Tuesday, September 18, 2018

Day 14: Misplaced Stress

I am very much enjoying my classes. I am looking at areas in which I can improve both my planning and my instruction, but overall, I'm pleased with how they are going.

With that said, there is a very brief period of my day, entirely unrelated to my classes or my students, that is the source of a disproportionate amount of my stress.  I am examining ways to mitigate this, but as of now I am unclear on what to do.

I had a conversation with a supportive coworker today that helped me to pinpoint some areas for improvement and shed some light on the reasons for my anxiety.  It's going to be a process.

I'm also noticing an interesting phenomena in my Algebra II classes.  I gave an assignment yesterday for students to practice solving equations with 1 variable.  With no exceptions, my students fell into one of two categories: they either blew through the problems (accurately) and asked what was next, or they became hopelessly stuck on the first one and spent 15 minutes trying to solve it.

There was no middle ground.

Looking at the questions, they were remarkably similar to those given today by the Algebra 1 teacher to his classes as well as bearing a striking resemblance to the ones I gave to many of these same students in Pre-Algebra.

Why do we teach the same problems over and over again and why is there no retention of this information?

A good friend who is working on her doctorate reminded me that OF COURSE there is no retention of this information.  The process of solving equations doesn't really make a ton of sense and is often put out to a level of abstraction that makes it difficult for students to relate to.

The process of solving an equation is disjointed and unconnected to any tangible concepts.  This goes slightly better when we use a physical representation, such as a scale balance, but ultimately this process has the same faults.

I am doing what I can to put in more concrete questions and lifting the restrictions of "write an equation that represents this situation."  Instead, my directions are that I want to come up with an answer that makes, using whatever process makes sense provided that it is mathematically sound and they explain it.

To illustrate this point, we did a series of problems today that read something like:
Stefan left school and drove to his friends house. Eugene left school 2 hours later. He drove at 40 km/h for 3 hours and arrived at the same time as Stefan. How fast was Stefan travelling?

The traditional process would want students to set up an equation relating the various speeds and times.

As a class, we talked about how they would solve this problem if they weren't forced to make a single equation. What if they solved it like they would solve any problem outside of school?

We solved it one piece at a time, picking apart what we knew and finding things we didn't.  When we finished, I showed what the traditional process would look like and we talked about whether one was better than the other. One way was better for generalizing and was more efficient, but the other helped students to understand why we were doing what we were doing.

I prefer the second way as I would rather they have understanding, trusting that efficiency will come later.  I wish I had more time. 40 minute periods are not enough...

Monday, September 17, 2018

Day 13: What's Important?

I put up a question on Twitter today, seeking advice from other math teachers on an issue of precision:

How important is this distinction?

Specifically, what I'm concerned with here is the amount of emphasis that should be placed on the distinction between congruence and equality.

My understanding has always been shapes/segments/angles are congruent while the measures of those objects (lengths, angle measures, etc.) are equal.

As usual, when I put this question out to my teacher community, I received back a wide array of constructive answers that forced me to think about my goals for the class.

On one end, one of the major focuses from the National Council of Teachers of Mathematics has been an emphasis on precision of both work and language.  As a result of this, it's important to teach students the precision language that is used by mathematicians.

On the other end, there is the work of students making themselves understood without bogging them down with the semantics.  As Brian pointed out, he isn't 5'10", he's a human being.  At the same time, when he says "I'm 5'10"" you'd be hard to pressed to find someone who would say "Nice to meet you, 5'10"! I'm Justin!"
Before the discussion, I was heavily leaning towards the view and reasoning expressed by Christopher Danielson:

Yes, the precision is important. Yes, students should be exposed to how mathematicians speak and express their ideas, but I also don't want to lose student interest over semantics.

I feel as though this may be the math-specific version of:
"Can I go to the bathroom?"
"I don't know! Can you??"

If what you're teaching is the difference between "can" and "may," then this is an important distinction to draw.

Along these lines, I want my kids to know and use proper notation, but I don't think it's the most important aspect of the lesson.

I've settled somewhere in the middle.

I, too, will model appropriate use, making corrections where I see them. I want kids to get used to seeing the difference between congruence and equality.  As the year goes on, I may become more strict about the usage, but for now, I think I'll simply settle for modeling.

I have enough things on my plate that I don't think I need to add this.

But I may be wrong...

Thursday, September 13, 2018

Day 11: Exhausted on Purpose

I noticed something interesting today.

My energy and attitude are staying at pretty high levels during my classes, but I'm finding more and more that as soon as class ends, that changes drastically.

I'm finding that the conscious effort of maintaining the high energy and positive attitude is emotionally draining.

With that said, I feel the need to clarify 2 things:

First, the energy and attitude aren't fake. I'm not putting on a show for the students, nor am I pretending to be happy when I'm not.  The effort comes from continuously looking for how to make the best of whatever situation may arise.  Rather than expressing disappointment when only 1 student has completed the assignment, I am using it as an opportunity to talk about decision making.

"I know that you all have things going on. I'm not going to harp on you about your homework because you need to be making choices based on your goals and needs. If you want to understand this material, you're going to have to practice it."

I am trying very hard to honor who they are as people and not just as students. In middle school, it seems more important to emphasize the school as, in theory, their parents are taking care of much of the rest.  As junior and seniors, many of them have jobs and activities, or are responsible for younger siblings.

In addition to this, we talk about getting students ready for the real world, but if we don't allow them to make decisions about what to prioritize are we actually doing that?

Second, the energy and effort spent to maintain this level of involvement is, in my opinion, 100% worth while. I am doing what I can to provide my students with an environment in which they feel safe and comfortable asking questions and taking risks. Students greet me in the hallway with a smile, a handshake, a high five or a fist-bump.

Teaching is about building relationships, otherwise you might as well be an audio book or video playing in front of the class.

At the end of each class, I am exhausted and am tempted to take a nap before the next group comes in, but I don't plan to change what I'm doing any time soon.

Wednesday, September 12, 2018

Day 10: Multiple Pathways

One of the major hurdles that we have to overcome in math class is the misconception that there is one right way to get the answer.

Unfortunately, due to the pressures of the amount of material we are supposed to cover and the limited time in which to do it, we often opt for the "fast way" rather than helping students to understand the how and why.

I've encountered several math teachers who require students to complete assignments using specific procedures and take off points if they find the answer a different way.

I'm not going to judge their pedagogy, or criticize this approach except to say that those are not the skills I choose to emphasize.

I'm lying.  I think this is a terrible way to teach. I think it fosters hatred and confusion for mathematics in students who think differently.  There is no one way to do mathematics and grading students on whether they use your preferred method assesses compliance more than mathematical understanding.  I understand that there are times for this, but if a kid can get to the answer and explain how they got there using mathematical methods, should I penalize them for that?

In any event, once this habit has been formed for students, it's very hard to break.  In my experience, this manifests itself in two main forms:
  1. Students asking "is this how you're supposed to do it?" Occasionally, this a conceptual question, but more often they are really asking "is this how you want me to do it," expecting a single path to be the right one.
  2. Students glancing at a problem and immediately giving up, claiming they don't know what to do.  What this frequently indicates for me is that they think there is a single procedure for each problem and they can't remember it.  In this world view, it makes sense to give up. If there is only one path and I don't know that path, what's the point of trying? Requiring a single procedure in mathematics discourages them from experimentation and trying to "figure it out."

Trying to combat this takes a serious effort.

Today, to help the Algebra II students remember what they know about working with proportions, I had an opportunity to do exactly this.

When presented with a proportion that contained variables on both sides of the equal sign, many of the students fell back on memorized procedures, some of which only work in specific cases, others of which would always work, but made the problem needlessly complicated.

We did the problem at least 5 different ways.  I wrote so that everyone could see the work and so that multiple students could provide feedback as needed.

Each time we went through the problem, students would ask what we do and I replied with "what do you WANT to do? What's your instinct? What do you see?"

No matter what their response was, I wrote it on the board (as long as it was mathematically sound). We played with numbers until we ended at the same spot.

I loved it! There was a ton of "what if" discussions and, since I was the one doing the writing, the risk to the kids was minimal.

There can be great value in working deeper on fewer problems, especially if it's being used to value student thinking.

Tuesday, September 11, 2018

Day 9: Perspective: A Play in One Act

Scene: A classroom
Students are working on pattern recognition

Mr. Aion
Tell me about this pattern

Student 1
It looks like a plus sign

Mr. Aion
Awesome! What else?

Student 2
Each time it's adding one block to each side

Mr. Aion
Say more about that

Student 2
It starts as just a square, but each side of that square has another square on it.  In each new picture, another square is added

Mr. Aion
Cool. What about the next one? What will that look like?

Student 2
It will be a square in the middle with 4 squares in a line on each side

Mr. Aion
Do we agree?

Students nod and murmur approvingly

Mr. Aion
Alright, so what about the 10th step? What will that look like?

Student 3
It will have 21 squares each way

Mr. Aion
**smiles cunningly** 21? Step 1 had 1, Step 2 had 2. Tell me how you counted 21

Student 3
It's makes a plus sign and there are 21 squares going up and down and 21 going back and forth

Mr. Aion
**looks around to rest of class** Thoughts? This isn't at all how we were examining the pattern. What do you think?

Brief discussion ensues where students ask clarifying questions.

Mr. Aion
Since day 1, we have been talking about how mathematics is a language. Everyone speaks it to a certain degree, but our goal is to become fluent. Student 3 had an idea that was different from the rest of ours, but she was able to make her thinking understood, turning an answer that, on the surface, was wrong, into a class discussion about perspective.  While there are answers that may be incorrect, it's important not to discount results that are different without examining the thinking behind them.

Math is more than calculation. Math is discussion, debate, argument and exploration.

**Bell Rings**

Students climb onto their desks.

Oh Captain, My Captain

**Curtain falls**

Monday, September 10, 2018

Day 8: Cracked Expectations

We are beginning Algebra II with a review of linear equations.  This was a topic that was covered in Algebra I, Pre-Algebra and Math 7.  This means it should be a breeze of a review.

It's not going as smoothly as I would like, which is alright.  I don't have a problem with the kids struggling with this topic because I understand how much material was covered in those classes and how long it has been since they worked with the concepts.

It does, mean, however, that I needed to readjust my Monday plans.

Instead of "Any questions on the homework? Cool! Let's start the next section" we ended up with "Anyone do the homework? No? Ok, let's talk!"

We had a brief conversation about how as they get older, the onus of learning passes more to their shoulders.  I explained how I don't want to give them absurd amounts of work just to prove that they know it, but if they aren't doing what I do give, then I don't know what they know.

There are a few reasons why students don't complete assignments on time:

  1. They don't understand the material and then give up
    • If this is the case, then I need to know about it. Students who are lost can become insanely frustrated. (I saw this last week with my own 2nd grader, but that's it's own post) I want a level of frustration that makes them feel they can accomplish a task if they just knew one more thing, and so they go find that thing. This is a difficult balance and I'm still working on it.
  2. They forgot
    • Bruh, get yourself organized! You have a school issued planner to write down your assignments and you can always shoot me a message over Remind. In addition, if they are subscribed to my class using Remind, they get notifications for any assignments I give.
  3. They don't want to
    • As a crappy student myself, I totally get this. I spent WAY too much time thinking that I knew what was going on in class only to get to the test and discover that I was wrong. I sympathize with this completely, which is why I've tried to minimize the amount of out-of-class work I'm assigning and making sure that what I do assign is relevant.  My plan has been to do mini-lessons and then give them time to practice the skills in class where I can help them.  This has been working well in theory, but the majority of the students who REALLY need the practice have been taking too long to get started and too long to complete the tasks, making themselves believe that 10 practice problems will take 6 hours.  ANY task can take forever if you don't start it!
  4. Extenuating circumstances
    • Mom/dad/grandparent/sibling is in the hospital, they are homeless, they have to work 6 hours a day after school, they have to babysit siblings, etc. etc..  I am, in no way, judging these reasons. Many of them are incredibly valid and many of our students live lives more complicated than I can fathom.  Everyone has their own struggle.

I think that, moving forward, I will need to be much more deliberate about my classroom structure. Today, I gave extension activities to the few students who completed their work for today.  The rest of the students and I went over some examples in great detail.  I worked on some problems using a very clear format and engaging my kids, rather than just having them copy the answers.  I also tried to make it clear that my going over these things wasn't a punishment at all.

It was an opportunity for them to learn note-taking and to create a resource for themselves that they could reference as we delve deeper into the content this year.

I will also admit that this strategy changed over the course of the day. I asked my morning students to work more independently and by the afternoon, I realized that they needed a bit more structure.

Unfortunately, we aren't a point yet where I can say "here's the assignment, go to it" and they will.  I have several students who need me there with them, giving them support and encouragement.  I don't mind doing those things, but too many need that at the moment for independent work to be productive for more than one or two.

We will get there! I have faith in my students and myself.

Friday, September 7, 2018

Day 7: Beep Beep!!

All aboard the struggle bus!

I started most of my classes with the following statement:

Today is day 7. For the last 6 days, I have tried my best to be upbeat, energetic and excited. Today, it's not working for me. I'm not sure if it's the heat, or the lack of sleep, or just the normal stress of starting a new year, but I'm on the struggle bus today.  I need you to understand that it has nothing to do with you.  I don't want you to think that my mood, which is obviously depressed from what it has been, has anything to do with how you have been participating, acting or working. You are all champs and have been doing great work since day 1.  Some days you just aren't feeling it and, for whatever reason, that's me today. I'm sorry for that and I will try to do better.
 In addition, the weekly warm-up sheets are coming in and I'm seeing student responses to "I wish Mr. Aion knew..."

The vast majority are either random and funny, or very encouraging.

"I am enjoying this class this year"
"It's ok to not be energetic all the time."
"I would smile at his jokes, but I cut my mouth and it hurts to smile."
"He is one of the coolest, respectful teachers."  (Seriously)

There are, of course, a few that are discouraging

"I used to think I was better at math than most people, but now I've been knocked down a peg."
"I used to think math was easy, but now I know it's not."

Since I am a human being who is currently working on myself, I am WAY over focused on the negative comments, despite the fact that they are vastly outnumbered.  Is it in the nature of humans to look at 99 successes and see 1 failure, or are teachers more susceptible to that mentality?

The goal for next week is to get kids up out of their seats, or working on activities as much as possible.  I want the Geometry classes to start doing constructions and I'm going to browse my resources for a good Algebra 2 activity.  I avoided it this week simply because it has been too hot to ask people to move around.

Overall, it's been a great week and I'm thankful for the kids I have in class.

Time to nap until Monday!

Thursday, September 6, 2018

Day 6: What the Buck-et

I've taught geometry several times before this year.  In all of that time, I would say that the aspect of the course that my students find most challenging is the ability to visualize the shapes without having a physical manifestation.

I'm not sure if this is due to the lack of abstract visualization in the K-12 curriculum, or it's simply an insanely difficult skill.

When we begin to study 3-dimensional shapes in 5th and 6th grade, we hit on the idea of nets as representations of objects.  When I did the cereal box project last year, this was, by far, the most difficult aspect.  Students had tremendous difficulty figuring out what their box would look like when they were unfolded and, similarly, struggled with how to create a net for a box of a desired shape.

We had a VERY long talk in geometry today that dealt with moving from lines to planes.  We talk about each wall in the classroom, as well as the ceiling and floor, can be seen as a plane. Things written on those walls are contained in that plane.

Where it usually falls apart is when we're deciding whether three corners of the room that are NOT all on the same wall happen to be on the same plane.

"Are points E, D and B co-planer?"

"They aren't on the same plane because they are on different walls."
"They are because you can draw a line between any two of them."
"They are because you can make a triangle between them."
"But that triangle isn't on any of the planes."

How do you get students to realize that ANY three points can make a plane? How do you get them to SEE that plane when one isn't there?

I tried drawing it.  I tried having them picture a mesh net between those points.

After they left, I realized I needed a bucket.

I have several small rectangular crates in my room to hold pencils and markers at each table.  I held one up as a physical representation of the space. It allowed me to turn it around and have the kids look inside.

I needed a bucket.

If I had a bucket, I'd bucket in the moooo-oooorning.
Get out of here, you three!
If I had had a bucket, I would have filled it with water.  Then I could put the crate in it, using the water level as the representation of the plane!  Is there a way to submerge the crate such that those three points are on the surface of the water?

I come up with great ideas for yesterday's lessons!

I'm taking a bucket of water to class tomorrow!

Wednesday, September 5, 2018

Day 5: Lesson Plans?

Well, that didn't take long!

Something about the best laid plans...

I managed to fall behind my lesson plans already.  To be fair, our schedule has been messed up by having early dismissals from the heat.  I was also reminded that I have a tendency to ramble when I find an interesting topic.

In geometry, we were defining terms, specifically points, lines and planes.  We were talking about what it means for something to be three-, two-, one- or zero-dimensional. I was reminded of one of my favorite ideas in geometry, which is that two-dimensional objects (like squares) can be seen as the projections or shadows of three-dimensional objects (like cubes).  Similarly, one-dimensional objects can be seen as the shadows of two-dimensional objects.

Extracting from this, it's a great way to ask students to visualize that they (three dimensional objects) are merely projections of some other four-dimensional object.

"Mr. A, what's the highest level math class you can take?"


In other, more reflective, news, I'm having difficulty realizing that my 3rd period class is my 3rd period.  I keep thinking it's 4th and I keep getting shocked when kids come after they leave.

No worries though. I have 175 more days to get it right.

Tuesday, September 4, 2018

Day 4: Lesson Plans

Due to concerns about heat, our district, as well as many of the surrounding districts, closed school early today and will do so again tomorrow.

As much as I'm disappointed to be losing some of my classes so early in the year, my intense desire to fall asleep in front of an open refrigerator tells me that it's a good idea.

In spite of the heat and humidity, my students worked very hard for me today.  I've been scouring various curricula and picking the things that I like in order to try to organize these classes that I haven't taught in many a moon.  I'm also making a concerted effort to be more organized than I normally am.  This is helped along by our new mandate to have lesson plans available for the next week should someone ask for them.

The plans themselves don't have to be anything absurd, just a basic outline and topic list so someone could grab the folder and teach the class in case of emergency.

This is the 12th year full time classroom teacher.  Including the two years that I spent working on my Master's Degree, this is the first time when I have been told to make lesson plans that meet the needs that I set out for myself.

In all previous years, when lesson plans have been mandated, a specific format was chosen and implemented district-wide, regardless of whether aspects of it were useful to individual teachers.

Anyone who has attended an education program can attest to the fact that the required lesson plans would make Tolstoy say "that's a bit much."

As a result of this, I have almost never taken lesson plans seriously.  I would spend hours on a format that was filled with information that I didn't find helpful and would only confuse me further.  In addition to that, they were almost never checked.

A former colleague went a year without changing the plans, only modifying the date before they were submitted.  She was told that her attention lesson plan submission was exemplary.

She even turned in several plans that were laced with profanity, or contained excerpts cut from fiction articles.

No one noticed.

I am not opposed to lesson plans.  I am, however, deeply opposed to assignments that serve no purpose except to check a box.

When we were told that we would be required to have lesson plans on hand in the classroom, I'll admit that I grumbled and planned to look for the old plans from years ago.

This weekend, I actually started doing just that and quickly realized it was a mistake.

Lesson plans in the past have not been helpful to me, but this was a chance to make them so.  With the ability to choose the format, I was able to make it what I wanted.  It is very minimalist, but contains what I'll need to keep my classes moving at a good pace.

I need structure.  I need time tables and check lists and to-do's.  I wander off task in my content, hitting things that are interesting, but not necessarily helpful to my students' learning.

I am glad to be writing lesson plans again.

Education programs, instead of forcing students to write 35 page lesson plans, should be teaching them how to develop their own planning style, selecting the items that are important and productive for them.

I stuck with my lesson plans today and will continue to do so.

At least until I see something shiny.

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