Assignment 3.2

Continuing to field test Day 1 of my Mathematics 1 Unit Plan on number systems and measurements, I tested out my lesson plan on number systems and types of numbers on a group of four adults (two male, two female) and the 10 year old, self-volunteered son of one of the pairs.

I found that my lesson was a lot shorter than I had intended. Overall, the core of it took about twenty to twenty-five minutes. This may have been partially due to the artificially small class size, but in an actual class, I would have certainly found myself running short. I'm not certain on how I would extend on the fly like that. If my pre-assessments, ongoing assessments, and post-assessments indicated a strong grasp, I could expand into the reals and complex number systems and possibly into the application of complex numbers as fractals. Another option would be to demonstrate some simple closure properties of the sets of numbers (e.g. adding two integers will always yield an integer answer). Alternatively, more exploration of the applications of these numbers in measurements and starting the next part of the unit plan could be more appropriate.

During the Graffiti activity, several of the adults expressed a reluctance to get up as they were, "pretty comfortable right here," but it seemed that once they did, they actually enjoyed being up and moving around. The son definitely enjoyed it and was very active in assigning the adults colors and putting papers and the whiteboard up. They also seemed to appreciate having the opportunity to discuss with their peers what the possible answers would be and the general sense of both collaboration and independence during the activity. While they were working on their own, they had the opportunity to bounce ideas and suggestions, as well as offer help to others. I happened to observe a small bit of peer teaching occur as well during the activity.

One student was still struggling with the idea of rationals and irrationals. His peer pointed out checking to see if it could be turned into something with, "just plain old numbers," instead of, "a bunch of stuff after a decimal point." Her suggestion seemed to help him grasp the idea better.

Doing an on-going assessment of observation and assistance in such an artificially small class size was easy, but also difficult to make it seem like I wasn't hovering over them and not letting them work independently. I feel like this dichotomy would balance out as the class size increased, but after a certain point, would become unbalanced in the opposite direction; monitoring becomes more difficult as the sense of independence becomes easier. This could easily tend towards directionless and chaos.

Sample of adult student's work

The use of the video seemed to be an excellent reinforcer and I ended up using that as the post-assessment instead. There was a brief struggle to get the Khan Academy YouTube video to run on the RaspberryPi that was hooked up to my T.V.; instead, we opted to run it through the YouTube app on the Xbox360. Making sure that the technology works correctly is something that should be handled before a lesson begins.

Using the rubric, I found that the students all succeeded. Only one of them got an answer wrong and it turned out to be a result of having been taught that pi = 22/7. Pi is irrational, therefore 22/7 must also be irrational. The student themselves caught the error when the answers were given and understood on their own where the misconception was.

Overall, I think the lesson was successful with this test class and provided me with some good starting points for making improvements and enhancements. In retrospect, I think I would more have liked to test a different day in the unit plan, however, I lacked sufficient materials to do so outside of a regular classroom. I think that would have let me test for potential struggle and chaos points during some of the more hands-on activities and shown where the plan was deficient.

Assignment 3.1

I decided to field test the first day of my lesson plan for Mathematics 1. In this lesson, no particular standard was being addressed, but it is designed as a lead in for studying measurements, units, and choosing correct units and being able to use appropriate quantities when using descriptive mathematical models.

Not helping the learning process.

The test class was held at my home and consisted of four adults, two male, two female, and the 10 year-old son of two of the adult students. The adults approached the lesson with a positive attitude and enthusiasm and seemed both willing and eager to (re)learn. The child was told that he didn't have to participate as the instructions specified 12 and older, but he insisted on joining in. He also had a positive attitude and enthusiasm, but was frequently distracted by the cat. While keeping him on task was a slight challenge, he still had good participation and seemed to enjoy himself for as long as his attention could be held.

I administered the pre-assessment where I asked them to recall the different types of numbers and either describe or give an example of each. The answers varied somewhat. Everyone remembered whole and natural numbers, most remembered negative numbers, half of the adults and the child remembered fractions and decimals, and one adult remembered the square root of 2. (Though it should be noted that this adult was someone who I had shared a number of math classes with at university and we had a running joke about the square root of 2*.)

What was interesting was the short argument over fractions and decimals being the same type of number or not. The child and half of the adults were insistent that they were different, while one adult said that they were the same, and the fellow math student (correctly) stated that "Well, it depends." This turned out to be an excellent lead-in to the lesson content since the point of contention was that even though you could write a decimal as a fraction and a fraction as a decimal, that didn't mean that they were the same type of number. However, rational numbers are defined to be those numbers that can be written as a fraction while irrational numbers cannot be. Irrational numbers can be written as non-terminating (i.e. never-ending), non-repeating decimals. However, this technical explanation did not help clear up the confusion of an adult and the child. (It should be noted here that the child has been struggling with decimal to fraction conversion and vice versa and that the material was somewhat above his level. This was one of the points when he went off to go fetch the cat.)

I was able to explain to the adult that many decimals can be written as fractions and that those were the ones we were generally used to. Irrationals aren't frequently noticed in everyday life for most people but could be encountered if, say, a 1x1 piece of wood or metal was being cut along the diagonal.

While I did not have a particular rubric for this pre-assessment, I did find that going through and discussing student answers helped open up a path into the lesson and show where a misconception was (fractions are never decimals and decimals are never fractions). Having to clear up and correct this misconception initially led down a path that, while technically correct, turned out to be confusing and misleading before additional clarifying information was provided. It also opened up the opportunity to explain that integers and natural numbers were also rationals because they could be written as fractions.

I feel as though this ended up being a good pre-assessment and lead in to the lesson. I got a good sense of where the students were at and an understanding of where some of the pitfalls were and would be for this lesson. I think a good addition would have also been to have the student gives examples of where they would expect to see the different types of numbers (although one volunteered that negative numbers are seen as his bank balance) and what the context would be. This would have helped tie it into the rest of the unit plan, were it an actual class. Students would have a much stronger sense of how it would all tie together and what "the point" was.

*In every math class we took, the professor would spend the first class going over techniques of proofs. Every time, every professor would, for proof by contradiction, demonstrate that the square root of 2 is irrational. This was to the point that when we were selecting math electives and asking each other what the class was about, the answer would always include, "that the square root of 2 is irrational".

I like to maintain as much privacy as possible, hence, the stick figure picture of me. Now, with a bit of work you can connect that back to my actual name, but no actual pictures of me, which is how I like it and loss of points over this fact means absolutely nothing to me.

I was teaching at a school, but I left due to substantial safety concerns. I started teaching because I wanted to help people and kids in particular. I still want to teach however.

I can't really like both?

I took the two personality tests. TeamTechnology seems to think I'm either an ISFP or an INFP while Kisa seems to think I am an INTP. So we can really only conclude that I'm introverted (shocking, I know) and "perceiving". TeamTechnology had some really nonsensical dichotomies, while Kisa's mostly made sense but had only three options.

Kisa thinks that math and logic is certainly the way to go for me which is great, because I needed justification from a script, and thinks that I enjoy word puzzles but doesn't know that I am literally the worst Scrabble player on the planet. Also, Kisa says that "They spend considerable time second-guessing themselves." because no one else, only INTPs ever do that. (Which is why you never see constant reposts of the aphorism about comparing oneself to other being like comparing your blooper reel to their highlights.)

Meanwhile, over in "Learning and Teaching Styles"...

NC State University rates me as around the middle between "Active and Reflective", "Sensing and Intuitive", "Visual and Verbal", and "Sequential and Global." Meanwhile, the Grasha-Riechmann Student Learning Style Scales rates me as high in "Independant", "avoidant", and "Collaborative" and Moderate in "Dependant", "Competitive", and "Participant", so I apparently am motivated to learn the content and completely unmotivated to do so at the same time. (It actually depends on content and my approach depends on the situation. I retain what interests me and leave behind what doesn't.)

So how does all this relate to how I interact with my students? I try to be strict, but understanding. If I know why a student is demonstrating certain behaviors or where they're having difficulty, I can better help them. I will deal with the reality of the situation, but I will try to find a way to help even if I have to come up with it myself. I like abstraction, but I know that concrete is often easier to grasp. For a student to see and demonstrate by using measuring cups that 1 1/4 = 5/4 can often be more useful than simply teaching the standard algorithm. Once they see why it works, they can understand why the method works. 

The approaches can be balanced and they have to be in order to maximize reaching the entire class. Combining learning styles and activities in appropriate and accessible ways will enable more students, if not all students equal access to the content. This is what we want to do and what I've been trying to do.

All things in moderation.