Desmos Discourse

Summary:

In preparation for a class discussion, I read an article entitled “Increased Participation and Conversation Using Networked Devices” by Christopher Danielson and Dan Meyer. It outlined some of the content provided by the online desmos graphing utility and how teachers can utilize it to reach the students to help them learn in the best ways.

While other online graphing utilities are adequate at best (like this one or the calculator app found here), desmos.com, or more specifically teacher.desmos.com, provides tools teachers can use to access previously created activities, or create their own. Two of these activities Danielson and Meyer analyze and promote. First, they outline Central Park, which is a simple virtual manipulative in which students can estimate and eventually perfect the exact amount of space needed between each of the parking space lines so that all four cars can fit. It walks students through guess-and-check strategies and eventually creating their own algebraic expressions to describe the situation. The second activity they describe is called Polygraph, where partnerships of students guess each other’s graphs by eliminating options based on a series of yes-or-no questions. The authors explored why this particular activity helps teachers gauge the level of vocabulary and mathematical understanding of particular concepts having to do with graphs.

In our class discussion, we extensively reviewed the strengths and weaknesses of the Central Park and the Polygraph tasks. We discussed appropriate amounts teachers should be able to allow students to guess and check until they find the correct answers. We concluded that the Central Park task could be greatly improved in a way that actually teaches the students how to write mathematical expressions in more productive ways than just trying to guide the student to figure it out completely on their own. The jump between guessing and checking their answers and developing an accurate mathematical expression is too big that this task alone cannot accomplish. Much of our class discussion revolved around teaching strategies using these two tasks specifically as well as more general strategies to be used when, students get stuck and cannot proceed on their own, hindering their learning. The Polygraph task is much better because it is set up in such a way that teachers can monitor student thinking based on their strategies for asking questions about the graphed polygons in order to eliminate them down to one chosen graph. Teachers can see all questions and responses to see not only different vocabulary words being used, but also different pathways of thinking about these polygons that students might use (i.e. how many sides, orientation, size, regularity, etc.). We discussed how we as future educators can help students learn through their independent (and partner) exploration by giving them just the right hint. The “right hint” will enhance learning, rather than be unhelpful or give away the solutions too quickly without sufficient student contemplation.

Critique:

Originally, after reading the article on my own, I thought the simple Central Park manipulative was a good way to help students learn how to create their own expressions after realizing guessing and checking the answers was no longer a good strategy. However, as the class discussed this task and its limitations (including the students too easily sliding into just guessing-and-checking instead of dividing, creating expressions, incorporating variables, etc.), my opinion shifted negatively towards this task, and my eyes were opened to this fundamental flaw. It seems to me that this task, developed in the early stages of Desmos.com, was sufficient for the time being, but since its technology has drastically improved in the last couple of years, it will no longer suffice as a helpful tool for instruction.

The class collectively agreed that the Polygraph task provided much better results. This is because this task informed the teacher about what kinds of vocabulary each student was able to use when describing characteristics of simple polygons graphed on common Cartesian planes. I agree with the fact that students are much more apt to engage in a mathematical problem if they see a need to do so. It is much harder to have students engage in a problem that they do not already have. This task puts them in a situation in which they need something (i.e. formal mathematical vocabulary), so they are more likely to learn the formal language required.

On a more general note, one suggestion someone made in our class discussion when students get stuck, is to scale back to an easier “back-up” assignment. While I did not agree with this strategy for my own teaching techniques, I do agree with the next suggestion: Instead, ask a question that makes the current difficult assignment a little more doable. This does not allow students to get in the habit of “playing dumb” or giving up too easily or often or soon because they know they will have to engage in the task anyway—it is already established over time in the classroom’s expectations for the learning environment. However, we teachers must be sure the cognitive demand of the task is not too high before administering it to the class if we do not have an easier “back-up” task to administer if students begin to struggle.

Connections:

A feature included in this website that thoroughly impressed me as a future educator was the ability to monitor all the students’ screens at the same time. In each of these two tasks described above, I will be able to observe all the questions students are asking, all the responses, all the expressions students come up with to try to describe the space each car needs in the parking lot, etc. By having all of this information so easily accessible, teachers can identify the specific students who might be struggling with the concepts, or who might just be guessing-and-checking. In the class discussion, we mentioned that there is a stark contrast between the students who are guessing-and-checking because they do not want to engage in the task (a motivation problem), and the students who are doing so because they do not know how to engage in the task (a mathematical problem that could be a result of the task shortcoming, a misunderstanding of a fundamental concept, or a misunderstanding of instructions). In addition, this also allows teachers to observe whether or not it is a class-wide problem or just an issue with certain individuals. Teachers can then decide how to address the issues at hand to facilitate the best learning environment within the classroom.

Students are also able to monitor their own progress as the program checks their answer and suggests new ways to accomplish the task. In the Polygraph task, students bounce ideas off of their partner to eliminate different graphs through the use of yes-or-no questions. The series of these questions and their affirmative or negative answers teach both students involved what the other might be thinking and how he or she might go about solving the task. This diminishes the busy work of the teacher going around individually to check answers and explain to each student where they went wrong and why. In my future classroom, I will give my students in partnerships the opportunity to talk together about the conclusions they reached on their own and the conclusions they reached as they worked together to complete the task.

Another point discussed was the benefit from helping students by giving a little hint—perhaps giving an idea of the direction a student might solve a problem. Often there are students who need just a tiny push in the right direction, and then they can hit the ground running, figuring out the answer completely on their own! How satisfying that is for them! This is a strategy I plan to use in my future classroom as I get to know my students and identify the specific students who need subtle hints every once in a while instead of the whole procedure, or the lack of any help whatsoever.