Graphing Understanding

One of the things I would like to do with my students is have them graph their understanding of different objectives and mathematical practices. This would help them visualize their progress at different points throughout the semester. It would also help me see where they know they need help. Both asking questions in class approaching a teacher after can be intimidating and a bit nerve-wracking for some, so I think this would be a very low-key and passive way for them to tell me how they think they’re doing. I imagine something like a y-axis of levels of understanding, including statements like “I do not understand this at all,” “I understand parts of this,” and “I fully understand this.” On the x-axis would be the different elements students have been exposed to.

While this would not be used to grade them, I would use it as an informal, formative assessment to help let me know what to reteach or which students needed intervention. I think another way this could help my students is to visually show them that they have, in fact, learned something over the course of the class. If they start with a graph at the beginning of the year with low levels of understanding and increase over the semester or year, they will be able to give a tangible answer to the question, “What have you learned this year?” It would also be useful to show parents during conferences as a way to illustrate (albeit self-reported) learning or during interventions to show which elements the student definitely struggles with. It would also be interesting to see if some students understand past elements better after applying them to newly learned material. There is no doubt that students need to learn certain information before moving on. But would the deliberate practice and application help increase that learning after they did move on? Also, I wonder if I were to measure their understanding immediately before and after breaks, would there be a drop in knowledge due to the time off?

This is an activity I would like to take class time to go over at least three times a year, though ideally it would be done once every month. I think it would be helpful to me as an informal, formative assessment and a good illustration of self-reported knowledge for students, parents, and administrators. Let me know what you think in the comments!


What Future for Education?

I’ve started auditing the MOOC “What Future for Education?” on the Coursera platform. It’s common to think of the future of educating and school in terms of what we have available and what we value today. This course admits that extrapolating what the future will hold has failed multiple times in the past (in everything from past teaching techniques to sci-fi films of the last century).

I hope this course will open my mind to techniques and tools that I haven’t considered, as well as bring up philosophical questions concerning teaching and schools that I haven’t been exposed to in my bachelor program.

My current views about education are a bit scattershot, but overall I have a few resounding beliefs:

  • Technology in education should be used primarily to teach standards and skills in the respective class or subject. A secondary goal may be to expose students to that tool. Technology shouldn’t be brought into a school unless the primary goal is served, as education is often lost when classes are forced to use objects that are “cool” and “fun”, but hinder learning.
  • Schooling should be teacher guided, but student-centered. At the moment, I believe school is student-guided (students are supposed to know what part of a lesson they don’t understand, then ask the right questions to get there without being given the base knowledge to do this), but teacher-centered (teachers decide what levels students should be studying, whether or not that student is above or below that point).
  • Students need to be given a reason to be in school that is useful to them. Many students have other responsibilities (young siblings or their own child to care for, work or parents’ bills to help pay, etc.) and education should offer those students skills that will help them in their alternative lives.

Future posts will follow my path in this class, where I touch on these topics as they are brought up in the lectures.



Class Discussions

In both of the EdX courses I took from Stanford, the professors often mentioned how important working with others and discussing justifications can be. I believed it worked– for others. It was one of my “I am awesome and independent” moments. Sure, I would encourage my students to pair up and talk about the questions and answers that came up in class, but it was never something that I had intended to practice myself. This changed Monday, when I swore during my Intro to Probability class.

We were asked how many permutations there were in the stringing of a necklace, consisting of three different-colored beads. We easily came up with six. Next, we were told to discuss within our table groups how many possibilities there were without repeats (Since the necklace would connect, rotations would fall into the same permutation). I came up with two. My partner claimed that there was only one. We argued a bit, then I finally raised my voice and said, “No. See, there are three rotations of this one [where I gestured to my model], and three reversed rotations of that one, and unless you flip the necklace– Shit!” My partner and professor grinned and some of my classmates snickered. My prof then waited as I explained to the rest of the doubters how there was only one unique way to string the necklace.

By justifying my reasoning that there were two options aloud, I was able to hear the flaw in my logic that my partner had seen. I was also able to solidify my understanding of the problem by tailoring my justification to make sense to other students who I had previously agreed with.

Communication is something my prof frequently encourages during class. There is a very small portion of class time spent watching him write on the board or explaining concepts without us interrupting with questions and comments. When we bring something up, he immediately asks us what our interpretation of the issue is and to try to come up with an answer in our own way and at our own pace. Class discussions continue to play an important part in my learning. When I believe I understand a concept, I can turn and explain my thought process to my partner, who then peppers me with questions about every step I’ve made. This helps me find and understand my mistakes and gives me a chance to practice my own teaching skills for the future. When he grasps something first, he’ll explain step-by-step, then ask me to re-explain it to him in my own way, to make sure I ‘get it’. These activities have deepened my level of understanding for the subject and helped me to double-check my work and fix mistakes early on.

While class discussion may seem overrated to introverts and independents (like me), it is a key element in the learning process– particularly due to the requirement of putting thoughts into words and listening to what comes out. I highly encourage classrooms to become less listen-and-repeat and more discussion-based. I was very surprised at how much of a difference it has already made in my education.