Author: samjshah

Girls and Math

Act I:

In one of our department meetings near the start of the year, we started talking about the representation of girls in our math club and our math team. In years past, there was a higher representation than this year. And although I suspect that the distribution of boys/girls in our math classes are probably relatively even — based on my own anecdotal evidence — I will readily admit that in all years past, there were fewer girls than boys in math club and on our math team.

We as a department brainstormed different possible reasons. One teacher (it may have been me? maybe not though) said we could just ask students. But we agreed that this is something we should be cognizant of. And we all agreed that by reaching out individually, we as teachers might be able to make a difference. So we all committed to doing so.

And so I did. I emailed the girls in one of my classes.

Hi [Stus],

I wanted to send y’all a personal invitation. Among y’all, I see a large amount of mathematical curiosity and intellectual firepower. I can say with complete honesty that each of you have different qualities that are important in doing mathematics, and you all have impressed me thus far. Some of these qualities include dedication and the ability to work through initial frustration, the ability to math intellectual leaps/discoveries/connections, and the ability to express abstract conceptual ideas in written form.
I strongly believe that mathematics is not and should not be seen as a “boys club.” Among the students who I look back over my teaching career and think “wow, they were original and deep thinkers,” the majority of them are girls. Which is why I was surprised to see how few girls are involved in our math club and our math team this year. I want to encourage and nurture mathematical talent in girls in your generation so that future generations can see more women role models. And for me, that starts with me reaching out to you.
I wanted to give you a special and personal push/nudge in case you were interested in joining either math club or math team to talk with me so I can tell you more about these activities. (And if you’re not interested — which is totally fine, this email comes with no pressure– I’d also love to hear why you might not be interested in joining them. That would help me understand things better so I can think more broadly about things.)
Mr. Shah
It didn’t really work. The two replies I got were from students who were interested. But their schedules seemed to preclude their participation.
Act II:
I was alerted to an essay contest run by the Association for Women in Mathematics that I thought I could entice some students in my class to participate in. I threw this opportunity in Google Classroom, but … nothing. (As I write this post, it inspired me to re-post the opportunity since the deadline has not yet passed!)

Hi all,

The Association for Women in Mathematics (AWM) has an essay contest. You can interview a woman who is a mathematician or in a mathematical sciences career, and write a 500 to 1000 word essay based on that. And if you need help, AWM will even help you find someone to interview! More information is provided at the link below (along with some winning essays from previous years). From my reading of this website, this opportunity is open to everyone — not just women. If you are at all interested in hearing what a mathematician does (or what higher level math actually is!), or how gender plays a role in a mathematical career, this could be an amazing opportunity for you to find out.

Later in this year, you will be doing a set of “mini math explorations” based on your interests. They are very open-ended. If you do end up doing this essay, which would be so awesome, it would count as two of these mini math explorations!

Mr. Shah

Act III:
Over winter break, I saw the film Hidden Figures, about black women mathematicians who helped put a man into space. I had quibbles with this and that about the movie, but my final judgment: I want all my kids to see this.
And then this week, I had an idea. I posted this on our google classroom site:

In case you haven’t heard about the movie _Hidden Figures_, I wanted to make you aware of it. I saw it over winter break and wanted to recommend it to all y’all! Check the trailer out:

I was thinking about reading the book it’s based on [] sometime in the 2nd semester. If there are three or more of you that want to read it at the same time and have an informal book club around it, with me, I’m totally down. Just let me know who you are and we can make a plan!

Mr. Shah

And… I got a bite.Two actually. One student emailed me saying she was interested in joining the book club. And she told me that another student in our class was also interested! I asked ’em if they knew of any others — in our class or outside of our class — who might be interested. They got back to me with some names, and they agreed to reach out to them.

So it looks like we’re going to be having a book club around a book that talks about gender, race, and mathematics! I don’t know if it will be large or small. But I’m psyched that it will happen. This story was going around twitter today, and it made me emotional. Because I saw the relevance between this post which I have been working on, and this story.


Act IV:
One student in my class received a book called Math Girls as a present from another student in my class. Yes, my heart skipped a beat. Because how deliciously geeky was this gift exchange?! I have a project called Explore Math which allows kids to investigate something they are interested in that is related to math. The student who received the book wanted to meet with me to see if reading this book could be her exploration for the project. That particular book would have been too much to bite off for the mini-exploration, but I gave her a different but similar book (by the same author) to read for the exploration, and told her that when she had the time, I would help her work through Math Girls.
Also making the rounds twitter tonight was a keynote address from the president of the Mathematical Association of America. He gave this talk two days ago. The purpose of the talk: Why do mathematics? And as I read, and continued to read, I welled up with emotion. There’s a lot there to unpack. He has a call to action for college professors.

Find one student and be their advocate!  Be the one who says “I see you, and I think you have a future in math.”  Be the one who searches out opportunities for them.  Be the one who pushes them towards virtue.  Be the one who calls them up when they’ve skipped class, and asks “is everything okay? what are you going through?”

I know what I’m asking you to do is hard and takes time. 

But we’re mathematicians… we know how to tackle hard problems.  We have the perseverance to see it through.  We have the humility to admit when we mistakes, and learn from them.  We have hopefulness that our labor is never in vain and that our work will bear fruit in the flourishing of our students.

Because what I am asking you to do is something you already know, at the heart of the teacher-student relationship, pushes us towards virtue.  I’m asking you to love.

But this call to action applies to us math teachers too, not just college professors. Except we don’t get to find one student. We are given many students. And being all of their advocates is harder, and takes time. And is Herculean. Perhaps Sisyphean. But people like Fawn and Rebecka and Annie and Sara remind us that gaining a deep student-teacher relationship with our kids–  having our connection go beyond math and to a position of mentor and trusted ally — is possible. Someone our kids can look up to, as a human being, not just as a font of knowledge. I have a suspicion that figuring this out separates us mere mortals from the master teachers we look up to.

Sequences and Series

A few days ago I posted a card sort I did to start my unit on Sequences. I figured I’d share my entire packet for Sequences (2016-10-31-sequences [doc] 2016-10-31-sequences [pdf]) in case it’s any help for you. This was designed for our standard precalculus classes, and I have to say it worked pretty well.

  1. It allows for kids playing with math at the start (card sort, some visual patterns, a 3-act)
  2. It doesn’t “tell” kids anything. They discover everything. And sometimes asks for a couple different ways to do things.
  3. Kids messing up notation is always an issue with sequences. So I introduced notation way after kids started playing around with sequences — and got a handle on them. I did still see some kids confuse n and a_n (the term number and the value of the term), but it wasn’t a huge number. I actually was pretty proud of that.

Note: Problem #17 on page 23 is actually ill-posed. So I didn’t have my kids work on it. I have a replacement warm-up question (included at the bottom of this post) which worked wonders!

Also in the packet, there is a blank page (on page 4) which simply says “A Pixel Puzzle.” For that, I used Dan Meyer’s 3-Act.


For me, the 3-Act was a mixed bag. Mainly because I thought it would be easier than it ended up being for my kids, so I didn’t plan much about how to help groups get started. At the beginning of the 3-Act, kids asked great questions, but not the one I was hoping for (when does the pixel hit the border). So I didn’t have a smooth way to deal with that. And I anticipated it would take less time than it actually did, so the 3-Act got split up between multiple classes. And when kids were working, I don’t think I was strong at facilitating them working. I should have added in a wee bit more structure to help kids out. Even if it’s something as simple as coming up with ways to get kids to think about making a table or a graph — and what would be useful/important to include in the table/graph. And I didn’t have a good way for kids to share their findings. So I’d need to think about the close to the 3-Act better. I’d give myself a “C.” Would I do it again? Yes. I saw the potential in it. It got kids thinking about sequences visually. It had kids thinking temporally. And had them relate tables/graphs/equations. All good things!

One last thing I want to include in this post… This is related to the ill-posed Problem #17 on page 23 I made note of before. There is one type of question which tends to flummox kids when it comes to geometric sequences. It reads something like “The third term in a geometric sequence is 6 and the seventh term is 96. Come up with a formula for the nth term in this sequence.”

The reason this is tricky is because there are two possible sequences! (The common ratio could be 2 or -2.)

Thus, I created this warm-up (.docx) for my kids to check if they truly understood what we were doing. The conversations were incredible. Some groups were done in 7 minutes… Others had a solid 15 minutes of discussion.




2, 4, 6, 8, what do we appreciate? A Card Sort!

This year I’m teaching both Advanced Precalculus and Standard Precalculus. (Totally confusing on a daily basis? Yup.) And I’m working with two other teachers to write the Standard Precalculus curriculum from scratch. Of course this is something that is daunting, but I love to do when I have the time and like-minded colleagues.

I was in charge of spearheading our sequences and series unit. In this post, I want to briefly share how we started the unit. Instead of diving right in, or doing something intense, I wanted to gently get some good conversations percolating. So I handed out this set of cards:


and gave them this set of instructions:


I debated having kids use Desmos for the card sort, but since I have kids work in groups (mostly groups of 3), and I wanted the entire group to be working together, and I wanted them to actually physically move and shuffle cards, I decided to use physical cards. I also had all kids stand up while doing the card sort. I had a feeling that would be magical, in terms of getting kids talking, moving, and engaging with each other (even thought they were all at the same table), and it was! So I highly recommend that.

These cards end up having three different types: arithmetic, geometric, and recursive.

Most kids got the arithmetic sequences quickly, but it was interesting to watch them struggle with the geometric and recursive. There were great conversations, and because I demanded the next number for each of the sequences, kids had to really think through what the pattern was (and in geometric sequences, how to find the common ratio). I had thought that kids would finish this really quickly, but I was totally wrong. It took about 20 minutes. So plan accordingly. (A few groups needed to do a little bit more at the end of class, so I had them take a photo of their card sort and use that photo to finish things up!)

One note: Card H which has the sequence 0,0,0,0,0,0 fits all three categories. So it’s great fun to watch kids try to place it.

I wanted to share this activity because I haven’t really done many card sorts before —  and I was so pleased that this particular one generated productive conversations. So I need to keep this teaching tool in my arsenal for generating conversations about something new. (Example: I just thought of giving a bunch of graphs of rational functions to kids on cards, before we start that unit, and say “find different ways to sort these!” There are so many features, so that could lead to so many different ways to sort the cards. I suspect that Desmos would be good for that particular card sort, since there would be many different ways to sort those pictures, and I’d want to project the different ways kids did it to the entire class. I bet through that sort, we could actually recognize vertical asymptotes, horizontal asymptotes, oblique asymptotes, and holes!)

Here is the .docx (2016-10-31-card-sort-for-sequences) and .pdf (2016-10-31-card-sort-for-sequences) for the cards.

I will try to write up some more about my sequences and series unit soon!

New Year, New Blog

Note: Julie Reulbach wrote this post on the ExploreMTBoS site! I’m copying it here for two reasons. One is that I want everyone to participate! Two, I have been in a crazy place this year, and I’ve thought “oh I should blog this” a bunch and never took the time to follow through. I am going to participate too! Join me!

Additional Note: Carl Oliver has also set up a blogging resolution challenge for 2017! And his post is inspirational.

Welcome to the Explore the MTBoS 2017 Blogging Initiative!

With the start of a new year, there is no better time to start a new blog!  For those of you who have blogs, it is also the perfect time to get inspired to write again!

Please join us to participate in this years blogging initiative!  To join, all you need to do is write just one post a week for the next four weeks.  To make it easier for you, we will post a new prompt every Sunday!  Once you have blogged, please fill out the form below.  Each week, your blogs will be posted on this site for all to enjoy!

This Week’s Theme:  My Favorites

This week, the blogging theme will be “My Favorites”, where you can post about one (or many) of your favorite things!  Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well.  It does not have to be a lesson, but can be anything in teaching that you love!  It can also be something that you have blogged or tweeted about before.  Some ideas of favorites that have been shared are:

  • A lesson (or part of one) that went great
  • A game your students love to play
  • A fun and/or effective way to practice facts
  • A website or app you love to use in class
  • An organizational trick or tip that has been life changing
  • A product that you use in your classroom that you can’t live without!

Blog Newbies!

If you are brand new to blogging, you can read Starting A Blog from the 2015 initiative.  This post will give you specific instructions on how to start a blog.

Hot Tip!  Don’t stress about your blog name!

The hardest part about blogging is often coming up with a title.  Do not let this detail derail you!  A great suggestion is to make your blog address your name.  Then, you can title your blog later – or change the title anytime you want!  To see what this looks like, check out Sam Shah’s blog.  His web address is, but the site name is “Continuous Everywhere But Differentiable Nowhere“.  No one cares about your blog name, they just want to read interesting, inspiring, and helpful posts!

Hashtag it!  #MTBoS #MtbosBlogsplosion 

Don’t forget to tweet out your blog link and add hashtags so other teachers in the MTBoS community can easily find your post!  If you are not tweeting yet, you should be!  There is an amazing community of math educators there just waiting to inspire and support you!  Check out How To Start a Twitter Account to get started!  Also, if you are brand new to Twitter or just want to get more out of it, there are more Twitter tips on Julie Reulbach’s blogpost, Tweet, Connect, Repeat.

This year, we are joining up with the #mtbosblogsplosion.  Special thanks to  Carl Oliver@carloliwitter, for jump starting blogging for many people in our community!

Hashtags to add to your tweets:  #MTBoS #MtbosBlogsplosion

Also, if you have a wordpress blog, please re-blog this post to get the word out!

Deadline: Press submit by the end of the day Saturday, January 7, 2017.

Yes, this is a quick turn around this week – but we don’t want you to put it off or delay!  Once you are finished with your blog post, fill out this form and your blog post will be featured on this site [meaning the MTBoS site this is reblogged from] next week!


Twitter Math Camp 2017

We are starting to gear up for TMC17, which will be at Holy Innocents’ Episcopal School  in Atlanta, GA (map is here) from July 27-30, 2017. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc ( It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 27 and 48 one hour sessions that will be either Thursday, July 27, Friday, July 28, or Saturday, July 29). That means we are looking for somewhere around 70 sessions for TMC17.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 16, 2017 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC17 – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Daniel Forrester, Megan Hayes-Golding, Cortni Muir, Jami Packer, Sam Shah, and Glenn Waddell

Teaching is hard work. Election aftermath.

Yesterday, I told one of my precalculus classes how it was an exciting day. I was setting them up because it was election day, and kids at my school are heavily interested in politics, so I thought they’d say “yes! Election!” And I would say: “Actually, it’s because one of my best friends from college is having a baby.”

Of course that setup didn’t work, because of course a kid asked “why is today exciting?” Thanks, kid. But I told the class about my friend’s baby.

Yesterday evening, as the election results came in, I got more and more anxious. And when it was clear that Trump won, I was destroyed. I am not going to use this blogpost to explain my love for Clinton, or why Trump makes my blood boil. Instead, I want to just share how my day has gone.

I teach at an independent school in Brooklyn, and the population of kids and parents we serve are (for the most part) liberal. The kids are politically active and aware and interested. Today, I came to school and kids were destroyed.

In my first class, I talked to kids a bit, and then asked them what they wanted to do. After hearing them, I came up with the following plan. The kids who woke up to the news and wanted to learn more and get informed could read articles online. (There were about 4 of those kids.) I just asked that before they started reading, they take 3 minutes to type out all the questions that they have — so help them start processing. (Like “How could this happen? What was wrong with the polling? Who was voting for Trump? What does this mean for issue X?”). For the others, we formed a circle with the desks and I let kids talk. At points, kids cried. I didn’t join in — I wanted this to be a space for them. They expressed real sadness, hopelessness, optimism, anger, frustration, embarrassment, terror, empathy. I really heard my kids, and when talking about this election, they were speaking their truth, about their hopes and dreams (and how those hopes and dreams were altering). It destroyed me inside to hear them. To see how much this election has affected them. I guess I hated the fact that my kids are feeling what I’m feeling. I don’t want that for them.

I went to my second class, that precalculus class that I told about my friend’s baby. The first thing a kid said to me was inquiring about my friend’s baby. That small gesture — that this student would remember that — lifted my spirits. In this class, more wanted to read the news, and a handful of us talked. This discussion tended to a bit more political punditry — about the what’s and the how’s and less about their emotional state. I suspect they got many of their feelings out in their previous classes.

In my third class, we watched Hillary’s concession speech.I teared up twice during the speech. One kid left to gather themselves for a few minutes after the speech. I didn’t know what to do after. Kids said they didn’t feel like discussing things anymore — they were discussed out — but they also didn’t see how they could focus on work. I made the executive decision to spend the last 20 minutes of class having my kids watch the pilot of the West Wing. I hoped that some optimism in politics might help.

I have one more class to go. It’s a 90-minute block. I’m drained, right now. I don’t have much more in me. I suspect kids are also drained, but I don’t know. I’ll suss out how things are, and try to get through it.

I’m exhausted. Yesterday I woke up at 5:30am to vote. Yesterday I didn’t get to bed until very late (maybe 1pm), and then woke up at 3am to watch Trump’s victory speech. I then read articles until I forced myself to sleep from 4-6am.

Teaching is hard work. Yes, there are lesson plans and grading and meetings and a zillion other things. But days like today, days like today keep me in check. And reminds me how hard the hard work can really be. Because the hard work is being an emotional support. To let kids cry. To let kids know you cry. And to get through the hard times together.

Update: My last class came in with bags under their eyes. I was also tired. I asked them what they wanted to do. A few wanted to continue talking, a couple wanted to do some math and do some talking about the election (a mix), and one just wanted to do math. I decided we would go over the nightly work first, and then talk about the election.

When going over the nightly work, kids were actually focusing better than expected. They asked questions. They were able to answer questions. It was going well. I then ended up going on a fascinating tangent about fractals (related to one of the questions we talked about). And when I realized kids had never heard of fractals before, I showed them a youtube fractal video. Then they wanted to know how it was made. So I gave a short 10-minute lecture on the complex plane, and how the Mandlebrot set is formed. Kids were entranced by the video. I gave a 5-minute break before we sat down to talk about the election. (During the break, kids were in the hall watching more of the fractal video on one of their phones!) When we returned, everyone was silent. No one spoke. I just let it hang there. Eventually one voice. Then another. It wasn’t a rowdy discussion. Not everyone was in it. But most kids had something to say. And then when the day was close to ending, and there was a natural lull, I used a comment about “voting systems” to show a video about alternative voting systems. And then I let kids go home.

I just made the first four slides for class tomorrow. They’re not fancy. I’m tired. But I think they encapsulate what I’ve taken away from today.



Now I must end. I now have to change all my lesson plans for the upcoming days, prepare for parent visiting day tomorrow, and write narrative comments. This feels impossible. But I needed to process today.


UPDATE: A student gave me a paper flower she made today, to thank me for facilitating a conversation about the election in our class on Wednesday. And that flower is going to stay on my desk all year to remind me of the other things we do as teachers that can be meaningful.


Waiters, Waiters, everywhere…

Today I was nerdsniped in the math office. My department head and two colleagues were working on this problem. I don’t know where it came from. But golly, did I enjoy it!

Imagine you have a row of waiters all facing forward. Each waiter has a beautiful silver platter that they are carrying. They have to choose: will they hold it directly in front of them, or on their left side, or on their right side? Here’s a diagram showing the three options (I imagine I’m looking down on the waiter.)


Okay, so there is one constraint. Remember the waiters are all standing in a row. So you can’t have the platters crash into each other. So here’s an example of an OK way the waiters could hold their platters, and then a NOT OK way the waiters could hold their platters.


So here’s the question… If you have n waiters standing in a row, how many different ways could they hold their platters?

I am not going to post the answer here, because I like to nerdsnipe! But if you want to check your answer, for 20 waiters, I calculate 267,914,296 different positions!

I bet you will have a lot of fun with this problem. One person in our office came up with many pages of work, and had a very complex approach which yielded some deep insights. She was super psyched about the intricate superstructure she was building. Another person got to review solving a particular type of thingie using matrices (I want to keep things vague so I’m going to use the word thingie to avoid giving anything away). I and another person had the same approach that led to a quick and elegant solution, but left me with rich conceptual questions to pursue. And as I started doing that, I realized that I had accidentally stumbled on the complex approach that the first person had taken.