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".