A Brave New World

If my classes are all built on the concept of reflection, metacognition, and relationships, how do I build and foster that in a hybrid and remote setting? Not going to lie, that has kept me up at night this summer.

We will be in a hybrid situation to start (I’ll see half my students for half my classes in the morning, then the other half in the afternoon; the next day I’ll see the students for the remaining classes in the same manner – but while trying to continue the curriculum), but will still need to accommodate students who cannot or choose not to be at school.

So here are my (very evolving and very flexible) ideas:

  • Start with “Getting To Know You” Google Slides (thanks @howie_hua!)
  • Some of my classes will have a Desmos AB each weekend – alternating between “What’s Going on in This Graph?” and Slow Reveal Graphs.
  • I used templates from @SlidesManiaSM (throw some coffee money their way – the templates are awesome!) to create two great writing tools:
    • My AP Stats students will do chapter summaries (if you use TPS, feel free to use!).
    • My students will still be able to do a weekly online journal (and still be able to handwrite their work, which is big for learning and understanding).
  • One of the classes I teach is a 4th year course for seniors who will not be taking calculus in college. I created the course last year; it’s a combination of intro to stats and personal finance. I made the curriculum based off of Stats Medic, Skew the Script, Next Gen Personal Finance, Practical Money Skills Curriculum, FDIC, Phillips Exeter Academy, Park School of Baltimore, Deerfield Academy (Nils Ahbel), Park City Mathematics Institute, and North Carolina School of Science and Mathematics. This year, the class will be entirely project based.
  • I will continue to do self-evaluations, but I’m still working that through. I may allow students to create a video and use FlipGrid or Google Slides. Still working that one out. Would love to hear ideas.

This is my foundation and I will build from here. The bottom line, my mantra: problem-solving, metacognition, and relationships – to make my students better mathematicians and better people.

I don’t even know how to title this…

…that’s how much I am struggling with this post.

My classes are about problem solving, metacognition, and relationships – and we use math to help us become better in those areas.

I believe in the power of writing in the math class. I believe writing helps students gain a better understanding of the mathematics, their thought processes, their metacognition, and – even more importantly – building relationships.

I use writing all the time: journals, unit summaries (in AP Stats), special writing prompts, correction reflections, self-evaluations, etc. All of these – to be done well – require students to be willing to trust me, to be brave enough to reveal their thoughts to me and to themselves.

Between the inquiry-based nature of my classes, the group work, and the reflective writing, my classes are entirely about relationships and trust.

I’ve been asked how I get kids to buy in. Honestly… I’m not sure. It’s certainly not my “warm and fuzzy” personality or my gentle demeanor. The best answer I can give: I am honest with them, I model what I expect, I treat them with respect and dignity, I spend a ton of time giving them fast, honest feedback, I think they see how much I sincerely believe in the benefits of writing and metacognition. And they learn to trust me.

So, it was upon this foundation that school, as we knew it, ended on March 12, 2020.

As with everyone, the pandemic hit home. And hit hard. And, as hard as it was hitting my family, I knew it was hitting my students.

I teach at an all-male, Jesuit high school, that has been in the city of Cleveland for 134 years, with students from 9 counties, 100+ feeder schools, a range of races, ethnicities, identities, and socio-economic levels. I had students who had family members who were first-responders, medical personnel, executives, essential workers in grocery stores, many had parents who lost their jobs and some of my students had to go to work, some had loved ones who got sick, some had loved ones who died.

Our school ensured every kid had internet and a device – which was huge. I used Zoom and was available to my students online 3+ times/week. At the end of the year, they said the face-to-face interaction was critical.

But, I also wanted them to write. Jesuit education has a philosophy “cura personalis” – care for the entire person – and that was desperately needed.

So, each week I had my students do an online journal. This was a Google Doc that they added to each week and shared with me.

I knew I had to be sensitive to traumas my students were experiencing, but I couldn’t ignore what they were going through. The relationships and trust we had built allowed them to write about what they were experiencing in a safe and meaningful way.

The journal entries were due Monday mornings. It was beyond challenging, but I responded to everyone of my 120 students by Tuesday. And, for some kids, that became a life-line. Just knowing that they could share how they felt, they would be heard, and someone would respond without judgement meant the world to them in a scary time.

And, it was still about problem-solving, metacognition, relationships … and that was the point.

Tomorrow I’ll write about how I’m going to do things this year. Thanks for letting me share this.


be brave …

When I ask my students to write reflectively, I’m asking them to be brave enough to be honest with themselves, willing to take the risk of doing something new, and I ask them to trust me with their inner-selves. And, amazingly enough – they do.

This summer, I was asked if I would host a session on Global Math Department. My first instinct was to immediately say no. Imposter Syndrome took over – and not just in a little bit – it hit me big time, like, on steroids. I’m no PhD, I’m not published, I’ve never conducted research, I’m not a “name.”

But… I ask my students to be brave, to take a risk, and to trust…

So, I said yes.

On Tuesday, from 9 – 10pm EDT, I’ll host a session on: “Thinking About Your Thinking: Writing + Math.” https://www.bigmarker.com/GlobalMathDept/Thinking-About-Your-Thinking-Writing-Math

In the words of Jim Halpert, “At the end of the day, you gotta jump.” I’d love it if you joined me –

Student Writing and Cura Personalis

I have written and spoken about the tremendous benefits I believe journaling and writing in the classroom have for students. Last year, as I was reading my students’ Problem Journals, their periodic special writing prompts, and their quarterly Reflections & Self-Evaluations, I noticed something unexpected – it wasn’t just the students who were benefiting from their writing. I, too, was being helped.

This year, I decided to conduct an experiment of sorts.

Let me give you a little background: I teach at an all-male high school with 1500 students. This past year I taught 5 senior-level, year-long courses (~130 students). As you might imagine, or know, teenagers can get antsy/grumpy/disinterested/lazy/surly/obstinate/etc. at times throughout the year – especially seniors. And there are times when it seems like students project all of those feels onto the teacher. And it’s hard to separate their projection from what is actually happening in your relationship with your students.

In the past, I would find myself “filling in the gaps” or subconsciously jumping to conclusions about a student’s behavior. For example, when the extremely popular, super-star football player, who has just committed to a huge college football program, refuses to put a problem on the board, or puts the problem but doesn’t cooperate in discussing the problem, his thought process, any potential error, other approaches to the problem, etc. – it’s really easy, and human, to jump to conclusions about what’s going on.

My thoughts immediately were: “Oh, Mr. Superstar is too good for math class now.” (Big soul-revealing honesty here…) Then, it’s easy for me to take his actions, his hesitancy, his feelings, and project them onto myself: “He’s being disrespectful (to me).” “He’s being rude (to me).” And, depending on the day, sometimes worse thoughts.

Then it’s only natural for your feelings toward your student to shift, to change. You now see all of your student’s actions/comments/responses in the light of their “disrespect,” their “rudeness.” And that impacts my relationship with my student.

This is where the experiment comes in.

This year, when I noticed a situation like this occurring with one of my students – or even one of my classes – I gave a very deliberate special writing prompt as a homework assignment.

When I caught my opinion of Mr. Football changing, I asked my class to write responses to two prompts. The first: “In all the different aspects of our class, where do you feel you are weakest?” This was my football player’s response:

Screen Shot 2019-06-05 at 9.38.43 AM

I do not present my answers without the fear of being wrong.

That stopped me. That shook me. My big 6’5″ superstar football player wasn’t being disrespectful to me, he wasn’t being rude – deep down, he was just a kid who was afraid. On the football field, he could do no wrong – and he knew it. He was confident there, he had swagger. In my classroom, he was way out of his comfort zone and insecure.

His response to the second prompt:

Screen Shot 2019-06-05 at 9.38.59 AM

What did this writing do?

  • Helped me find the source or root of my student’s issue in the class
  • Helped me see that he wanted to do well (be confident) and allowed me to seek out opportunities and resources to help him.
  • Helped save my relationship with my student.

My students did the writing, and I know that helps them – but I was helped too.

I continued my experiment throughout the year with a variety of writing prompts:

  • Screen Shot 2019-06-05 at 10.15.54 AM
  • Screen Shot 2019-06-05 at 10.16.01 AM
  • Screen Shot 2019-06-05 at 10.16.08 AM

Not only did my students grow in self-awareness and metacognition, but my relationship with my students was able to grow through the insights I gained from their answers.

I teach at a Jesuit high school. Within our pedagogy, there is an idea called “Cura Personalis” – care for the whole person. Their writing in my classroom allows me a window into their thinking, their concerns, their successes – and helps me care for their whole person.

For me to be the teacher I want to be, it is not enough to just care about my students’ math skills; I need to have them write in my class to understand how to care for their whole person.

Student writing helps me provide cura personalis.

This year wasn’t the easiest with Mr. Football, but our student/teacher relationship was solid. Here is his final quarter Reflection & Self-Evaluation:

Screen Shot 2019-06-05 at 10.29.29 AM



AP Stats – blowing things up and picking up the pieces

I’ve taught AP Stats for 12 years. My students have always done well on the exam and write me notes from college letting me know how well they are doing in their stats classes and how well-prepared they felt. And, you would think I should feel great about that – but I didn’t.

About 5 years ago, I had 100% of my students pass the AP exam, and I was miserable at the end of the year. I hated the way I was teaching – lecture, drill, and kill. My students were doing great regurgitating the material – but I really didn’t get the sense that they were understanding at a deep level, nor did I feel like I had a way of measuring that potential knowledge (or lack thereof).

That summer (2014), I went to the Anya S. Greer Mathematics Conference at Phillips Exeter Academy. I attended the week-long session taught by Floyd Bullard (from North Carolina School of Science and Math) on Lab-Based Statistics – and it was an awakening. As we were working through some of the labs, I felt like I understood some of the concepts better and deeper – and if this is how I felt, how much more would my students?

So I brought his labs back to my classroom, I researched others, I used ideas from the AP Stats forum on the College Board page, I started following some stats folks on Twitter and incorporating some of their ideas, plus I got brave and I made my own labs. I took all of this and I made my own curriculum (I used the textbook, the labs, and a lecture hybrid). And things were … good. Better, for sure – but still just … good.

At the end of the 2017-2018 school year, I realized my stats students were annually deficient in 3 areas: 1) Reading and understanding the textbook/technical writing; 2) Keeping track of the large amount of vocab and definitions; 3) Reflection to develop and incorporate deeper understanding and foster metacognition.

That summer, at Twitter Math Camp, I attended the session taught by Bob Lochel (@bobloch) and Glenn Waddell (@gwaddellnvhs) on “Embrace Variability: ‘Big Ideas’ in Statistical Understanding.” Bob and Glenn encouraged us to follow a philosophy that I had seen others, like Luke Wilcox (@wilcoxl22) and Lindsey Gallas (@MrsGallasMath) and their Stats Medic (@TheStatsMedic) curriculum, use: Experience First, Formalize Later. I loved this idea – this was how I taught all my other classes, why not AP Stats too?

So, I blew the whole damn thing up.

My main goals were to keep the integrity of my students’ learning, focus on the idea of “Experience First, Formalize Later”, and address the 3 areas of deficiency. NBD, right?

Integrity of Student Learning: I teach with the belief that all my students are capable of learning and will never apologize for holding them to a high standard. I will also knock myself out to make myself available to help them. All assessments have both Multiple Choice and FRQ sections, and the FRQs come straight from prior AP exams. The FRQs are graded according to the AP rubrics and grading scale. This does lead to some pretty shockingly low scores for some of them initially. However, on every assessment, they are allowed to do corrections to earn back 1/2 the points they missed. Their initial score is rough, but the points back from the corrections ends up giving them a very fair grade. More about the corrections below.

Focus on the Idea of “Experience First, Formalize Later”: Using the Stats Medic curriculum as a backbone, I took their idea for templates and ran with it. I took all the labs I had used before, added a bunch, and created templates for everything. Now, we had labs for every topics, templates to refer back to, PLUS – each template had a reflective re-cap element (more about that below). Obviously, with a discipline like statistics, there has to be some explanation, some teacher guided discussion, some textbook reading – but that was now occurring after the experience of the lab.

Address the 3 Annual Areas of Deficiency: First, the issue of reading the textbook and understanding technical writing. I used to assign my students sections from the book to read, then problems from the back of each section, then collect those to make sure they had read the section. Borrrrrinngggg! And ineffective, and promotes cheating. Bleh. This year, I gave each student a composition notebook and when a section from the book was assigned, I asked them to answer the following prompt:Screen Shot 2019-06-03 at 11.01.03 AM

Before each assessment, I would ask the students to look back to their notebook and see if we needed to talk about #3, 4, and 6. This led to some great discussions. It also helped the students learn to read technical writing more carefully. When the students were taking their chapter tests, they would put their composition notebooks on their tables. As they worked on their tests, I would walk around and look at their notebooks. They received a grade each chapter for effort and completion. (I am open to any ideas on how to make this better; don’t hesitate to share!)

To help the students keep track of the large amount of vocabulary and definitions, I created a Google Sheet template. Each student used that to create their own guide that they shared with me. At the end of each chapter, I could just look at their sheet and give them a grade for effort and completion. When we finished with all materials and reviewing, I printed out their lists for each of them – their own, self-created review guide. Here is an example from one of my students:Screen Shot 2019-06-03 at 12.36.05 PM

Finally, how to add reflection to develop and incorporate deeper understanding and foster metacognition. I tried to be as deliberate and organic as I could with incorporating reflection. But reflection had to be/should be a part of everything we do in the class. As I mentioned previously, I added reflective elements to each template. Notice #3 and #6 in the Book Summary prompts – for sure appropriate and natural in that context, but also building a pattern of reflection that can be looked back at to determine patterns and foster/build metacognition.

With correction on assessment, students must meet with me to discuss what they missed. Then, they need to answer the following:Screen Shot 2019-06-03 at 12.44.37 PM

At the start of the year, I need to help students understand the “reflection” part. I do this when I meet with them to go over their errors. After I am sure the student understands the concept, I will ask the student – very deliberately – to explain to me what they did wrong, why they were wrong, if they could go back in time and change how they prepared for this assessment what would they do, etc. I sometimes need to push/prompt their thoughts as they are explaining their thoughts out loud. Then, I tell them – that’s a reflection! I encourage them to think about having a conversation with me as their reflection. That often helps students be able to go deeper and more personal.

Finally, at the end of each quarter, I have my students so a Reflection/Self-Evaluation.  Students need to answer the following questions:Screen Shot 2019-06-03 at 12.49.56 PMBased on a rubric (that I stole from Carmel Schettino (@SchettinoPBL) – the metacognition guru):Screen Shot 2019-06-03 at 1.04.12 PM

The students do this in a Google Doc and share it with me. Throughout the next quarter, I encourage them to go back and review the goals that they set for themselves and challenge them with the question: “Are you meeting your goals? If not, what are you going to do about that?”

Here is an example of a student’s reflection on the first question:Screen Shot 2019-06-03 at 1.09.32 PM

Now, I’m at the end of the 2018-2019 school year and reflecting back. Blowing up an AP class and starting over was an incredible amount of work, but so, so worth it. I think my students, not only mastered the curriculum, but did so with a depth of understanding and level of metacognition none of my classes have had previously. Plus – it was so much more fun! For them, and me.

Bottom line: I don’t have this all figured out. I am still a student myself. However, I trust my gut and my students to let me know when I’m on the right path – and I feel like I’ve finally found it.

Thoughts on Graded Homework

There has been a lot of discussion within my school about grades, the value of homework, how much of a student’s grade should come from homework, etc.

This year I taught two sections each of two senior level classes (PreCalculus and Honors PreCalculus). As I have done for the past 7+ years, I do not collect or grade homework. (I also taught AP Statistics, but I teach that class in a very different way – more on that later.)
My students do a lot of journaling and reflections on their learning, meta-cognition, self-assessment, etc. For their end-of-year self-assessment, I asked them to write words of advice to next year’s students.
Of my 43 PreCalculus students, 36 emphasized the need to do homework: PC advice on homework
Of my 50 Honors PreCalculus students, 44 discussed homework: HPC advice on homework
The most frequent comment was the realization that homework helped them learn, and that they were responsible for their own learning.

I’m also not saying this will work in every classroom. These were all senior-level classes, which, in theory, should have a different maturity level. Also, my class is very unique for a variety of reasons. But, I really believe de-emphasizing points for homework. I will also mention this completely takes away any motivation a student has to cheat on their homework.

Now, I’m not foolish enough to believe that all these students did their homework daily (or, for some, even regularly). However, they have been able to reflect back and gain insight into what could/did help them. Hopefully, they carry that insight and self-awareness with them next year.


As much as I feel prepared and organized in August, by the end of September (usually ~ week 6 of school), I feel like the wheels are coming off the train, the control for the brake has broken, and I am hurdling non-stop into the school year.

Don’t get me wrong, I love the ride! But, sometimes … whew! I can only imagine how the students’ feel. I have to keep reminding myself that writing and reflecting about their thinking, their learning, and their math problem solving skills is all new. As tired as I am from grading, they are tired from trying. We need to support each other.

I wanted to share some of the feedback I give to students as they are learning to journal and write reflectively. I’m including some samples from my “regular” PreCalculus class and Honors PreCalculus class. I tried to include a mix of students who need a lot of coaching and students who really get it.

(PS – I never write comments in red ink – just seems to “judge-y.”

Screen Shot 2018-09-24 at 9.43.40 PM

Screen Shot 2018-09-24 at 9.44.00 PM

Screen Shot 2018-09-24 at 9.44.14 PM

Screen Shot 2018-09-24 at 9.46.29 PM

Screen Shot 2018-09-24 at 9.45.18 PM

Screen Shot 2018-09-24 at 9.45.33 PM

Screen Shot 2018-09-24 at 9.45.51 PM

Screen Shot 2018-09-24 at 9.46.08 PM

Each week, in addition to all my other grading and prep, I’m offering feedback to ~100 journals. I’m not going to lie, it’s sometimes exhausting. But it’s gems like this that make me eager for next week:

Screen Shot 2018-09-24 at 10.25.18 PM

It’s More Than Math

[Thanks to Sam Shah and The Virtual Conference of Mathematical Flavors (https://samjshah.com/mathematical-flavors-convention-center/) for challenging me to reflect on the flavor of my teaching.]

Today was the first day of school. We are a faith-based school, and in order for the school to have a special Mass today, our classes were only 30 minutes long … that’s not a lot of time. At all. Especially when it’s the first time you are meeting your students.

So, I gave my PreCalc classes the 4-Square problem:

Screen Shot 2018-08-15 at 10.36.33 PM

Working together in their groups of 4/table, using only straight lines (but as many as they need), divide the unshaded area in Square A into 2 equal parts. There’s an obvious answer, and I praise the beauty of the simplicity.

Next, they are to divide Square B into 3 equal parts. Most kids see that they can divide the square into quarters. But, then we discuss other methods and how can we tell if they are correct.

Then, Square C has to be divided into 4 equal parts. This isn’t as obvious. Working together, students come up with answers – some are correct; some that are on the right track, but need a little assistance. So we work together as a class figure out how to make those ideas into solutions.

Finally, Square D is to be divided into 7 equal parts. The obvious solution is to divide the square with 6 equally spaced vertical or horizontal lines – but most students try to find a complex solution. And that led to a great discussion about trusting ourselves and appreciating the beauty of a problem – even the “simple” ones.

I concluded the class by telling the students that our class is collaborative, we become better thinkers by making mistakes and learning from them, and I can’t wait to work as a team to help each other learn and grow.

I wrapped up class, and honestly – I moved on. I was already thinking about tomorrow.

Until – out of the blue and unsolicited – I received this email from one of my students.

Screen Shot 2018-08-15 at 10.58.00 PM.png


Three things jump out:

  1. This kid may or may not have previously experienced the type of class he’s describing – but is desperate to experience it now.
  2. My students aren’t just “students” – they are individuals. And are worthy.
  3. I cannot underestimate the importance of relationship and trust within my class.

And, that the flavor I work so hard to create in my class – relationship and trust.

My main purpose of my class isn’t to teach math. My goal is to help students become better thinkers, better learners, better people – and math is one of the tools I use toward those goals. I know I’m not going to be able to reach every student, but I need to have an environment in place for those who are ready.

This is precisely why I loved having my students write – they share with me things they would never / could never tell me to my face. They gift me (and themselves) with a glimpse into their true thoughts and reflections.

At the end of last year, a student wrote this as part of his year-end reflection. I save it so I can read it after those long, hard days that we all inevitably will face during the year. This, this will remind me of the importance of the flavor of my class:

Screen Shot 2018-08-15 at 11.13.04 PM