Dave Hewitt & mathematical awareness

As for the dvd clip we watched in class, it was really interesting how the teacher taught polynomials to grade 8 students without doing much talking. Instead, the teacher used actions and let the students think and answer each other's questions. When a student made a mistake, the teacher will let other students correct them instead of he himself doing it. I found it fascinating because students were able to think and self-learn the material while remaining engaged in class. Connecting it to the previous reading, I felt that this was a good way to learn necessary information because contrary to arbitrary content, necessary content requires student understanding.

Dave Hewitt Video:

I thought that the way he explained being unable to add two things together unless they have the same name was a really profound method to teaching grade 8 polynomials. I have never thought of it that way and would otherwise just use the textbook to teach the material. It is often these interactive activities that help students both understand the practicality of and visualize their learning.

While thinking about what he said about being unable to add things without a "common name," I thought that we could even use the idea of a common name to teach substitution methods. For example, if we have (x+y)^2+(x+y)+1 and we call z=x+y, z is now a common name tand we can substitute in z for x and y. In his example, he had 2 pencils and 4 brushes. He could not add them together because they are different objects. However, if he called the pencils and brushes "things" or even more specific, "stationary," he can now add them together. Dave mentioned that this promotes awareness that in order to add things together, you have to find the same name. I found this idea really cool and applicable in my classroom. 

Key awareness allows people to understand an idea without memorizing a method or a mathematical rule. This was an interesting idea. It is kind of like the idea of using common sense to teach mathematics. For example, it is obvious that 3 apples plus two apples is 5 apples, but 3 apples plus two oranges is not 5 apples. On the other hand, 3 apples plus 2 oranges is 5 pieces of fruit. For students who struggle with mathematics or have math anxiety, we can often try to reach them by allowing them to see its practicality in the real world and using common sense to teach the subject (at least at the lower grades when math is not as rigorous). 

These examples were interesting, in the sense that if you find the least common denominator between7 and 4 and compute the two corresponding fractions, there is no integer between the numerators. Hence, we need to change the denominator in order to find a nice fraction in between those two numbers. Such a question would make students think outside of the box and help them understand the relationship between numerators and denominators and how they work.

Overall, Hewitt's ideas are very useful for teachers to think and reflect on how the material can be best communicated towards students. Maybe teaching them that apples and oranges cant add up is a better way than teaching them that x and y cant add. This speaks to the face that awareness is really important for both teachers and students.