#NCTMRegionals PHX

After school on Wednesday, I high-tailed it to the airport to fly to Phoenix for the National Council of Teachers of Mathematics Western Regional Conference.

When I took off, it was 50 degrees.  When I landed, it was 92 with 0% humidity.

Phoenix is gorgeous! The city itself is almost perfectly flat, but it's surrounded by mountains.

What sessions did you attend, Justin? What did you get from them?

What a great question, fiction reader!  Let me tell you.

When I opened my laptop, I was pleased to find a picture from my older daughter:

Those red marks on our faces are kisses.  Also, I assume that my daughter thinks that, under my beard, I have a butt-chin.

I began the conference with Karim Ani (@KarimKai) from Mathalicious talking about the differences between procedural, conceptual and applied understanding.  I have seen several presentations from Karim and Ginny Stuckey on the amazing programs, activities and applications that Mathalicious uses to make math real for students.  I always love how engaging they are, both in terms of the activities themselves and as presenters.  I am a firm believer in their vision and how it's executed.

My session went as expected.  I had 7 attendees, all of whom were engaged and asking questions, but it solidified in my mind that I need to stop presenting about blogging.  It just isn't a topic for which people are interested in attending a session.  It was suggested that I make it a workshop, where I help people sign up for Twitter, create a blog and get them started on the path, but I'm not sure how interested I am in that.

Rather than blogging, I've been thinking more and more about doing workshops on how to teach all levels of mathematics using the interlocking hexagonal building blocks that I've been collecting.  Workshops tend to draw more people and a considerable amount of interest has been expressed about this.

Looking around at the exhibitor hall, I noticed that there are tons of manipulatives for various grade levels, but not really any that are used from K-12.  I think this is an area that could use some exploration.  I was encouraged further on this point by a fairly important mathematician (whom I'll talk more about in a bit.)


Following my own session, I attended Megan Schmidt's (@VeganMathBeagle) session on teaching statistics and social justice.  The room was very well attended and the presentation was excellent.  We had a discussion afterwards about the kinds of people who would attend such a session.  Social Justice and Mathematics can be a fairly polarizing topic.  Either you understand the importance of it and feel the need to incorporate it into your lessons, or you feel that math class has no need to address these issues.

We encountered a few gentlemen in the presentation who felt the need to question the validity of the statistics that were used because they didn't conform to their worldview.  They felt the need to voice their concerns as though they had just attended a seminar entitled "Mansplaining is Like Nike: Just Do It!"

The first time this occurred was when Megan presented the raw data that showed the birth rates by gender paired up with the gender breakdown of elected law-makers in each state.  There was no analysis, simply a presentation of data.

"Well, actually, male babies have a higher birth rate than female babies," he interjected.

Later, when looking at breakdown of school performance by race, another gentleman called out that Asian students were not on the list and had been marking their race as "white" so that they could get into medical school.

At our morning session on Friday, this man sat with us and attempted to engage Megan in conversation. We were not having it.

A third gentleman questioned the bias of the raw data obtained by the national department of education.

Megan did an incredible job redirecting the conversation back to the presentation and it was, ultimately, an engaging, fascinating and important conversation for us to have.


That evening, I attended a cookout with Stephanie Bowyer (@Melomania) and several of her colleagues.  We spent the evening talking math, teaching, nerd culture and having generally a great time.  I FINALLY got to meet Daniel Schneider (@MathyMcMatherso) and we hit it off like gangbusters!  Dude is awesome and brilliant!

When I got back to the hotel, I opened the program book for Friday and promptly fell asleep in it.


I attended 3 sessions on Friday, all of which were AMAZING!

Lisa Bejarano (@lisabej_manitou) started us off by walking us through her warm-up procedures for the week.  We worked through several of the exercises ourselves and it gave me a considerable amount to think about in terms of my own warm-ups and how directed they are.

Monday: Which One Doesn't Belong
Tuesday: Visual Patterns
Wednesday: Estimation 180
Thursday: Number Talks
Friday: Find the Flub

I've been using Estimation 180 as my daily warm-ups, but the diversity of problem types here makes me want to switch it up.  She even had a sheet for students to fill out each week to provide them with guidance and direction.

I left her session feeling energized.  It only went up from there.

Next, I went to James Tanton's (@JamesTanton) session on Exploding Dots and a deep discussion of understanding place value.

Everything he said and showed us made me lean further forward in my chair and say "Holy crap!!"

Halfway through the presentation, I sent a text to my colleagues back home that said "ASK ME ABOUT DOTS WHEN I GET BACK!!!"

What really hit home for me was how simple his explanation was to understand and how I can smoothly it goes along with the work I've been doing with hexagons in my classes.

After the presentation, I went to speak to him, to show him the hexagons that I had and how I had been implementing his practices in my class without realizing it.  When I pulled my hexagons out of my bag, he asked me how many I had.
"About 6300 at this point."
"Fantastic! Keep collecting them!"

James Tanton told me to continue feeding my hexagonal addiction.  Be still, my beating heart!


Still running high from these two sessions, I went over the "big hall" to watch Robert Kaplinsky (@RobertKaplinsky) talk about Depth of Knowledge.

His talk was engaging and disturbing, the kind of disturbing that makes a teacher question if they just spent 10 years thinking that they were teaching when really they were just passing students through the classroom.

He spoke about 3 different levels of Depth of Knowledge and how we can determine whether students TRULY understand the concepts we are teaching.  He talked about the different types of questions that go with each level.

DoK 1: What is the perimeter of a rectangle that is 4 by 8?
DoK 2: What are the dimension of all of the rectangles with a perimeter of 24?
DoK 3: Of all the rectangles with perimeter of 24, which one has the largest area?

All of these questions assess an understanding of the definition and execution of perimeter, but they get more and more complex as you increase the depth of knowledge.

Another incredibly interesting thing that he said was about the difference between "complicated" and "complex."  He said that things which are are complicated are difficult, but easily to replicate, while things that are complex are difficult, but very hard to replicated.

This is the difference between rocket science and parenting.  Rocket science, while incredibly difficult, can follow a series of steps.  Building the FIRST rocket is infinitely more difficult than building the 1000th.  Raising a child, however, has no set of steps to follow.  Just because you've successfully raised your first child, or second, or fourth, doesn't mean that you will be any better at raising the next one.  There are simply too many variables.

"People think that cooking is complicated, that you can just follow the recipe, but in reality, it's complex. Just following the recipe isn't always enough."

I've followed Robert on Twitter for a while now and have talked with him a few times, but actually getting to meet him in person was fantastic.


The list of people that I was honored to spend time and speak with is WAY too long to mention and I'm afraid that I would miss someone vital.  It was a pleasure to speak with so many incredibly insightful educators and learn from them.  I have a few days to mull these things over before NCTM Eastern Regional Conference in Philadelphia on Monday and Tuesday and then back to work on Wednesday.


I'd love to talk further about my experiences to anyone who is interested and made it this far in my post.

Time to board a plane.  It's 94 degrees here. It will be 53 when I land.

Day 44: Hexagears

Every student who reassessed skills today showed improvement.  I am incredibly pleased.

While they were working, I put the finishing touches on my substitute plans until my return from NCTM Phoenix and NCTM Philly.

I also started playing around a bit with the hexagons to try to incorporate some movement into my creations.  I developed a rudimentary set of gears with some string I had around my classroom.

These are much more impressive when they turn

I will be bringing those, as well as 600 other hexagons, with me to Phoenix.  I would bring more, but that's all I can fit in my carry-on.

Knowing that I will be spending a considerable amount of time waiting for and sitting on airplanes over the next few days, I went over to Math Equals Love and got some pretty awesome puzzles to nerd out to next to strangers.

Man! Those strangers are going to be so lucky to be sitting next to me!

I'm so jealous of them.

Day 43: Poor Grade Prep

I want to avoid giving homework as much as I can, but the students do need practice.  To that end, I try to make time in class and today was that time.

While they were working, I went around the class to see who would be reassessing skills tomorrow before the end of the marking period.

For a week, the list of requirements has been on the board.  They've boiled down to:
1) Do corrections on your last test.
2) Get test and corrections signed.


Every student had this opportunity.

7 will be taking advantage of it.

Several students claimed that they didn't know they had to do test corrections in order to be eligible.  I pointed out where it had not only been written on the board, but where I had them copy it into their notes.

I am bracing myself for a barrage of angry emails and phone calls when students receive less than ideal grades on their report cards.  I have been considering my response:

Thank you very much for your concerns.   I believe in learning from mistakes and am more concerned with student improvement than perfection.  To this end, my grade book will remain open for the entirety of the school year and students may return to any skill to demonstrate mastery.  Students learn at different rates and I do not believe that students who take longer to learn a certain concept should be penalized for it.  As of this moment, your child has demonstrated mastery in several skills and has the opportunity to improve in several others. 

In regards to the grade on the current marking period, the percentage that you see may be lower than what your child has traditionally brought home.  This is partially due to the the increased level of rigor in the middle school and partially due to the difference in my grading system.  The percentage grades that are displayed on the report card are not set in stone, but rather are fluid and may change as your child demonstrates their mastery of the material.  The students have had three opportunities to demonstrate their understanding of the concepts.  As we progress through the year, they will be given more. 

While learning is a life-long process, and shouldn't be limited by calendars, the school year does have such a limit.  If your child is unsatisfied with their mastery of the material, we have discussed in the class the steps that need to be taken to improve their understanding.  For example, the reassessment that was offered on October 26th needed to be preceded by test corrections and a parent signature to keep parents and guardians informed of the process. 

I have explained to the students that their grades are only final when the year is over or when they choose to no longer reassess the skills.  The Report Card is merely a snapshot of their current demonstration of understanding and can be improved up until the end of the school year. 

Thank you very much for your time.  I know that this process is different from the norm and can be confusing.  If you have any further questions, please don't hesitate to ask.  

Thank you again,
Mr. Aion

Here's to hoping I won't have to send it out.

Day 42: What's Going Wrong

I don't want to go back to rows.  I don't want my classroom to look like the traditional class.

I want my class to be based in discussion and discovery.  I want my students building ideas together and working to understand the idea that other people built.

I just don't know how to get there from here.

I feel, in fact, as though we are moving in the wrong direction.



I'm not sure why is happening, but I do have several theories.

1) The novelty has worn off

The district is entirely under one roof and the graduating class is less that 80.  The majority of the students have been here for not only their whole lives, but for several generations.  A large portion of the teachers have children in the district, or are alumni themselves.  It's very tight-knit.

Being a new teacher in any district comes with benefits and drawbacks.  The major drawback is that you don't know the kids as well as you would like and it takes longer to build up the rapport that makes teaching what it is.

As a new teacher, however, you can do all sorts of new things in terms of pedagogy and the students will go with you.  Ideally, they see benefits from these new methods before they slide back into "school sucks" mode.

It takes 28 days to build up habits and some have taken root, but not all.

2) I am not familiar with the curriculum

I have taught multiple different subjects using different curricula over the years, so this really shouldn't be an issue.  I should just be able to follow the text that was chosen by the previous teachers and go through it.  I'm having a problem with timing and sequence.  In addition to this, I find myself having to teach or reteach concepts that would consider to be pre-requisite.

Since the curriculum is new, the pre-requisite skills are also in a very different order than the one with which I'm familiar and different from that which the students have covered.

3) I haven't adjusted to the maturity level

Part of the purpose of having a middle-school model for education is because the drastic change between elementary and high school.  7th and 8th grade are supposed to be used as transition years, helping students to develop the habits that they need to be successful.  Many of these skills and habits are social in nature and are designed to help students not only deal with the academic rigor of high school, but the social pressures of life.

It took me almost an entire year to adjust when I first started teaching 8th grade.  That age level suits me well.

The 7th grade is entirely a different animal.  The kids are goofy and high energy.  Many of them still have an elementary mentality and I haven't adjusted to that yet.

Since this is my first year teaching kids so young, I'm still not sure what is developmentally appropriate behavior and what should be addressed more seriously.

I've been dealing with "he's touching me!" more than I ever have before and I can't tell if it's the kids, or the age.

"Phillip, please cancel my 4:15. I made a boom-boom in my Armani."

Greg, we need to talk.

The reality is that it's clearly a combination of all three of these, as well as several others that I haven't thought of.  The cause is only important since it will help me to identify a solution more quickly.  I'm getting frustrated and that frustration is feeding into a cycle.

The first quarter ends on Wednesday, so I need to figure it out soon.

Day 41: I Made A Thing

For the last day or so, I've been fooling around with Geogebra in an effort to make a 3D geodesic dome to show my class.

I started with a polygon inscribed in a sphere.  At each vertex of the polygon, I added another sphere with a radius of the side of the polygon.

Then I marked the intersections of the smaller spheres with the larger one.

Where those projected circles intersected with each other, I dropped a point.

Finally, I connected the points into equilateral triangles!

I made one from a 10-sided polygon, but didn't love how it looked.  It was good, but basic.

After a few more attempts and several program crashes later, I managed to get the 16-sided base the way I wanted it.

Each triangle is created by using the intersection of small spheres with the large one, then finding the intersections of THOSE, marking a point and connecting them together.  And then doing that again.

And again.

And again.

I'm not super pleased with the top, but with the sheer number of objects on the screen, the program couldn't handle what I wanted to do.

The resulting dome looks pretty awesome.  It was my first time playing around with the 3D capabilities of Geogebra and it was fun.  Several of my pre-algebra students were fascinated by it so I'm going to use a computer lab day to have them play around with it a bit.

I maaaaay also have recreated a simple version of it in every class to show off my skill...I mean...to show off the math and simplicity of it...


I also think that I need to do a better job of structuring the greenhouse project.  The groups are deep into the design phase with several teams already building models.  This is before they've blueprints, calculations or budgets.  We will talk about it more on Monday, but I'm worried if they do those things AFTER the model and things need to be changed, they will be too frustrated with the effort they put in and either fudge the numbers, or give up altogether.


I'm glad it's the weekend.  This week has been trying.

Day 40: Half Day

Last night, while eating chips, I broke a molar.  Today, I took a half-day so I could get it fixed.

In Math 7, we worked more on area models for fractions.

In Pre-Algebra, since we are moving into transformations, I handed out geoboards and had the kids find congruent triangles.  They seemed to truly enjoy the activity and they only broke a few of the rubber bands.

This is one of the shortest posts I've written.  That's mostly because during my down time at school today, when I normally write, I was busy tinkering with Geogebra, constructing a geodesic dome for the integrated math class.

It's taking a while, but it's pretty awesome.

Day 39: It's Escaping!

After a nice long walk to get fresh air and catch a bunch of Pokemon yesterday, I'm feeling a bit better.  I'm still tired and was up, for some stupid reason, at 3 am.

In spite of the lack of sleep, I was feeling more engaged in my classes today.  In Math 7, we began talking about rational numbers and I'm VERY excited to talk to them about the area model of fraction operation.

I'm a pretty big fan of physical representations of abstract concepts, which is probably why I think, speak and teach through analogy.

In Pre-Algebra, since we are starting to talk about transformations, we used mirrors to examine what it looked like to reflect a point across the x- and y-axes.  I even build a small GeoGebra applet for quick and easy examination of the concept.

The Integrated Math class is really what has my attention at the moment.  This Greenhouse project is MUCH bigger than I had originally planned.  We have permission from the principal to build the thing and we scoped out a location.

I called the local borough office to find out what permits we would need to obtain.  I spoke with the shop teacher about the logistics of what we would be able to do in-house and what we would have to outsource.  He talked to me a bit about material selection and planning since he has a fairly large greenhouse of his own.  I asked if he would come and speak with my students at some point soon to talk about planning.

I went and spoke with the director of food services for our cafeteria.  The previous director had actually started the process of getting a grant to build a greenhouse, but never completed it.  The new director told me that she would help look into the requirements that need to be met if we want what we grow to be served in the cafeteria.

Things I need to do (A non-comprehensive list):

Look into fundraising from local businesses
Obtain necessary permits
Set up a time with the computer teacher to get a CAD tutorial for my students
Ask local gardening center to come speak to them/organize field trip to local garden center


The more people in the district that I tell about this project, the more excitement is growing. I'm really fascinated to see how this turns out.



On a totally separate note, the superintendent sent an email to the faculty today advertising EdCampPGH and encouraging everyone to go.  When I wrote back and told her that I have been involved in organizing it for the last 3 years, she asked me to organize one for the district.

While it is not without challenges, I'm so happy in this district.

Day 38: Tired

I'm not feeling it.

We've been preparing our house to sell while looking for a new house and I'm trying to prepare lessons that are engaging and solid.  On top of all of this, I'm presenting at NCTM next week and the week after where I'm taking lead on manning a booth in the exhibitor hall.

Yesterday, I spent 10 minutes trying to explain to a frustrated 8th grader how place value works.

I haven't been able to do any serious exercise recently and I haven't been feeling well enough anyway.

I'm tired and stressed and my mental state is making it very difficult to do my job well.


I'm not sure what I need, but I think it involves sleeping well and staring at the stars.

In Pre-Algebra, we are starting to talk about transformations.  In Math 7, we're talking about rational numbers.  We had a LOOOOOOOOONG discussion about how turning fractions into decimals using the trick of "the fraction falls on it's side and that's how you know which number is inside the division box" is utter nonsense.

There's more to write, but I don't wanna.

Day 37: Corrections

A majorly important component of Standards-Based Grading is the opportunity for students to reassess skills when they feel they have mastered the material.  Many teacher who allow reassessment require students to demonstrate the work that they have done to learn the material

My grade book will be open for the whole year to allow students to do just that, but with the marking period ending soon, many students (and parents) are concerned about the grades.  All of my students took a skill assessment on Friday and now have multiple skills to represent their learning so far.

I began class today by putting the grade book on the board, sorted by overall grade.

"What do you notice about this?"

One of the first things they noticed was how much green was on the screen.  They noticed how the class as a whole was doing was fairly well.  They also noticed the rather wide gaps between the classes.

We also talked about what it meant that several students didn't finish.  We talked about taking your time, but also knowing the material well enough that it didn't take forever.  When I took the quiz to make up the answer key, it took about 5 minutes, which means it shouldn't take them more than 20.

Before I handed back to the assessments, we talked about what steps they needed to accomplish before they could reassess.

Skill Corrections!
1) Use a separate sheet of paper
2) Correct answer with ALL work clearly shown
3) At least one sentence describing the mistake that was made, leading to the incorrect answer
4) Test and corrections signed by a parent
5) Test and corrections stapled together

I then gave them the rest of the period to work on their corrections.  Several asked if they could work together and I strongly encouraged it.

"If you are able to teach another person, you'll be able to understand it MUCH better yourself."

I made sure to tell them how proud I was of their work and their effort.  Time to start working on the next skill list...

Day 36: More Assessment

The skill assessments did not go as planned.

The Math 7 kids stumbled over questions that we have been covering for two weeks.  They didn't do as poorly as they could have, but it was very clear that they were not comfortable with the material.  When asked to write how we've been talking about multiplication, they wrote about division.

I had even left the wording on the board and kids who looked up still used the wording for division.

I'm not sure how to get them to examine their work and ask themselves "is this giving more or less than I started with?"

Over the last few days, I've been emphasizing the importance of using diagrams and our review yesterday was full of them.

The students who used diagrams on the quiz (including my English language learner) performed consistently better than those who didn't; somewhere in the range of 40% better.

There are issues with focus, attention, maturity and retention.  I've already had to move students in two separate classes because they can't seem to stop touching the people around them.  I point these out not as a way of excuse or blame-placing, but to identify the challenges that I need to overcome.

For the first time in a LONG time, I have faith in my lessons and methods.  It may be time to say that my seating arrangement (in groups of 4) isn't working for these classes and it's time to go back to rows for a while.

In Pre-Algebra, I can see a clear correlation between the attention that they pay to class and grades.  In addition, the kids who ask me questions on a regular basis are showing huge amounts of improvement.  This is no surprise.

There's also a MASSIVE difference in my classes.  Two classes have an average in the mid 80's while the third is hovering around 60.  Part of this is because of the time of day when the classes are held.  Another has to do with the make-up of the classes themselves.  The third class has a much higher percentage of students with special needs and IEPs.  That class is also bigger.

All of those are irrelevant since they are out of my hands, but it does show me that I need to attempt different interventions in that class.

The breakdown in Math 7 was similar, although not as extreme.

I'm incredibly fascinated by these charts.  They are sorted by overall score and not by name or other identifying marks.

What do you notice?

What do you wonder?


I wonder what I can be doing better for my students in the afternoon.

Day 35: Dot Placement

Tomorrow will be the next major skill assessment for Math 7 and Pre-Algebra.  We've spent a few weeks going over the material and refining our language and tomorrow will be their chance to demonstrate proficiency.

I am lazy.  I don't like making up my own assignments when someone else already has one I can use.  I don't like making up tests because that takes effort.

Since I've been using Standards-Based Grading, many things have become easier.  I don't grade homework and the classwork that we do isn't for points.  It's been two weeks since a student asked "is this being graded?" because they know that only the skill assessments are worth "points."

The trade off is that I have to spend a TON of more time and effort on my assessments.  I can no longer copy and paste questions from an online book because they don't line up with the skills in a cohesive way.

Rather than tailoring my lessons to the assessment, I need an assessment that is tailored to my lessons.  Since those don't exist, I need to create them.

There are some truly amazing resources online and in the math teacher community to which I belong.  I'm able to pull from those places, but nothing is exactly what I want.  So, grudgingly, I have to create my own assessments.

And I'm LOVING IT!

Not the actual "creation" because I'm still lazy but the level of control that I have is excellent. I'm phrasing things in very specific ways and recreating concepts in the specific fashions

At this point, all teachers reading this are wondering how I managed to go this long without regularly creating my own tests and are rolling their eyes at me.  I get it. I deserve that.

I was pretty pleased with the review on which we worked, since it looked very much like what will be on the test tomorrow.

Put the colored dots on the number line that match the corresponding number 

This was the work that one student did for her warm-up. It's awesome!

Near the end of the day, one of my students had a meltdown and stormed out.  I had a LOOOOONG talk with the rest of the class about their actions and behavior.  We talked about how behavior has consequences, whether or not we see them right away.  This student's meltdown wasn't their fault, but they didn't help.

Kids can be pretty cruel to each other, even when they don't realize it.

Day 34: Team Contract

After splitting my students into groups of 4 yesterday, we spoke about the long term plans for the Greenhouse Project.  They each received a Team Contract and Team Calendar and spent the class filling them out.

The contract is an outline that lays out the rights and responsibilities for each member of the group.  It needs to be filled out and signed with the consent of all group members.  This means that each group contract will look slightly different, but is a consensus of all parties.  I REALLY like this idea because it prevents certain discussions later.

If I assign the roles, there is benefit to having them work outside of their comfort zones, but (as anyone who has carried a group through a project in college) the job still needs to get done whether someone does it well or not.

The contract seems to bring a level of maturity to the process where each team member must actively determine for which role they would be best suited and can benefit the team.  The assessment at the end will be a group grade, but there will also be individual grades looking at whether or not the members completed the agreed-upon tasks.

They also, as a group, decide what happens with members who do not live up the bargain.  They are responsible to their peers, rather than to me.

The scope of this project is...enormous.  One teacher has me thinking about sunlight and pH of the soil.  Another teacher has me thinking about planning a field trip to a local gardening center for research.  I'm thinking about how to get stones engraved with local business names so we can honor sponsors.

In addition to all of this, I'm struggling with how much direction to give them and how much to let them run.  We will have multiple conversations over the next few weeks, but even as someone who likes to say "here's the idea, now run with it" I'm debating about the amount of structure.

With that said, the groups worked VERY well today.  All 3 were engaged in discussion over the contract, the roles they would take on and the timeline for the entire period.



I also had a conversation with the students from last week.  They asked for my help on a problem and then immediately jumped all over everything, derailing my explanation before I could even begin.

Me: "I know you're excited and curious and want to ask questions..."
Them: "If I have a question, I need to ask it!"
Me: "Yes.  And you should! Curiosity is how we learn.  But I also have a plan and a path. It's likely that I'm going to answer your question in the next breath, but I don't get a chance to get there.  On top of that, not everyone in the room thinks the way that you do and they need to be allowed to think and process.  So here's my suggestion for you.  When you have a question, ..."
Them: "Just shut up?"
Me:  **pausing to allow the irony sink in a bit** "No.  Don't shut up.  When you have a question, write it down. Let me finish my thought and I may be answering your question. OR, better yet, you may figure out the answer on your own."

This entire conversation was held in quiet voices while I was sitting at the desk with them.  I REALLY like these kids and I don't want to stifle their curiosity.  I NEVER want them to think that I don't want them to ask questions.  But I also want them to try to solve those problems themselves and to realize that they are part of a learning community.

Communities need to work together and respect the needs of all members.

Day 33: GeoGarden

I have spent considerable amount of time thinking about what I want to do in my Integrated Math class.  We spent a week or so working on coding and did a survey project before that.  Today, we started the project that will take about 4 weeks.

Congratulations! You are now all employed at Aion Engineering!  We have been contacted by GeoGreenhouses, a small startup focused on urban gardening.  They have asked us to help them to design a geodesic greenhouse that can be assembled in the yard to allow the growing of fruits, vegetables, herbs and flowers all year round.

As design teams, you are to create plans and blueprints for a greenhouse kit that can be purchased and assembled at home.  It must be as efficient as possible, using a minimal amount of heating/cooling elements.  It must be rigid and must maximize volume while minimizing surface area. 

Your team will then design a presentation, including a design for the interior layout of planting beds and a proposed itemized budget.

We talked a bit about the history of the geodesic design and Buckminster Fuller.  We talked about greenhouses and watched a brief video about a family in Colorado that built one in their backyard.

I assigned groups randomly and handed out the Team Contracts.  Tomorrow, we will be assigning roles and discussing how to keep the team together.


One of the first questions they asked was about whether we were actually going to build the designs.

I emailed the principal laying out my idea and talking about how it would benefit the school as a whole.  So far, my ideas have been well-received by my administration, but this one goes a bit further than previous proposals, so I sat with my fingers crossed for a while.

My principal was incredibly supportive!

He copied the shop teacher, the Earth science teacher and the cafeteria manager (who has been working on getting funding for such an endeavor) and suggested ways that we could integrate various other aspects of the school community.

My plan is to have each group create a design and proposal on which (hopefully) the rest of the district will vote.  The winning design and budget will be the one that we build and utilize.  We can ask local business for sponsorship and donations.

I see incredible amounts of potential for this, now and in the future, including integrating the elementary in the growing and measurement of plant growth.

More important than any of this, the kids are enthusiastic about the idea.  I will need to work to keep that momentum going, but I hope I've hit one something that they will enjoy and work passionately for.

Day 32: Ruining The Story

My annoyance and frustration at having my students answer questions either before I ask them, or that I'm not asking at all sent me down an odd path today.

"For this next example, I don't want to hear any voices at all" I said, making direct and pointed eye contact with each student individually.

"I'm not asking you any questions or looking for any input.  I'm showing you my thought process.  If you have a question, you can ask that, but don't give me any answers, don't shout out numbers or steps or ideas.  Just be quiet and listen."

"For the next example, I'm being asked to give an estimate for the square root of 23.  I'm going to start by..."

Three hands immediately go up.  Two different students yell out "BETWEEN 4 AND 5!"

I'm having some deep internal struggles with this class.  I love the enthusiasm that some of the students are showing.  They love the problem solving aspects and are excited to move quickly through the material.  The issue, however, seems to be one of empathy and understanding of others in the room.

As with all classes, not everyone is at the same pace.  The students who are excited and ahead of the game are constantly running back and trying to drag the rest along.  It reminds me of children running through a museum or zoo, not allowing others to appreciate what they want.

The way that I design my lessons and my questions are in the form of a story.  I try to build scenery, characters, suspense and drama before coming to the conclusion.  When I teach a lesson, it's for a group and I attempt to bring them with me on a journey.  I have stops planned, sights to see and things to discover.


When The Sixth Sense was released, I went to see it with a group of friends.  In the first scene where Bruce Willis and Haley Joel Osment interact with each other, one of my friends looked down the line of us and said "something's weird with the mom. She's not talking to Bruce Willis.  I'll bet he's dead."

This was a pretty innocent statement and was just a theory.  However, since it turned out to be true, I watched the entire movie in a completely different light than I otherwise would have.


I don't want to crush the enthusiasm of my excited students.  I want them to be excited to come to class, excited to participate and excited when they discover things.  Added to this is the complication that they are middle school girls.

Middle school is typically the age where female students lose their excitement for math and science and I refuse to contribute to that.  These young ladies are talented and I want to push them forward.


I'm just not sure how to help them understand they can do those things without spoiling the surprises for anyone else.

Day 31: HexaSquares

With the success that we had in Math 7 yesterday with using the hexagons to talk about addition, subtraction and multiplication, I wanted to keep the process rolling.  Today, we added division.

As before, I started by having them talk about the operations in the language we've been using, talking about the concepts as physical quantities.  This is a much easier prospect when you have physical objects in front of you.

"15 divided by 3 is asking us how many groups of 3 can we find in 15."

Again, all of the students got their stacks of 48 hexagons.

"What is '48 divided by 8' asking of us?"
"How many groups of 8 are in 48?"
"Show me!"

And they did!

We did several different combinations, having them explain them each time.

I think I'm getting better at the kinds of questions that I'm asking and they are getting better at understanding what I'm asking.  Since we have stacking, interlocking blocks, tomorrow, I'm going to ask them what 4*6*2 would look like.

It also occurred to me that I could use the hexagons to talk about irrational numbers and square roots in my Pre-Algebra class.

So I did!

Each student got a set of 50 hexagons.  We discussed again what questions are being asked by squares and square roots.

"5 squared is asking us to find the area of the square that has side lengths of 5."
"The square root of 25 is asking us to find the side length of a square with an area of 25."


They played around with different combinations for a while, making squares from 16, 36 and 49 hexagons.

Near the end of the period, we began looking at how was could make a square from 17 blocks instead of 16.  Could we make a square from 24?

We talked about how we couldn't do it with whole hexagons, but instead would need pieces of hexagons.

Them: "For a square with an area of 17, side lengths of 4 are too small, but side lengths of 5 are too big. It has to be somewhere in between."
Me: "So let's estimate. Do you think it's closer to 4 or closer to 5? Give me as an estimate."
Them: "4.1 or 4.2."

I built a GeoGebra applet that showed us exactly what it would be and, since we've been working with estimates for 30 days now, it was a pretty easy sell.

We also began looking at where to plot those square roots on a number line.  That will continue tomorrow.


After class, a student who has been particularly distracted and distracting for the past month came up and told me how much he liked using the blocks.  He was on point today and I made sure to tell him so.  He said the blocks help him to understand what's going on, especially with the multiplication and division.

Nothing works for everyone, but everyone has something that works for them.

Day 30: Assistance

After posting about my frustration yesterday, I had some excellent conversations on Twitter.  There were many suggestions made about how I could improve the lesson to hit my desired point a bit harder and create a solid impression on the students.

Traditional mathematical education starts with the abstract and moves towards the concrete.

"Here's the formula. Now let's use it."

More progressive education moves in the opposite direction.

"Here's a pattern or characteristic of physical objects.  Let's figure out what's going on here."

The former of these approaches is often much easier on the teacher, but rarely is helpful for the student.


During one of the conversations last night, I realized that I needed to do more with the physical aspects of multiplication and division and less with the abstract (for now).  This is doubly true if I want my students talking about these operations using a specific language of those physical objects.

@getting_sharper they didn't. Any suggestions on how I could frame those questions?

— Justin (@JustinAion) October 5, 2016

@JustinAion in other words keep the whole the same so less attention is on finding "the answer" and more about communicating actions

— Charlotte D Sharpe (@getting_sharper) October 5, 2016

@JustinAion like, "if we have 48 apples, how could they be arranged on 2 tables?"

— Charlotte D Sharpe (@getting_sharper) October 5, 2016

@JustinAion revoicing student responses as addition (40 and 8 more) and multiplication (2 groups of 24) and division

— Charlotte D Sharpe (@getting_sharper) October 5, 2016

 It is by will alone I set my mind in motion...

Or, more accurately, it is by incredibly helpful suggestions from brilliant educators alone that I set my mind in motion.

Since I wanted them to work on these either individually or in pairs, I had a logistics problem.  I don't have 48 apples for each of my students.  I also decided against stopping to buy a giant bag of M&Ms on my way to work for obvious reasons.

Thankfully, I happen to have a giant tub of interlocking hexagonal building blocks and my OCD has been nudging at me to sort them anyway!

I spent the morning putting them into stacks of 10 and then groups of 50.

Each student got 50 hexagons and was told to put 2 aside so they had 48 (because there are tons of great factors for 48!) I emphasized that since we all had the same number of blocks, the answer was always going to be 48, but what I wanted to see was the representation.


"Show me 40 + 8."
"48!!"
"...Yes. I know. Nicely done. Now show me how you know that."
Second Student: "48!"
**slow blink** "Good. Show me."

Then we got the hang of it and they were really into it!

We moved from addition to subtraction to multiplication.  Each time, I reminded them to talk about it in the language of groups.

"Show me 6*8."
**they do**
"Tell us what you have there."
"We needed to make 6 groups of 8, so I made 6 stacks of 8 blocks."
"Nice. What about you?"
"I made 6 rows with 8 blocks in each row."

We got a slower start than I would have liked, so we didn't get to division today, but that will be tomorrow.

When my 5-minutes-left alarm went off, and they were cleaning up, I asked them what they thought of the activity.  The majority expressed that they thought it really helped them to understand what was going on and, particularly, the language that we're using.

I have a student who is an English Language Learner and I watched her thrive more today than in any other activity we've done so far.

I am indebted to Charlotte for pushing my thinking in this direction!