Day 112: I AM PRETTY!

I started my day by putting on a tiara that was donated by a 7th grader.

I looked fabulous!

A large portion of my students really enjoyed it.  I got compliments on it all day.  I had the following conversation more than once:

"Mr. Aion, why do you have a tiara on?"
"Because I'm a pretty pretty princess. Don't be jelly!"

I found it amusing that throughout the day, students who were wearing skin-tight leggings imprinted with images of emojis kept telling me how absurd I looked.

I was just as strict in the hallway when it came time to moving kids along, scolding them for profanity or expressing disappointment at inappropriate behavior.

I just did it while wearing a tiara.

The Math 8 kids worked (or didn't) on any assignments that they needed to complete, including the test review that I gave yesterday.  I was able again to sit and work with the kids who needed and wanted the help.

In geometry, I came to the conclusion that we weren't actually doing enough practice.  We talked too much about theory and I wasn't giving the kids adequate time to put that theory to work.  So I gave them more problems from Five Triangles.

They started (and finished) one of them yesterday and were supposed to work on the second for homework.  It was apparently more confusing than I had thought it would be, so we talked about it as a class.  I let the discussion go down some dead ends so that we could talk about various approaches.  All in all, it was an excellent exercise.

The third problem was a bit harder and stumped all of them.  I pushed them in a few different directions and explored the problem myself.  I LOVE the problem, but I think it may have JUST over the border of intrigue and frustration for most of them.

Dylan Kane first introduced me to this one when he was toying around with it at Twitter Math Camp this summer.

It's a great one!

The triangles have an aligned base and are equilateral, each with an area of 20cm^2.  Find the area of the shaded region.

It's the weekend. I'm gonna binge watch House of Cards and not get out of my pajamas.

Day 111: The Good, The Bad, The Ignored

Teachers are never allowed to give up on a kid.

More accurately, teachers are never allowed to ADMIT that they have given up on a kid.

After yesterday's quiz in Math 8 (with class averages of 45% and 36%), I became concerned that a large portion of my students were picking answers without thinking about them or checking them.  SMP6, which we have been reading daily, states "I can work carefully and check my work."  SMP3 states "I can explain my thinking and try to understand others."

I don't think I've been doing enough to emphasize these in the assignments that I've been giving.  So today, I devised an assignment that would cover both.

I picked 5 problems from the test with which many students had difficulty.  I put each one on its own page with lots of white space.  The directions at the top of each page read "Use complete sentences, diagrams or equations to explain why each answer is right or wrong."

On the front cover, I put a 6th problem that I had done as an example.  I went through and gave thorough reasons as to why each set of equations worked or didn't and why each answer was right or wrong.

I need to make sure that I don't forget the students who did what I asked of them.  There were several groups of kids who worked VERY hard.  They kept coming up and asking me for my feedback.

"Does this answer make sense?"
"Is this enough to say what I want?"

I sat with two students who deeply struggle with my work because, in all honesty, it's not developmentally appropriate for them.  They worked incredibly hard and through their frustrations.  I was able to spend the time on them that they needed because I wasn't willing to fight the rest of the students.

When the period ended, only 1 student out of 27 had completed the task I assigned.  For the most part, this was because she was the only one who was working constantly.  While I found this infuriating, I realized later that it was because I was focusing on the students who weren't working rather than those who were.  Thinking back on it, the majority of the students were on task for most of the class.  My experience was skewed, as usual, by the few who were off task because they were significantly louder.

The kids who were working were having quiet conversation while they worked.  When I managed to get a few others back on task, I noticed how their volume dropped dramatically.  Their conversations didn't change much, but it was no longer possible to hear them from 2 area codes away.

In 8th period, I moved between two groups of students and was incredibly pleased by their work.  They took the work seriously and I was able to get them to explain their reasoning very thoroughly.

I focus WAY too much on the negative.  I've talked about this before that when lessons go poorly, I blame myself.  When they go well, I attribute it to the kids.  Neither of these is fully accurate.

In addition, I have been doing a wild disservice to the majority of my students who DO what I ask of them by focusing my energy and frustration on those who don't.

I pulled two of my students aside today to tell them how much I appreciate them.  I need to do this more often and with more students.  My frustrations and focus on the few who are disruptive is modifying my attitude and making my classroom a place where students feel as though they are walking on eggshells.

I can't have that.

I can only control me.  I can do my best to provide an environment that is safe and educationally sound.  I can do what I can to encourage my students, but ultimately, the choice is up to them.

I can't give up trying to convince them, but neither can I sacrifice the rest of my students on the altar of perceived neglect.


At lunch, one of my students asked me:

"Mr. Aion, if you found out that someone was talking about you, what would you do?"

I explained to her how much trouble I had with bullies, minor and major in high school and middle school.  I told her that it took me years to realize that there was no way that I could beat them.  I told her about my favorite Twain quote.

It warmed my heart that she trusted me enough to be able to come to me with this concern.  I am doing good and I need to remind myself of that more often.

Day 110: Re/PreTest

As part of the Pennsylvania Teacher Evaluation Process, we have to set objectives for ourselves using a group of students that we select and markers that we choose.  We've been told that if we find ourselves falling short of the goals, we will be able to modify it later.

I selected the Geometry class and our section on similar triangles and proportions.  Today, they are getting the end of chapter test and we'll be using it as a pre-test.

I told them about it ahead of time because that group has a good amount of test anxiety and I didn't want them to freak.  My plan is to score the tests, but not put the grade in.  I want them focusing on the content rather than worrying about how a test for which they aren't prepared is going to drag their scores down.

I'm debating whether or not to even show them how they do until the end of the chapter.  I want them to see their growth, but I don't want them to fixate on the starting point.

I also sent out a Remind text to all of my students last night.

Tests are not a measure of your value, nor grades a measure of your worth. They are merely a still frame from the movie that is you.
— Justin (@JustinAion) February 24, 2015

The Math 8 students also took a test today.  It was a 20-question, multiple choice test on functions, a topic we've been talking about for almost a month.  I don't always make my own tests, but I did this time.  I specifically picked questions that should have been familiar to my students.

The test did not go well.

On the bright side, of all of the students who attempted the assessment, everyone did better than they would have if they guessed.

I am, however, very disappointed in the lack of identification of reasonable answers.  Too many of the problems were ones where they could have plugged in all 4 answers and seen which worked.

They didn't.

The strategies that I've been emphasizing in class (problem solving, checking work, estimation, etc.) mostly seem to go out the window when it comes time for assessments.  This is true in all of my classes.

There are, of course, some pretty wonderful exceptions to this.  There were several students who used the white space on their tests to recreate strategies that we talked about, and I'm very pleased about that.

Even though I talk about these skills and emphasize them in class on a regular basis, I don't think I'm doing a good enough job of having them practice those skills.

Tomorrow, I'll be picking four or five problems from the test and having them write a sentence or two for each answer about why it's right or wrong.  I have them justify their answers in class, but not often enough on tests.  If I were to do that, perhaps they would be more likely to check their work and realize mistakes before they hand it in.


When I was college, I had a professor whose teaching I didn't understand.  The class was complicated and often confusing, but when it came time for papers, I always earned a better grade than I thought I should have.

A friend of mine clarified how this worked.  He told me that this particular professor "taught hard, graded easy."

This reminds me of a conversation that I had a few weeks ago where it was suggested that the tests should NEVER be a source of a stress.  If I'm doing my job well, when it comes time for a test, the students will not feel as though they are trying something new.

Assignments and homework are for practicing skills, testing theories, experimenting and exploring, while assessments are for demonstrated what you've learned.

I need to be doing a better job of creating directed assignments so my students know EXACTLY what to expect on the tests.

Day 109: Flipping the Birdman

Another 2 hour delay.  I got to sleep in a little and see my family before I came in, which was a great way to start a work day.

I have been lax on checking the work that I've asked my Math 8 students to do, so I gave them a goal and a deadline today.  Then, as they worked, I was able to circulate, helping students who needed it.  I wasn't in a rush and was able to help them much more thoroughly than I have before.

For the majority of my students, interpretation and getting started are the major problems.  The best way that I've found to help them, rather than giving the next step, is to ask guided questions.

What is happening in this problem? What's the story?
What are we being asked to find here?
What do we need to know in order to find that?
How do you know?
What does that answer mean in this problem?

I feel as though I made a good amount of progress with the students whom I was able to help.  They showed greater confidence in their ability and were willing to take some risks with their answers.

The rest of the students (a minority in 1st period and a majority in 8th) were off task for the period.  When I walked near them, they flipped pages in their books and suddenly put their heads down as though I wouldn't know that they had been goofing off.

I asked if they needed any help and asked what I could do for them.

"Nothing. We're good."
"Great! Show me what you're working on."
"Oh.  ...We weren't working."
"I appreciate the honesty.  How about you start? Let's take a look at number 1."
"Nah. We're good."

It's incredibly hard to justify spending precious class time trying to bring them around when I have a ton of other students who want my help as well as need it.

I have this discussion with myself frequently.

I don't have an answer and I don't know if I'm even closer to one than I used to be.

What I DO know, is that it tears me up inside to ride a kid to do work when another is begging me for help.

In #MSMathChat last night, I said that not every person likes every movie.  We want our students to enjoy what we do and be engaged with it, but when they aren't, we try new tactics.  If someone didn't like Frozen, maybe it was because they would rather watch a Shoot-Em-Up action film.  If they didn't like Big Hero 6, maybe it's because they prefer romance.  If they didn't like Birdman, maybe it's because they prefer not to see Michael Keaton in his underwear.

You can't unsee this.

But this analogy depends entirely on the students being interested in movies at all.  Some prefer books, or walks in the woods, or staring at the stars, or drawing, or having coffee and conversation.

We need to recognize that school can not, and probably should not, be all things to all students.  We have a role, but we seem to refuse to acknowledge what it is, almost as though doing so would automatically exclude some children.  Instead, we expect our teachers to work miracles without a basic understand of the needs or desires of our students.

Should James Cameron be held accountable for the fact that I didn't care for Titanic?

Actors and directors are not expected to make EVERYONE love their movies.  It would be nice, but they know their target audiences and they play to their strengths.

Teachers have a target audience of everyone. We do what we can do, but we need to be ok recognizing that can't be all things to all students.


I should teach a course to pre-service teachers.

Day 108: Attending to the Attentive

My being out sick on Wednesday, followed by snow days on Thursday and Friday, meant that the students were a bit nuts today.  Not just my classes, but the halls were buzzing with pent-up energy.

I knew this was going to be the case and I mentally prepared myself for it.

I didn't lower my expectations of my students, but I made a mental note to remember that they were a rambunctious group of 13 and 14 year olds who were coming back to school from 5 days without me and 4 days without school.

Some might argue (and have) that I should do this every day.  They are right.  One of the weird things about teaching middle school is that the students are at a very bizarre stage in development where they show bursts of maturity and responsibility, but are still, in reality, children.

In any event, I came in today with a very high level of patience and tolerance and made a conscious effort not to let age-appropriate behavior annoy me.  So what did I do?

I kept my voice at a slightly lower volume than normal.  I kept a patient and sincere smile on my face.  I took 5 minutes to answer questions of the task I left them to do on Wednesday.  I assigned a new task which allowed them to work (or not) at their own pace and told them I would be available for any questions.

Then I sat with a group of students and we got to work.  I helped them with a question or two, encouraged them to explain their thinking and reasoning.  I let go of their hands and moved to another group to do the same.

I checked in with all of the groups.  Some were working better than others.  Some weren't working at all.  I encouraged them to attempt the problems and asked if they needed help starting.  This worked for some, but not all.

I allowed myself to be ok working with the students who wanted my help.

I strive for 100% engagement, but I need to be ok with less.  Much like what I expect of my students, I need to do my best and be proud of the effort rather than just the results.

I will not get every student every day.

But if I can get most kids, on most days, I think I'm doing alright.

Day 106: (Un)Accountable Talk

It's hard to get back into the swing of things after 4 days off.  This is true for teachers, but especially true for students.

In the cold and ice, we didn't have a 2-hour delay or a closing, but apparently, no one told the bus drivers that.  As a result, my 1st period had about half the students that it normally has.

This half-population class still had difficulty focusing to the point where I'm considering moving students out of groups.  I want them to be working together, but they are having tremendous trouble focusing on the task at hand unless I'm standing over them.

There seem to be minimal issues when we are working as an entire class, but it's possible that I'm just not seeing them.  It's possible that it takes until I say "now you try the next one" for me to notice who is checked out.

I think, however, that they think I'm not paying attention when they are working in groups.  Perhaps they think that the din of conversation will cover the difference between accountable talk and not.

It's also possible that I have been slack in my duties of defining the difference between accountable talk and not.

The reality is that I don't really care WHAT they talk about as long as the tasks get completed.  If a group of students can do what I've asked them to do while discussing their plans for the weekend, I don't see why they shouldn't.  But they aren't.

When I ask them to try an assignment on their own, many of them just sit there and wait for me to go over it.  Or they attempt one problem with minimal focus and when it takes 10 minutes because they are only putting in that minimal focus, I get cries of "this is taking forever!!"

Me: "Of COURSE it's taking forever!  How long does it take you to walk from here to the bathroom?"
S: "Like 20 seconds."
Me: "And if, instead of walking at a reasonable pace, you were to lay on the floor and inch-worm your way there? How long would that take?"
S: "Like an hour."
Me: "Yes. Yes it would. Right now, you are inch-worming through these exercises, not because you don't know how to walk, but because you refuse to stand up and do it.  Not everyone walks at the same pace and that's ok, but laying on the ground crying that you're helpless is something that toddlers do.  Stand up and help yourselves."
S: "That's too much..."

I'm reminded very much of my own children, my 5-year old in particular. (And if I want to be completely honest, I remember this behavior from myself as well.)

When I ask her to get dressed, if she's in one of those moods, she'll flop her clothes around, loose-armed, claiming that she doesn't know how to put them on.  She will make half-hearted attempts to put her arms in sleeves and when it doesn't happen, she collapses onto the ground.  I encourage her to take deep breaths, calm herself down and them we work on the task at hand.

At 5, she doesn't quite understand the idea of "if you would just do it, it would be done and we could move on."

I have distinct memories of my own mother telling me that if I had spent a fraction of the energy I used trying to get out of something on actually doing it, I would be a genius and everything would be done.  I remember sitting at the table, crying and tearing at my hair, refusing (or incapable) of completing an assignment instead of just doing it.

Even now, I lack the kind of focus that I wish I had. (In the time it took my two write that last paragraph, I checked Twitter twice and my email once.)

So, if this is a task that I WANT to do, that is important to me and I have trouble focusing, how can I fully expect my students to do that on anything I ask of them.

I get frustrated because I don't know how to get them to focus and I can't seem to make them understand that if they would just DO the assignment, we could move on.  We have trouble getting to anything more meaningful than calculation because it takes too long.  Not being able to focus on the assignment means that it takes much longer to complete, meaning that the purpose and goal are lost in the shuffle, meaning that by the time we GET to the end, no one remembers why we were doing the problem in the first place and the connections are lost.  Then it might as well be a worksheet.

It's insanely hard to balance my frustration at my students with my empathy of their situation.


One of my geometry students spent the day at The Ellis School last week.  This is an all girls school that specializes in STEAM.  She would be a remarkable fit to their program and a tragic loss to our own district.

She wanted to tell me about her visit today.  This sparked a 45 minute discussion of sexual harassment, victim-blaming, gender roles, societal expectations and double standards.  This group of geometry students has several VERY strong female students who are constantly pushing their own thinking about race and gender, identifying problems and seeking solutions.

I deeply cherish my ability to make my classroom a safe place for these discussions.  I also find it fascinating that during normal class, the boys are chatty and involved while during these discussions, they sit quietly and listen.  I think part of it is that they may not be used to having the girls be so vocal about important issues.  I like to think it's because they are listening and gaining a perspective to which they are not accustomed.

Day 105: Assignment? No Thanks.

The plan for Math 8 was to have today be a practice day.  The students were to have completed the next page of practice problems and be ready to discuss them.

Not a single student had completed the assignment.  I find this frustrating, but wasn't surprised.

So I modified my plan.

Instead, I had the students use the class time to complete the assignment that I had wanted done last night.

Less than 10% did.

I walked around, redirecting students back to task, encouraging them to read directions and restate them in their own words.

The first problem was "Which of these two functions has the greater rate of change?"

It suddenly felt to me as though I were teaching another language, that none of these topics had ever been covered and that it was completely unreasonable of me to expect them to remember anything that we had done over the past 100 days.

And how absurd of me to expect them to attempt it.

There were 4 students in the class who were putting anything more than token effort into the work.


I let my frustration get the better of me.  I took a chair into the hallway and put myself in time-out.

A few students came out to ask me questions and I very politely and patiently helped them.

In relaying this story to my coworkers, I found that almost everyone else is experiencing the same issues.  There are huge problems with retention and effort, as well as organization, all throughout the grade.  With this not just being an issue in my class, I need to find a way to address it both in my room and outside.

It is clear that a concerted community effort needs to be put forth to figure out how to combat this if we are to do the best for our students.

What happens between 7th and 8th grade that changes the dynamics so drastically? 

In 8th period, I wasn't willing to sacrifice the learning of the class for the 4 students who were unable to control themselves.  I sat in the middle of the room, taped two whiteboards back to back, wrote on both sides and taught with patience and calm.

My students turned on each other.  The section of the class who has been trying to learn backed the other 4 into a corner and shouted them down every time they talked.

At one point, one of the disruptive students accused me of refusing to teach him.  He claimed that I wouldn't let him see the whiteboard where I was doing the work (even though it was directly in front of his face.)

I didn't have to say a word in my own defense.  The rest of the class screamed at him.  I hope that it will maybe translate to some introspective thought on his part.  In any event, this method of instruction was the best kind I've found with this group, so I'll do it again until it stops working...


I managed to snag the library computer lab for a period today, so I had my geometry students exploring GeoGebra.  I gave them some basic guidance but let them run wild with it.  They really seemed to enjoy it and were sad when it was over.

When we got back to the room, I pulled up the new shapes book from Christopher Danielson and we had an amazing discussion about properties of shapes.  A group of 4 shapes is on each page and we talked about which shape didn't belong and why.

It was VERY interesting to see which properties were noticed by which students.  Some kids looked for the number of sides, while others looked for whether they were concave or not.  Others looked for what jumped out at them with some students looking for lines of symmetry.

"The one of the bottom right doesn't belong because it's the only one that couldn't be a head."

"The bottom left is different because it's the only one that can't be packed like cheese."

Kids are weird, but pretty cool.

Day 104: Making Problems Better

Irrelevant, out of date, overly complicated, boring: These are just a few of the ways that I could describe the problems that are in our text book.  I found a great example today that I wanted to share:

The function c = 0.5m + 1 describes the cost c in dollars of a phone call that lasts m minutes made from a room at the Shady Tree Hotel. Graph the function. Use the graph to determine how much a 7-minute call will cost.

**yawn**

Dan Meyer talks about ways that teachers can take bad problems and make them good.

This problem is pretty bad.  Other than the incomprehensible wording, confusing presentation and forced premise, I can't figure out what skills or content this problem is supposed to be practicing.

Is the purpose to get students to see that functions are used in real life?  Is it supposed to imply that phone bills will arrive covered in algebra?  Does it want students to feel that world is a confusing and incomprehensible place?  Does it just want them to be able to graph a function with the basic direction of "Graph the function c = 0.5m + 1" cleverly disguised as a practical, real-world problem?

Does it want them to hate and fear mathematics, making them test averse, driving down scores on the standardized tests designed by those same companies, forcing districts to spend their scarce resources on quick-fix curricula designed by the companies that instigated and perpetuated the "downward spiral" of those scores in the firsts place?

I don't know. I'm just a humble math teacher.

What I DO know is that this problem is terrible.

So I used it as an opportunity for my students to practice the skills that I value.

"What in the world are they asking here?"
 **crickets**
"Does Verizon put a formula like this in their commercials?  Have you ever seen it written that way?"

They had not.  We had a discussion about how we could reword the problem to make it more real and less absurd.  We came up with several ideas.

In order to call out from the hotel phone, the hotel will charge $1 to connect the call and $0.50 for each minute that you talk.  Create a graph that shows how cost and minutes on the phone are related.  What equation would describe that graph?

Not perfect, but much better.  I want my students analyzing situations that they will encounter, using wording they will see.


In geometry, I've been following the guided notes that they use at the high school to ensure that we cover the same content.

Today, the activity was one where the students explored the possibilities of sides that will or won't form a triangle.  The activity in their guided notes had them experimenting with strips of paper that were 2, 3, 4, and 5 inches long, asking them to form a hypothesis about necessary side lengths.  Picking 3 of these sides, there are 4 possible combinations.

I wanted something a bit more than this.  So I made more strips for them of lengths varying from 3 centimeters up to 24 centimeters.  There were 15,600 possible combinations, making the group work a bit more interesting.  It opened up the possibility for one person to get all combinations that made triangles and another to get none.

While they were working on that and discussing it, I build an applet in GeoGebra with sliders that let us input side lengths and see whether they made a triangle.

With the visuals, the conversation got much deeper into WHY the sum of the two smaller sides had to be larger than the third side.  The other upside is that my demonstration, along with a brief video from Pixar about how they use geometry to make their movies, has my kids champing at the bit to play around with GeoGebra.

With it loaded onto the student profiles, I hope I'll be able to get them into the lab in the next few days and let them play around with it.

Day 103: Planning to Fail

When I am asked in interviews about my biggest weakness as a teacher, I always lie.

My biggest weakness is in planning.

I get bursts of inspiration, find cool activities and tasks and engage my students.  But I could do a better job if I were willing to spend more time planning.  One of my biggest strengths is my ability to take teachable moments and run with them.  I can start with a basic activity and turn it into something deep and interesting.

I have come to the conclusion that this is not nearly enough for my Math 8 classes.  I don't need a script for them, but with 90 minutes, I clearly need a more involved plan than I've been using.  My basic plan is usually an outline of the topics that I would like to cover with specific examples that I want to talk about and I let the lesson flow organically from there.

This causes a very negative feedback cycle for me.  I don't plan quite enough, so the lessons don't go as well as I would like, so I just want to put it behind me and think about other things, so I don't plan enough for tomorrow, so the lessons don't go as well as I would like...and so on.

I did the same thing as a student and many of my current students do it as well.

"I got this answer wrong, so I'm going to put my head down and tune out."

This causes exponential problems!  In a race, if you stumble, you can't just lay on the ground.  You HAVE to get back up and keep going.

So I'm making a conscious decision to get back up.

I made a plan for Math 8 today.  It was a solid plan that involved students writing, explaining and working in groups.

It didn't go as I planned.

It went better.

My original plan was to start out with some selected entries from Visual Patterns.  We would talk about the relationships, input, output and how to relate them to functions.  Then, once a level of comfort was developed, students would break into groups, develop their own patterns and have other groups find the functions that describe those patterns.  I originally wanted this to be almost a game or contest with group competing to find the best function.

With this plan in place, I had a goal and a path.  Instead, the discussion moved in a different direction and, while I had them develop their own patterns, it became about identifying slope and y-intercept of the function from the chart.  We talked about how the difference between consecutive values of the output related to the slope of the line.

8th period did just as well, but was a bit more talkative.  I was very pleased with the discussions that we had, even if there wasn't as much depth.  Different groups have different capabilities and limits.

All things considered, I was pleased.

Day 102: Meta Feedback

I have had a toothache for about a week.  This afternoon is the earliest that I was able to schedule an appointment to get it fixed.  As a result, regardless of painkillers, I've been unable to get a good night's sleep.  This means that my resolve to fight my students' disruptive behavior has been minimal.  The results of this have been interesting.

Since my enthusiasm and energy have been low, the students have split into two very distinct groups; those who are taking advantage of my lack of engagement to use my class as social hour thinking I'm not noticing, and those who are now leaning forward in their seats, both figuratively and literally, to hear what I have to say.

What's even more interesting to me is that this latter group is primarily composed of the students with disabilities and their friends.  It occurs to me that with the "faster" students out of the way, I'm able to give the rest of the students the attention they need.  They write down what I ask and try problems on their own, knowing that I have the time and patience for them to move at their own pace.

I wish I could find a way to get those "faster" students to remain engaged.  I have yet to hit on a task or activity that will do so.  Their engagement seems to be random, or at least affected by things I have yet to identify.

The sad reality is that I'm not as good at differentiation as I would like to be.  I don't know how to make the diverse activities that my diverse learners require.  The best I've been able to do is to engage one group at a time, alternating based on days.


I've also been slack lately in having my students write about math.  The activity we did in geometry hit some of those points.

I handed out a sheet with three questions on them.  All three were about students making assertions about triangle congruence based on given information.  My students were instructed to determine if the statements made on the paper were true.  I told them to explain their reasoning however they could to make themselves understood.

I didn't let them write their names on the papers.

I wrote numbers.  Then I collected them.

Then I handed them back out to different students.

Part two of the activity was to have them critique the reasoning of others.  I gave them 7 minutes to read the answers and give feedback.  We talked about what is helpful feedback versus not.  We gave examples of both.  It boiled down to the question: If you received this feedback, would you find it helpful?

In part three, the critics gave the papers back to the original students.  Again, we talked about  what makes feedback valuable, useful and productive.  Then things got REALLY meta!

I asked the students to give feedback on the feedback.

Was the feedback helpful? In what way could it have been more so?  Did it leave you wondering what they liked or didn't like?

We had more of a discussion about the goals of the class and the differences between procedural and conceptual knowledge.

I'm still tired...  I need to get this tooth fixed.

Day 101: Arm Wrestling!

First period was on point today!  I gave them some practice problems to work on around functions and let them go.  The problems were low-floor problems that afforded them easy success.  While they were working, I put up a make-shift number line on the wall.

This group of students has been struggling tremendously with adding and subtracting negative numbers.  When I have the write a number line, they get it without confusion.  Once they have it drawn, they are able to ask themselves two important questions: Where do we start? Where do we go?

Me: "What's -4 + 5?"
S: "9?"
Me: "..."
S: "-9?"
Me: "...  Let's draw a number line."
**draws number line**
S: "We start at -4."
Me: "Then?"
S: "Then we move to the right because we're adding and we move 5 spaces."
Me: "So?"
S: "1."

I've been meaning to put up a number line for a while, but the layout of my room and the various objects hanging on the wall make placement a bit awkward.  I really wanted it to extend across the entire room, but I couldn't, so I made it work.

Before the end of the period, we had used it at least 4 times.  They worked while I put it up and finished just before I did.  So they started an arm wrestling contest.  The girls.

After a particularly tense match, I jumped in.  I toyed with the one girl a bit, letting her think I was struggling, then I CRUSHED HER!!!

The second girl, however, gave me a run for my money and I honestly wasn't sure I was going to win.  She had a solid grip and was VERY strong.  I beat her too, but only barely.

That's right. I beat two 13-year-old girls at arm wrestling.

I'm amazing!


I bring this up not to brag (although I am amazing and every should know) but because as soon as the contest was over, all of the kids went back to their seats and we went over the problems.  I didn't have to wrangle them or beg them to get back on task.  I didn't even have to ask.  I picked up my copy and they got back on task.

This was quite unusual and I chalk it up to the fact that I've been working solidly on building relationships with the kids.  Arm wrestling with those students showed them that I value what they value.  It would have been so easy to yell at them to get back in their seats and to stop goofing around.  There are times when I do that.

But not today.


Two or three times in the past week, my own children have wanted to do "experiments."  This involved them mixing various liquids they found in the fridge.  The idea of this makes me cringe, just thinking about the mess and the clean-up.  But this past week, I not only let them, but I set out a tray and helped them pick out some instruments for the experiments.

I try not to think "Oh god! This is going to be terrible! It destroys the order of my kitchen and makes things sticky."

Instead, I think "Why do I need to fight this fight? They aren't really hurting anything and it makes them happy."

I had that same thought today in class.  They had completed their work and I wasn't quite ready to move on yet.  They weren't screaming or disturbing the kids who were still working.  I was able to let them be kids.

And I could be a kid with them.



In geometry, we started a supposedly simple question that I had thrown up on Twitter.

Incenter: center of inscribed circle Circumcenter: center of circumscribed circle Centroid: center of mass Orthocenter: ...? #mtbos
— Justin (@JustinAion) February 6, 2015

After a prolonged conversation and internet search, I got it!  Here's the tidbit that blew my students away:

The orthocenter is the center of the circle that is inscribed in the anticomplementary triangle!

This is the GeoGebraTube sheet that I made for the kids.  We played around with it for an hour, looking at various other points on concurrency.


Exploration is fun!

Day 99: PLN Love and Pizza!

I LOVE my Professional Learning Network!

The people with whom I interact with over Twitter, Voxer and this blog are some of the most amazing teachers I know.  It is no exaggeration to say that they have helped me to change my life and outlook on teaching.  They provide me with resources and push my thinking in directions I would not have otherwise considered.

Today's activity in geometry was a prime example of a PLN provided resource.

On my post yesterday about Conceptual Knowledge, one of my colleagues from California, Jed Butler, linked me to an activity on Illuminations.  It's an extension of the "finding an apartment" activity that they enjoyed with a slight twist.

Today, I posed the following situation:

You own a chain of pizza places in Squareville.  Orders are taken by a central switchboard and sent out to the franchise which is closest to the order.  Your task is divide the town into regions so that you will quickly know which franchise will handle which order.

They start out with 2 franchises in the town, then 3, then 4 and finally 5.

This exercise pushed the outside limits of what they were willing to do.  Admittedly, it may have been more productive as a culminating activity when they had more confidence with the compass.  Regardless, it was interesting.

I still struggle with timing.  In this class, but also in my other ones, I want to give time for students to struggle, but not so much that the kids who are fast get bored.  With such a diverse group of learners, it's a difficult line to walk.

In this activity, one group was on to the exercise with 5 franchises while another was still struggling with 3.  If they get too far apart, the slower students get frustrated and give up, which I don't want.

If all kids were robots, it would be so much easier!

Maybe that's why the educational system tries to make them into robots...

Day 98: Conceptual Knowledge

I grow weary writing about Math 8 all the time, so I'm going to devote today's post to Geometry.

Last night, Michael Pershan (@mpershan), Geoff Krall (@emergentmath) and I had a discussion about the difference between procedural knowledge and conceptual knowledge.  It was a very lengthy conversation with increasing frustration for me because I was having trouble making my view clear in 140 characters.

My stance is that conceptual knowledge is what is used to set up a problem and procedural knowledge is used to solve it.  Even this explanation doesn't adequately state what I want, so instead, I'll use the geometry class to illustrate.

Over the last three days, we have been doing geometric constructions.  They have been using compass and straightedge to find the incenter, circumcenter, orthcenter and centroid of triangles.  They had to define those terms and tell me what lines or segments were used to find them.

"Find the incenter of the triangle" is a task that requires procedural knowledge.

Today, I handed them a map of Los Angeles.

Me: "You and two of your friends have graduated from college and been offered jobs in Los Angeles.  It's an expensive city so you've decided to get a place together.  What kinds of things do you need to think about?"
S: "Expenses, rent, groceries."
Me: "What else?"
S: "How far we are from our jobs."
Me: "Why would that matter?"
S: "It wouldn't be fair if one person was next door to their job and the others have to go across the city."

I gave them the addresses of their jobs and told them to find a place to live.

I didn't give them any other criteria or directions.

S: "Will we need a compass for this?"
Me: "If you choose a method that requires a compass,  I suppose you will. If not, no."
S: "What method are we supposed to use?"
Me: "I don't care. I care about results and reasons.  Be able to justify your choice."

What I was hoping for was "Our apartment will be here because this point is of equal distance from all of the jobs.  We found the circumcenter of the triangle formed by the addresses because the circumcenter is equidistant from the vertices."

I happily accepted several different answers and reasons as long as they were justified.

This is the kind of problem that I consider as one that requires conceptual knowledge.  Yes, they would need procedural knowledge to find the circumcenter, but to know what they were looking for is my goal.

I don't want my students to ask "when am I ever going to use this."  I worry that we focus heavily on procedure to the exclusion of the conceptual knowledge.  It's great to know how to find the circumcenter, but it's better to know when you would need to and why.


In my opinion, procedural knowledge is the how, where conceptual knowledge is the when and why.

Conceptual knowledge and understanding is where the fun and beauty of mathematics lives.

Day 97: Race

Class started today with a question:

"Mr. Aion, what are we doing for Black History Month?"

As the students started arguing about how this is math class and they should be doing something in history, I suggested that we do a research project about black mathematicians.  Instead, we had a lengthy discussion about racial inequality in the public education system.

We talked about white privilege and why, with 350 faculty members in the district, we could only come up with the names of 5 teachers of color (not including administration).  We talked about what it means for them and their families to see primarily white faces as educators.

We talked about what it means for them to leave their neighborhoods to come to a building where white people tell them what to do, how to behave and how to speak.

We talked about how, even though I have their best long-term interests in mind and do my best to act accordingly, I recognize that they don't always believe me.  I expressed my understanding for their reluctance and how I hoped that over the past few months, I have proven that.

We talked about decision making and trying to do what's best for themselves in the long-run versus social pressure now.

It was an important discussion and I was happy to take the class time to do it.

When we got back to content during the second half of class, things were status quo.


Change can't happen overnight, or during a single period.  Each of us is on our own journey and I need to be aware that my students won't always move at the speed that I would like.

That doesn't mean they aren't moving.



In amazing news for me, my speaker proposals have been accepted to the regional conferences of the National Council of Teachers of Mathematics that will be held in Atlantic City in October and Nashville and Minneapolis in November!  I'm pretty pumped!  I'm not sure I can afford to do all three in terms of money or time off of work, but I'm going to try to swing it!