+<figure class="figure">
+<img class="figure-img img-fluid" src="../Images/Tangram2.png" alt="Using Tangram pieces, with feedback">
+<figcaption class="figure-caption">Figure: Measuring areas with Tangram pieces.</figcaption>
+</figure></div>