Dave Hewitt and Mathematical Awareness and Fractions Reflection

Here was my attempt to the activities proposed by Dave Hewitt in the video:

For the first task, it required my awareness of how converting fractions worked. As a result, the trickiest part was ensuring my mental math was correct, without being able to check a calculator. 

For the second task, part 1: I started off with trial and error. By using 3/4, I tried to get as close as possible to 11, which brought me to 12/16. I subtracted 1 from the numerator and denominator to get 11/15, and it was honestly a fluke that that worked! The process of checking was the most difficult part, because the mental math could not be verified with a calculator. 

For the second task, part 2: I wrote down all possible fractions with numerators less than 11 that were somewhat around 70 to 75%. There were only a few options for each. So for the numerator 6, none of the denominators could be <=6 otherwise that would be > 1. Hence possibilities were 6/7, 6/8, 6/9... I could reason that 6/9 was definitely < 70% and 6/8 was = 75%. That left 6/7 as the only option to try, but that was in between 5/7 and 7/7, which likely would not work. This thought process was applied for almost all numbers. Ones like 8/11 that was difficult to reason needed scratch paper to do the common denomiator comparisons in Task 1. 

For the second task, part 3: The scratch work was roughly laid out in part 2, so based on that thought process, I don't think it is possible to have consecutive fractions, IF the traditional fraction is followed. Dave Hewitt didn't mention any restrictions on continued fractions which should open up the possibility for this to work, however if it is to follow his analogy of paintbrushes and sticks, then continued fractions are too abstract. 

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Three things that made me stop during his videos: 

1) I like how he gave the examples of 2 + 3 with various objects, and seamlessly integrated fractions right into it. In terms of mathematics teaching, this was a good example for explaining how fractions can be added or subtracted when the base is the same. However, in one example he had blue and black paintbrushes, and I had a harder time thinking of how to use regular objects to demonstrate equivalent fractions or fractions with integers. It would be really difficult to split 3 paintbrushes in 3, for example, to demonstrate that 1/3 = 3/9. This is where the abstract thinking in mathematics needs to be clarified, so students understand where the analogy boat stops sailing. 

2) The idea that students can "get a lot from a little", resonated really nicely. I have a hard time grasping what students are able to imagine and what is too much of a stretch, so it was comforting to hear about believing in the power of their imagination, especially in math. That said, on my practicum, I gave my ninth graders a puzzle that seemed tough but doable, and unfortunately it was not doable for more than 50% of the class. Since the students were not engaged enough in mathematics and lacked a lot of fundamentals, they don't feel comfortable being curious, and immediately freeze. It was also tough given how diverse the math ability in the classroom was. I'd like to see how this phrase can work once I understand how to target the average ability in the classroom. 

3) His mention of awareness was my favourite stop in the video! I always tell my students that in order to be helped, they need to know what they know, and what they don't know. Students frequently remark that, "I can't do this", "I don't know what's going on," "Nothing makes sense," "Everything is confusing," "This is too hard," etc. and none of those phrases are helpful to their understanding. When students are able to break down their comprehension, their learning journey becomes more familiar, especially in math classes. It also helps them shift their mindset, because they can see that there is something comprehensible to them even though there exists something that isn't. 

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How I think Hewitt came up with the fractions problem?

Given that there are plenty of numbers that can lead to this same problem, I think it's possible he could have stumbled on this set of 4 numbers and took advantage of their properties. That said, I think it's likely he started with the goal of wanting students to understand fractions and equivalent fractions (which would explain the restriction on decimals and calculators), and the first two tasks would have derived fairly naturally from it. In trying the problem (by having the student bird lens), he would have also realized more directions the problem could have gone in, and experimented as such. 

I think that teacher-created problems are significantly more effective than borrowed problems from textbooks. During the practicum, I noticed the difficulties in teaching lessons that aren't your own. Often, the designers of the activities have a number of thought processes and goals and tangents related to the activity, and it can be tough for someone else to pick it up without carefully understanding the rationale. Textbooks that are designed for practicing tasks don't usually allow for creative interactions with numbers, so laying the foundation for tricky questions like this can be powerful as a learning activity. 

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What do I take away from Hewitt's ideas for my own teaching?

I've been really inspired about his ability to scaffold lessons and take students to complex conclusions. From both of his videos (on mathematical awareness and the algebra one we watched in class), I think that if we remain authentic to our lessons, we'll have better efficacy. This is because as teachers, we can also have awareness over what works and what doesn't work, and that's not easy to determine when we aren't authentic in our teaching. I also really like how he uses non-traditional lecture styles, and encourages students to seek correctness from eachother and not from him. The idea of Hewitt being a "guide on the side, instead of a sage on the stage", is where all math teachers should aspire to be.