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.