Unit Plan

Here is the link: Link 

Textbooks

(Catching up:

It's interesting that the first textbook I opened (Social Arithmetic) began with an explanation of why a child might want/need to be learning the contents.  In the Math 9 and Pre-Calc 11 workbooks I've looked at in the schools, most of the sections give a similar sort of "where could this math be used" talk.  This textbook is also very much a guide for real life math and has very explicit examples about farming, etc (which makes sense given the title).  This book also opens with a segment about Native Americans that manages to explain how the writers (definitely white) saw the Native Americans' pictorical way of conveying numbers (they saw it as very much inferior) through their explanation of it.  Their own method of explaining math was a lot of sentences.  The book read a lot like an actual book that was telling a story, with questions embedded within the story.

I also took a look at MathQuest8, which has a lot of pictures and a more whimsical feel to it.  There's a lot of different things on each page, although they do pretty well to split it up via lines and boxes.  Each mini-segment does not do a lot of explaining, mostly giving an example and then practice opportunities.  This book is a lot more explicit as compared to Social Arithmetic with mathematical language fully being used.

Personally, I've used Khan Academy a lot in my own learning— it's quite different given that the medium is mostly videos rather than writing, but I think that it allows for a lot more flexibility in what the instructor can explain.  Also, if you have questions, the community might have already asked/answered them.)

While I was in high school, I actually did not use math textbooks in my classes other than to check answers to practice problems we were given.  So I can't say in particular how I was affected as a student then.  I use textbooks much more often in college, although I've never particularly thought about the language they've used.  I believe most that I've seen have used 'we' while writing, which I write as well when I'm writing proofs for homework (I'm not sure if I picked this up from the textbook or from my classes).  I like 'we' because to me it creates a sense that both the author and the reader are involved— you are not being directed to do something by yourself, but to follow along hand-in-hand a path paved for you.

I understand completely why someone would use a textbook, especially as a new teacher, because I think it gives the teacher something clear to follow (you don't have to make it all up).  It can also potentially help you align with other teachers in your school on what you're teaching.  I also think that the workbooks I saw in the schools can help students keep everything in the same place for math class and provide them with structure (explanations, examples, exercises).  On the other hand, I think it can be easy to get stuck in the routine of using the textbook and therefore not do any other sorts of activities.  Not all students learn well from a book.  However, I think it can be helpful at the very least as a reference.

I will say that I talked to a newer math teacher (Max from Hamber) on an observation who did not use the workbook at all; he also did not have students take any notes (and I think he didn't post notes either, but I can't be sure if I'm remembering correctly).  It was a classroom where he had them figure out what they were doing on whiteboards and then gave a quick synthesis at the end of the class.  To be honest, this really threw me for a loop because notes are so fundamentally part of school in my mind.  His argument was that students write notes and then never look at them again, so there was no point.  I personally don't agree with that because I reference my notes (at the very least!) when I do my homework and realize I've forgotten something.  In any case, it was an interesting class to observe.

 

 

Flow

The idea of flow is certainly interesting.  I've played soccer for my whole life at a fairly high level, so I'm very familiar with the concept.  I find that sports very much turn off my brain.  However, this also happens when I encounter an intriguing puzzle (absolutely can be math related) or even when I'm talking/thinking/writing about a subject that I'm invested in.

I may be misremembering, but I thought that the speaker said that flow only happens when one is particularly good at the subject at hand— like with the orchestral conductor.  This is definitely not how I've been thinking about it, as I find I can get into a "flow state" where I am locked into a subject, even if I am not the most skilled at it.  Then again, by the graphic in the post, we define flow as happening when we are sufficiently skilled as well as challenged; this must be the definition we be moving forward with in terms of our classrooms, as probably most of our students will not be "masters" of the subject.

Something that I have noticed in terms of when I can "flow" is that flowing is not necessarily an independent state, but that if other people are not in a similar mindset, they can take you out of it as well.  This would be important to consider if trying to get kids into a flow state in the classroom— if they are not all engaged in what you're doing, there may be a cascading effect that gets everyone working in that group out of it.  Thus, we as teachers must be able to place students that work well together or make sure that our activity is interesting enough that everyone gets invested (either through topic or type of activity).

I do think it is possible to get kids to flow in the math class— I have experienced this in some myself as a student.  Of course I am probably more interested in math than the average student, but this is just to say it is possible.  With a certain amount of buy-in from the kids and an interesting activity, I think it is possible to achieve more generally too.

The Giant Soup Can of Hornby Island

The problem: given the size of the actual Campbell's Soup can (of normal size) and the height of the bike in the photo (my own medium-sized hybrid bike), what are the dimensions of the volunteer fire department's water tank? What is its volume? Does it hold enough water to put out an average house fire?

 

My thought process:

1) How tall is the large can?

Given a "medium sized hybrid bike" and that a bike's height typically comes up to someone's waist, I'd guess the bike is about 3 ft from ground to saddle.  Then the left side of the can looks to be about 3 saddles tall -> around 9 ft.  The right side seems to be a bit shorter, maybe 7-8 ft.  Since the slope seems pretty steady, I'll just assume a regular cylinder with an 8 ft diameter for ease.

2) What are the details of the regular Campbell's soup can?

According to the internet, a soup can has a height of 4.5 inches and a diameter of 3 inches for a 10.75 oz can (based on images).  Based on a soup can I measured, it has a height of 4 inches and a diameter of 2.5 inches for a 284 mL (~9.6 oz) can.  I'll go with the one I measured in case the internet is lying to me.

3) What is the proportion between the cans?

If the small can has a diameter of 2.5 inches ~ 0.2 ft and the large can has a diameter of 8 ft, the large one is 40 times the small one.  Then it's easy to say that the length of the large can should be about 40 * 4 in = 160 in ~ 13.3 ft.  Based on looking at the width of the bike (4-5 ft?), this doesn't seem like a crazy number.

4) What is the volume of the large can?

If it has diameter 8 ft, then it has radius of 4 ft.  So the cylinder is pi*(4 ft)^2 * (13.3 ft) ~ 670 ft^3 ~ 19,000 L ~ 4,173 G.  You could fit ~130 of these into an Olympic size swimming pool, which also feels okay as a guesstimate.

5) Does it hold enough water to put out an average house fire?

Google has some quite conflicting numbers for the water needed for an "average house fire," ranging from 300 gallons to 3,000 gallons.  Either way, it appears that the soup can does hold enough water.

 

Thinking about my process:

For one, there were a couple parts where Google told me something that I then decided to go off my own knowledge instead: how tall a bike is, how tall a soup can is.  However, I genuinely had no idea how much water is needed for a fire, so I was forced to accept what was given to me there.

In basically every step, I fact checked whether or not I felt my number was reasonable (both with a friend and with a comparison to something else).  I tend to be bad at feeling out how big a random number is, so comparing to something that I've actually seen is very helpful for me.  I think that one of the most valuable parts of estimating something is checking whether or not what you have makes sense.

I briefly got stuck on when I thought about the ratios: I was thinking that I could just use the volume of the small can to get the volume of the big one (and you should be able to), but I tend to get confused when converting area/volumes through ratios, so I figured it would be best to just recalculate it.  In fact, if you go off the 284 mL volume of the can and multiply it by 40^3, you get about 18,000 L which is now another way to double check my math.

 

Going beyond— a new puzzle:

Something that I've found people enjoy in the past is the similar question: How many of [this] would fit in [that]?

A basic example: The room occupancy is [x], but if we didn't care about personal space, how many people do you think could fit in the room?  In this universe, people can fly.  How about cats?  How about ants? 

(no picture since the classroom you are in would be most applicable) 

A classic: How many [candy] are in the jar?

 

Not real life, but kind of fun: If we were in a zombie apocalypse, is there a vehicle that we could all successfully get away in?  

Further restrictions can include: 

- I must be able to drive it without practice (ie. no trucks etc)

- What's the smallest one we could get away in?

- Okay, we're driving across the country so maybe we all need some personal space.  What now?

 

 

 

Arbitrary and Necessary

 Hewitt has put into words with this article something that I have thought about before, but not exactly gathered together into coherent meaning.  Previously, I have learned through various tutoring and TAing that we often want to push students to figure things out themselves while perhaps helping them through guiding questions if they're not getting it.  This appears to adhere fairly closely to how Hewitt describes presenting "necessary" information to students.  On the other hand, "arbitrary" information brings back a memory for me from one of the previous articles we have read where the author proposed that so much of definitions are needless, something I took exception to.  Hewitt captures my thoughts on the matter much better- in math, we do have set definitions for things that we have essentially made up, but they are unlikely to change and so must be conveyed as is.

In terms of how to bring this article into my lesson planning- I think that what it brings to mind is the necessity of scaffolding for students at different levels.  I already had as a general concept that I would want to have students try to connect dots themselves if the situation allowed, but in thinking more specifically about lesson planning, I would likely want to prepare for the case that the students wouldn't necessarily all be able to do that.  Hewitt discusses this, saying that something being "necessary" information doesn't mean that every student could figure it out, just that somebody could, which of course seems obvious in retrospect, but I hadn't considered how I might adjust to that in terms of lesson planning.  For example, incorporating review of a subject previously learned before going into a subject that builds on that could be helpful or coming up with different levels of questions to get students at different levels to think about the subject more.  These are things I do while I'm TAing, but I think they are relevant to lesson planning as well.