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.

Pro-D Day - FASD Seminar

I attended the online seminar on FASD (Fetal Alcohol Spectrum Disorder).  We went over what FASD was, which is an often undiagnosed disorder that may result in students being behind developmentally and struggling with sensory input, transitions, memory, executive function, and more.  We then covered what we could do as teachers to support students who may have FASD such as having routine, building relationships with them, and coregulation.

It was interesting to learn about coregulation- essentially using yourself as a calm basis to help calm the student down.  It makes sense that if you are not calm as a teacher, students will be able to tell and subsequently will match your emotions.  I think it's something to keep in mind, especially since teaching can be so stressful at times, that we also need to be practicing methods to stay steady so that we don't pass on our disregulation to our students.

What surprised me was how many people probably have FASD- they gave the statistic that 4% of people likely have the disorder.  I had actually never heard of FASD before attending this session so I wasn't expecting that high of a proportion of the population to have it.

What I would want to follow up on is some of the methods that they gave for helping students with FASD and how they balanced supporting those students' needs with the other students.  For example, they mentioned that one solution they had for a student was to allow them to pace in the back of the classroom to help them concentrate, but I feel like that could be immensely distracting to the other students in the class.  I know that there will always have to be some sort of give and take in the classroom with different students; they can't all have the same needs.  On the other hand, it's true that finding certain accommodations can actually help all students, as you never know who might be having an off day, who might have an invisible or undiagnosed disability, or just otherwise would benefit from having extra support.

Curricular Micro Teaching Reflection

There were both positives and negatives about this curricular micro teaching experience.

Our activity was engaging/fun and we were able to graph our data to form connections for the students between the hands on part and the math.  In general, I think that our teacher presences were also fine.

However,  I think there was space for us to grow in terms of fully teaching the topic at hand.  The way we set up the lesson was as an "after" to a lesson where students had already gone over exponential functions, but I don't think it was fully clear to our class on what they were using the activity to learn about and that was reflected in some of our feedback.  I think that we could've gone over the lesson together more as a teaching team beforehand so that we would've been more clear on what was going to be covered and what we wanted to emphasize because we ended up going in slightly different directions in our thinking mid lesson.

Feedback posted below:

 

 

Curricular Micro Teaching Lesson Plan

Curricular Micro Teaching Lesson Plan by Ben W, Doreen, Minami
Lesson: Pre-Calculus 12 - Exponential Functions: Exponential Decay
Objective:  Understanding exponential decay using a dice rolling activity.

Content Learning Standard: exponential functions and equations (solving problems in situational contexts)

Practical details: 15-minute lesson, 9 students

Materials needed:
200 Dice (the more the merrier)
Computer to graph (desmos) (one computer per group)

Structure:
Introduction (3 minute):
Gauge prior experience with exponential decay. (ie. what should the graph look like?  What are the details of what an exponential equation looks like?)
Briefly explain what it is and where we can see it in the world (ex. radioactive decay)

Dice Rolling Activity (8 minutes):
Learners will break into groups of 4-5, splitting the dice between them.  Explain rules in groups.
Each group will roll the dice together, removing the specified values rolled from the dice pool. Then record the dice that remain in the pool. Then students repeat until a specified number of rolls have been completed.
Example:
build pool of dice, and appoint record keeper
roll the pool of dice, removing the specified values (ie all 1s) and returning the rest to the pool. Record the dice remaining in the pool in our graph.
Repeat step 3 as many times as desired (ie 8 rolls)

Connection:  this activity models exponential decay of the form y=100(⅚)^x, where 100 is our starting number of dice, and each time we are keeping ⅚ of the roll.  Each time we roll, we are keeping (roughly) ⅚ of dice from the previous roll (aka our new y is the previous y*⅚)

Note:  we can let the rolling go until time for closing without regard for specifying a number of rolls.  If they roll through all of the dice early, we can ask them to try it again and see if they get the same thing, adding more data to the graph.

Closing (~4 minutes):  
Pose questions to the whole group as a discussion to have them consolidate their learning.

What is the expected formula of our dice experiment?
Did the groups get the same result?  Does it match up with our formula? -> can plot w desmos (exponential fitting) very easily
Why is the result not the expected result from the formula?
What could we change to make this experiment more ”accurate” to the formula?

Extra Qs as needed:
Would the dice that we removed also model something?  If so, what?  If not, why not?
What would happen if we started with a different number of dice?

Battleground Schools

 The first thing that this reading had me pause on was the cultural assumptions of what someone who likes math may be like- although I agree that what was written is accurate, I think that there is also another dimension to it.  For me, the first thing that leapt to mind was that Asian kids are often expected to like and be good at math.  I definitely also saw the stereotypes of nerd and such playing out in my schooling, but there was also a certain profiling where for many people not being good at math was accepted, but the Asian kids in particular were questioned if they struggled.

Another part that struck me was the mention that they wanted to ban diagrams and geometry from being taught.  I'm a very analytical person where algebra is my forte over geometry, but even I can admit that having a diagram or some sort of picture is one of the most helpful things to think through something and see it in a different way.  I honestly had to stop for a second and try to process learning math without geometry.

The last thing that I'm left wondering about is the state of things today.  Of course, having been in high school just a few years ago, I've perceived that there is still an awful lot of direct lecturing and testing about what you can do and not for understanding.  However, I also went on a school observation in the spring where I briefly talked to a new math teacher who ran his classroom in a way where students didn't take notes and there was almost no lecturing.  Instead he had them working on whiteboards every class to put together patterns themselves before he consolidated their knowledge at the end of class with the more mathematical terms.  I also have watched a few other teachers who occasionally did work on whiteboards, but nothing to the extent that that one teacher did.  It was very interesting to watch.

Lockhart's Lament

 This article was very interesting to read.  I agree that a lot of what we teach students is not because they will specifically use it later in life, but to develop their problem solving skills.  I also agree that just teaching students formulas doesn't necessarily push them to think, rather just encourages them to memorize a process.  This is exactly what Skemp brought up with his relational vs instrumental understanding argument, where there are two ways that we can understand.  Lockhart says that we only really have instrumental understanding in our schools, but that we should really emphasize ways of teaching that lead students to relational understanding.  Now, I do think that it is difficult (as Lockhart acknowledged) to fully switch to something like he described, but what we can do in my opinion is to try and teach with the Socratic method when we can.

One point where I am less inclined to Lockhart's argument is his thoughts about notation.  Although he states that he is only disinclined to "excessive" notation, I feel like the notation that is introduced is for clarity in communication.  I agree that students should be allowed to understand the subject in any way that makes sense to them, but part of math is also about communicating your thoughts to others.  Using the same notation is an easy way to stay on track with what everyone is talking about.  As I wasn't entirely clear on what was meant by "excessive," I'm not sure that I fully disagree with Lockhart here, but it suffices to say that I appreciate notation and I think it's part of learning any new subject.  Even non-scholarly subjects have notation- for example in volleyball, you soon learn about bumping, setting, and spiking, although as a newcomer you may simply be thinking "ah yes I need to hit the ball."

Micro Teaching Reflection

 

My micro teaching lesson was on how to fold a paper shuriken (throwing star).  Overall, I thought it went well.  There were a couple main things noted by everyone about the lesson.  

 The first (which I was very aware of) was that we ended up being a little short on time.  Although I had judged throwing stars as being quick to complete, I didn't account enough time for re-explaining if someone missed part of the explanation, nor did I account (enough) for the fact that the students were origami beginners.  When I made one to practice, it only took me a few minutes so I figured that about double that would suffice for everyone else, especially since we had just done some other origami in class.  I'm honestly not sure if I could've picked anything easier for origami that would have been satisfying, but I will try to keep in mind for the future that my guesstimated timings still need work and so I should plan for situations where I run short on it.

The other main comment that I got was that the explanations were clear.  I particularly appreciated the note about giving warnings for tricky parts.  I hadn't specifically planned on doing that as anything more than a thought in my own head that those parts would take longer, but I think that it also helped everyone feel more calm about trying out the new fold because it had been acknowledged to be difficult.  I think it also can help in general to warn students to pay close attention to those particular instructions and I'll try to keep incorporating that into my lessons.

Although we ended up cutting it very close, everyone got their throwing star done at the end and one person even mentioned that it was the first time they had really accomplished folding a piece which made me happy to hear.

Micro Teaching Lesson Plan

Lesson: How to fold shuriken (throwing stars)

Objective:  Students will be able to fold a throwing star from a sheet of paper

Practical details: 10 minute lesson, 4 students

Materials needed: one sheet of paper for each person (I will bring an extra sheet for everyone in case they finish early and want to make another); 10 sheets total

Structure:
Introduction (1 minute):
Gauge prior experience with origami and ask if anyone has made a throwing star before (if most of them have, ask about if they have made cranes.  Whichever has more people learning something is the one we will do)
Pass around papers

Instruction (5 minutes):
Lead the students through the origami folding process.   
The first step (tearing the paper in half) is precarious so they need to make sure to crease it very well
Step two: fold one of the halves in half both ways (note we generally keep the color on the outside)
Step three: fold one side up at the half and the other down (note that you need to remember which side you have folded up and which you have folded down) to make a Z-like shape
Step four: repeat steps 2 and 3 with the other half paper, but this time we need to fold the sides in the opposite direction (will make sense with a demo)
Step five: place them back to back in a cross shape such that the sides with openings are facing out (will make sense with a demo)
Step six: fold in the end triangles (it can be a bit finicky) and we are done!

Practice + Closing (~4 minutes):
In this time, students can make a second shuriken with the backup paper (to test if they can do it on their own).  They can also (if there is space in the room) try to throw them.
If there is excess time, students will be asked if they know how shuriken were used, if they think this is the only shape they came in, what they are typically made out of.
If instruction runs over, this section can be dropped

Note: if we end up making cranes instead, the instructions take longer so it will likely fill the ten minutes more fully.  Should there be excess time there, we can talk about how cranes are a symbol of peace and hope due to an atomic bomb survivor who folded 1000 cranes to make a wish for health.
 

Math Art Write Up (Individual)

This project was very interesting to me.  I have done a lot of origami growing up and I find the way that we can twist one single sheet of paper into so many shapes to be fascinating.  I have made the icosahedron that we made in class before by myself in high school, but the artwork by Neel Shrestha caught my eye because although it was a similar shape, I couldn't quite figure out how it all attached.

After we had selected the piece, I told my group that I would help them learn how to make it since I had the most experience with origami and I found Neel's instructions online of how he created it.  I haven't done anything complicated with origami in years so figuring out what each part of the instructions meant was a little satisfying puzzle.  In particular, there was a part where we had to use creases to "twist" two triangles shut which was immensely gratifying to have click into place.  This piece also quickly showcased the math built into origami- the first step was folding a square into an equilateral triangle which was super cool.

Figuring out that first piece took me over half an hour and while we were making the rest of them as a group, we realized quickly that there would be no way that we could teach the class how to make that particular piece in five minutes because the base unit was too difficult for a beginner.  This was obvious from the fact that it took me several minutes to help Doreen and Yikang with their first pieces, and that was a two on one instruction ratio.  Based on this restriction, we decided to go with showing the class the Sonobe unit, which has a much simpler folding process.

Doing the demonstration for the class taught me a few things very quickly.  For one, using descriptive language is very important and will help everyone keep on the right track.  I hadn't fully thought through the fact that students wouldn't be able to simultaneously look at my paper and their own, so coming up with analogies such as the double doors or precisely describing which point on the paper we were folding from and to was key.  In addition, emphasizing points where it would be easy to go wrong was important.  The thing about teaching origami is that every step is important and needs to be executed with a fair amount of precision in order for the whole piece to fit together.  I can see how this could be a problem with classes that aren't listening as closely.

I really wish that we had been able to have a bit more time for the demo so that we could've asked the class to start playing around with the pieces that they made and see what they could create.  The Sonobe units can also be put together in structures of 3, 6, and 12 before you need the 30 to make our icosahedron.  It would've been very interesting to see if they could've made them and I think that this would be an awesome exercise to do with actual classes as well.  It gets them thinking about geometry, but in a way that's almost like playing with legos.

Overall, I think that making this art piece and doing origami in general is a fun (and cheap!) way to get students thinking about geometry.  You can scale the difficulty of what you're doing to ensure that everyone is capable of contributing.  I believe that this would be a fun activity to do in the first couple of days of class, where you end up with an icosahedron with everyone's names on it.  It creates a sense of community and reaffirms that everyone is one part of the whole class, as well as creating a memento for you as a teacher to remember your class by.

Math Art Write Up (Group)

Group: Doreen, Minami, Yikang
Original Artwork: Neel Shrestha - Origami Snub Dodecahedron

Remake & Issues:
We aimed for a complete replica, but with more colors to better demonstrate the number of shapes required to enclose the volume. One major issue in our process appeared on our first few attempts at assembly, the dodecahedron required a very specific orientation of each piece, where mirrored or chiral pieces would not fit in. In the end we realized all pieces require the same orientation, and remade several to complete the assembly.  Below are pictures of the two mirrored pieces.

One interesting aspect of making this structure was seeing how the 2D papers linked together to form the final 3D structure. The folds we made needed to be precise and creased thoroughly, otherwise the locations of connection wouldn’t align with each other perfectly. You can see this in the pictures below. With just two pieces, the origami still lies relatively flat. When we begin to link them all together, it introduces the curve that we see.

 

 

Alterations:
After the snub dodecahedron was finished, we realized that this piece would be too difficult for the class to make, both in individual making and the assembly stages. So we decided to use another polygon as the class activity topic for demonstration. During planning phases, we believed that adding names to everyone’s own pieces would be a nice addition to encourage group work spirit, so it was added to the plan.
This second structure that we made was an icosahedron, which was an interesting counterpoint to the dodecahedron that we had previously made, since their faces and vertices were swapped. Additionally, it had faces that rose up in pyramids as opposed to being empty space like the snub dodecahedron. Below is an example of the icosahedron we made. Another interesting observation about this second creation was that it was structurally quite sound. This is actually common for origami, but it can be strange to think that all of these pieces of paper interlock to form a ball that can be handled more roughly.

 

Interactive Activity:
We decided to use our interactive activity as our alterations to show the class what they might be able to do with their own classes in the future. This included a walkthrough of how to fold the basic piece of a Sonobe unit, as well as a little exposition on how they might talk to their class about playing with origami. When designing the activity, we tried to make sure that it would be fun and accessible to both beginners and experts, as well as creating a sense of community by contributing a small part to end up with one cool structure. If we had more time, it would have been nice to give the class a chance to try and explore to put some structures together themselves.
 

The Locker Problem Notes

 

Sorry, not sure how clear this picture will come out.  Essentially, I started with the facts given to me and tried to reason with the first few lockers about if they would be open or closed.  That led me to thinking about how primes would be open.  Then I started thinking about primes being primes because of their factors and realized that even factor having numbers would be open.  So I tried to think of what would have an odd number of factors and realized it would actually only be the perfect squares.

 

My Favorite and Least Favorite Math Teachers

 Among my favorite math teachers must be Ms. W, Prof H, and Prof A.  Ms. W's class I enjoyed because she worked really hard to make the class fun.  It was my senior year of high school and so she treated us as adults, but she also worked games into her class framework.  It was a class that I could say without a doubt that I wanted to attend every day.  Prof H and Prof A, whom I also liked just from their personalities, pushed us harder in class.  They were the profs who gave us all the pieces and asked us to put it together ourselves in class, which was difficult but ultimately I felt like I had a much better handle on the material because of the practical experience in class.  

On the other side of the spectrum, my least favorite math teachers all share one quality— solely lecturing without interaction with the class.  I find that listening to a lecture for an hour or more gets extremely tiring and I lose focus.  I don't learn the material very well nor do I end up finding it interesting.  This is absolutely something that influences me as I teach and I try to educate in a way that is more engaging than a straight up lecture.

Three Curricula

What Eisner says early on in the reading about schools offering more than simply class material brings to mind something I learned in my EDUC 401 class last year.  In that class, we discussed three different instances of students who were not necessarily acting like "good students" in that they often did not follow school rules nor did they do much work in class.  However, we were given context to their situations and in multiple cases, we rationalized as a class that those students were actually acting in ways that would give them the skills they might need for the future (e.g. social relationship building).

Something else this reading brought up for me was the discussion of implicit curricula, which teaches the values that the society values.  I've attended school in both the US and Japan, but they're very different.  For example, in Japan, the teachers move between classroom as opposed to the students.  The students are also expected to help serve lunches as well as clean up the classroom every day.  Culturally, doing things for the community is important in Japan and those values are reflected in what students are expected to do.  On the other hand, a more extreme example of parents/community getting involved in the US can be seen in legislature such as the bills to try to put the Ten Commandments into every classroom or the bills that try to ban schools from mentioning LGBTQ+ matters.

The explicit curricula that Eisner mentioned is most prominently connected to the BC curriculum through the content learning standards and the big ideas.  However, I do think that BC is trying to address the implicit curriculum and some of the values that we are teaching students through their curriculum by including the curricular competencies and the core competencies, which focus much more on what students are expected to do.  As for the null curriculum, there are some things that were previously ignored that are now trying to be incorporated (such as First Nations perspectives), but I have heard from current math teachers that actually managing to include First Nations perspectives in their lessons without it feeling cheap is quite difficult. 

Relational & Instrumental Understanding

    When Skemp defined the two different kinds of understanding, there was an instant connection in my mind to how I've approached mathematics over the years.  When I'm learning, I quickly gain instrumental understanding, but in order to feel good about the subject I need to reach relational understanding.  I find that one of the best ways to do this is to try to explain it to someone else- thus my tutoring over the years has pushed my relational understanding.  Skemp's descriptions of potential mismatches between students and teachers also caught my eye- although in high school, all my teachers looked for was instrumental understanding, university finals caught me off guard with questions that pushed beyond exactly what we had been taught and it forced me into relational understanding if I wanted to succeed.  One last thing that arose for me during reading was about the BC curriculum.  Although I don't know too much about it yet, I believe that a teacher I worked with last semester mentioned that they had switched from having content expectations to having "curricular competencies" where they are able to do all sorts of things like explaining and reasoning as opposed to knowing specific content.  However, she also mentioned that this made it a bit of a struggle, since the content is so closely tied to the competencies.

    Overall, I agree with Skemp that relational understanding serves us better.  Understanding the material instead of simply regurgitating rules not only helps making connections within math class, but also might help students think about math out in their own lives and to be able to apply lessons they've learned.  Many students don't continue in math past high school, but it can still help them adapt to whatever is happening in their lives.  While a certain level of instrumental understanding makes math speedy, relational understanding gives a much more complete picture.