The first thing that I realized was that the pupils were very attentive. Pupils back then were in my opinion more disciplined and didn't have a lot of external forces and influence pertaining to their lack of attention/self control. They were not chatting amongst themselves and followed Hewitt. Or maybe they acted more polite because they knew they were being recorded. The uniforms caught my eye as well, and those can be a whole topic on its own (on whether or not school should have them as a practice. . . not speaking about some merch clothing that the school may sell). I think it's because the pupils don't have to think about what they wear each day, and that they don't compare their outward clothing appearance with each other (so that everyone is represented on the same level). It is interesting to dive deeper into the topic of having school uniforms in schools. From what I know, independent schools have school uniforms and not public schools. There is probably some school pride associated with it as well. Secondly Hewitt has a unique teaching style. I don’t prefer his method when he bangs the wall with the stick since this creates a lot of noise pollution which irritates me. It might be a way to wake up some sleepy pupils but overall I think it is disruptive, especially if there are classrooms on the other side of the wall. He could’ve used his hand/arms and point/touch those places instead, since it is a nice prelude to the main topic. Also, this was a classroom setting in the 80s, so digital projection/smart boards didn’t exist. So to teach on the chalkboard is something rarely done nowadays since there are usually premade notes for pupils to fill out, and the teacher projects those notes onto the screen/smart board. However Hewitt used that space for pupils to work on problems together, which allowed collaboration and teamwork. I see that he asks and goes over lots of questions verbally. He puts finger by ear for a stronger response from the pupils answering in unison. This is nice since it isn't just the smart kid that shouts out the answer. He also says that he isn't a fact checker and won't really answer "yes" or "no". In fact he asks questions back to the pupils "is this correct". There is also a wait time for the pupils to respond, as the gears in their brain turn as they ponder on the questions. So being patient is crucial.
I think Hewitt created the fractions problems by taking two fractions that are close to each other in simplified form to see if there can be a fraction with particular numerators that is between those two fractions. This is an existential problem which is used in many areas of math. When we know something exists that satisfies a property, then that object may be further used to continue a problem or inquire newer problems that branch off to multiple areas. For me these problems are interesting and it makes one dump all of their math knowledge out of a box and find which tools to use. They make people think deeply and understand the underlying methods to tackling these problems.
Task #1 Fraction that is between 5/7 and 3/4:
Using the lowest common denominator of 7 and 4, then by making equivalent fractions we have 20/28 and 21/28 respectively. Since there is no whole number between 20 and 21 we can multiply top and bottom by 2 for each fraction to get 40/56 and 42/56 respectively. Thus, we can have the fraction 41/56 s.t. 5/7 < 41/56 < 3/4.
Task #2 Fraction that is between 5/7 and 3/4 where the numerator is "11":
Consider 5/7 < 11/x < 3/4 s.t. x is a positive integer.
5/7 < 11/x < 3/4
⟺ 4/3 < x/11 < 7/5
⟺ 220 < 15x < 231
⟺ x = 15 ∵15*15 = 225
∴ 5/7 < 11/15 < 3/4
Task #3, 4 Fraction that is between 5/7 and 3/4 where the numerator is less than or greater than "11":
This part took me a long time to figure out as it's one of those problems I see in number theory classes. To deal with a sufficient condition that gurantees a fraction falling between 5/7 and 3/4, first take a look at fractions 0 < a/b < c/d.
0 < a/b < c/d
⟺ ad < bc
⟺ ady < bcy ∧ adx < bcx x,y positive integers
⟺ ady + axb = a(bx + dy) < bcy + axb = b(ax + cy)
∧ adx + cyd = d(ax + cy) < bcx+ cyd = c(bx + dy)
⟺ a/c < (ax + cy)/(bx + dy) < c/d
Given this result, 5/7 < (5x+3y)/(7x+4y) < 3/4. So for example numerators 8, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23 work.
Hi Michael, thank you for sharing intriguing observations about the classroom dynamics and Hewitt's teaching style! I agree that his emphasis on verbal interaction, collaborative problem-solving, and Hewitt's encouragement of critical and independent thinking is noteworthy. His method of posing questions and waiting for student responses allows for deeper contemplation and active engagement, fostering a more inclusive learning environment.
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