July 5, 2019

Dreaming and Deliberating

In May I announced I was starting a job search. Since then I've had lots of time to think more deeply about what I want to do. The big things are still true:

  • My values are community and making a difference.
  • I'm only looking at jobs I can do remotely or commute to from Salem, MA.
  • I'm trying to cast a wide net as I consider options and opportunities.

The other things I've refined:

I believe in the power of a public education. Public schools are essential to the kind of society I want to live in. Our current public school system is broken. I spent over a decade working within the system, and I just can't imagine going back to that right now. I miss having students, I have so many ideas I want to test out, and I really want to see what the curriculum I spent the last year and half writing feels like from the teacher perspective. However, I don't want to be part of a testing culture that prioritizes compliance or graduation rates over students as human beings. And I just don't have any faith left that a local public school is doing that. Alternative public or semi-public schools (thinking ones for students with disabilities) fit within my realm of acceptable options, charter and private schools are off my list. I need to learn a bit more about how necessary a PhD is for the variety of college teaching options that exist.

I would really love to just play math with kids and adults all the time (see my last 3 posts and a growing folder of ideas and resources). I have made some progress in figuring out how to take this from a hobby to a job (see that same folder) but it feels like a giant leap and I'd rather approach it as an incremental transition from individual beta testers, to a library play group, to a thing I try to find funding for. But it sure is distracting to see this wide open public space next to an empty retail shop downtown. I would rather spend time playing than job searching any day.

So to pay the bills I'm applying to all sorts of curriculum jobs. I'm good at that. I enjoy doing it. But again I'm very picky about where I'll work. I'm not going to help someone make their computer program intended for rows of kids staring at screens mathematically correct, but I will gladly help someone make their math content more pedagogically sound.

If I don't find a full time job soon I'll probably write a third edition of Nix the Tricks (anyone want to pay me to do that?) this summer. I do really like writing...

Math Play Mat - Elementary Edition

Yesterday I went to a barbecue at my friend's house. A friend who has kids ages 7 and 9. So I brought along some math play mats and pattern blocks. The 9 year old had a friend over and they spent most of the day inside - he asked what the pattern blocks were when he came out to eat but I decided to prioritize food over math play at that moment, and then he disappeared before I could invite him to play after dinner. The 7 year old and I had a great time playing though.

As soon as I dumped out the pattern blocks she started building things. At one point she told me to make a big shape out of small shapes. So I made a square composed of 4 squares. Then she said I needed to use more than 8 shapes, so of course I made something using exactly 8. She counted the pieces in my design and said, "No, MORE than 8!" In addition to testing her definition of "more than 8" I also tested her definition of "shape." I wondered if she'd restrict me to mathematical shapes she knew the name for, but a house was fine for her definition of shape. It did, however, need to have a triangular roof. We live in a place that gets lots of snow, so this is definitely the smart choice.

We also played guess my pattern. The first pattern I made was yellow/red and her first guess was about color. Great minds! So then I made hexagons composed of different shapes that alternated with squares. She guessed rectangle/square at first (pointing at the hexagons as she said rectangle) but I told her my rule was big/little. Kudos to her mom for not jumping in to say, "That's not a rectangle." during that conversation. Later when she described one of her patterns as rhombus/square, I told her I was impressed she knew the word rhombus and then said, "this pattern could be rhombus/rhombus" which got me a funny look. But then she was accepting of my explanation that a square is also a rhombus since both of them have four equal sides. She even grabbed the tan rhombus to include it in the pattern. She was all about precision - once I guessed a pattern was big/small and she said no, big/little. Here's another example of precision - take a guess what the pattern is before scrolling past the photo.

My first guess was flat/upright, she said it wasn't upright but sideways. I wasn't entirely sure what that meant and it took a few tries to get there.

When she told me that my first guess was wrong her argument was about it not looking sideways from where she was sitting. Fascinating. The second photo was an acceptable match to her pattern (though it wasn't my second guess, I forget what my intermediate steps were). To note: we didn't talk about "red/yellow" patterns but described the pattern as red, yellow, red, yellow, red... while pointing to each shape. I wonder if identifying the unit of red/yellow would have resulted in more patterns with a consistent repeat. She made patterns that didn’t always repeat the same way (big, little, big, little, little). She also made a set of shapes that were increasing in size. Without any prompting she proved to me the square was bigger than the triangle by placing the triangle over the square and pointing to the parts of the square that aren't covered up. While I was appreciating that proof she moved on to the next thing so I didn't get to ask about the square vs. the rhombus.

We played a wide variety of games during the afternoon (including a washer toss where she helped us keep score!) but she came back to the shapes several times, including eagerly nodding when I asked if I should take them out again after we'd cleared the table for dinner. It was fun to play with an older kid and compare to the toddlers. While we certainly didn't need a mat to make up patterns, it was nice to have - putting shapes on the mat was a signal that we were making a pattern so the other person was ready to pay attention.

FYI: The pattern blocks are a foam version from Hand2Mind (yay for conference giveaways!) I had a set of multicolor squares in addition to the standard pattern blocks which made it easier to test color vs. shape patterns. The felt shapes I usually use with the math play mats solve this problem too, but they don't invite building composite shapes in the same way that pattern blocks do.

June 15, 2019

Following a Pattern

After making a few more math play mats I returned to the other pattern I was working on. (By the way, surface crochet works great! So I set up an info/order page here.) The project requires making lots of squares and then eventually joining them together. Since I was going to be repeating the same pattern so many times (42!) I decided to write it on paper so I didn't have to pull the pattern up on a device every time I took out my yarn, and also so I could make it clearer.

These are the squares I've crocheted so far. Lots more to go!
The pattern is freely available from a well known site. Lots of people have used it. It's completely accurate, but it reads like this:

Rnd 6: Slip st in next dc, ch 4 (counts as dc, ch 1), * (dc, ch 3, dc) all in next ch-3 space, [ch 1, skip next dc, dc in next dc, ch 1, dc in next ch-1 space] twice **, [ch 1, dc in next ch-1 space, ch 1, skip next dc, dc in next dc] twice, ch 1; repeat from * around, end at **; ch 1, dc in next ch-1 space, ch 1, skip next dc, dc in next dc, ch 1, dc in next ch-1 space, ch 1; join in 3rd ch of ch-4.

Even if you don't know how to interpret ch as chain, dc as double crochet, and st as stitch (or what any of those look like as a series of loops on a hook), you can still see that there are parentheses, brackets, and asterisks to navigate what gets grouped and repeated. Do you know what that entire paragraph is saying? All the way around the entire square I'm supposed to alternate a tall stitch (dc) and a single loop (ch) that will create a space between the tall stitches. That's it! So I rewrote it as:

Rnd 6: Slip st in next dc, ch 4 (counts as dc, ch 1),
In each corner: (dc, ch 3, dc)
Around the edges: ch 1, skip next st, dc in next st or space

The original instructions aren't wrong. They tell me exactly what to do every step of the way. I don't mind brackets or asterisks in patterns in general, it's much easier to be told to repeat something than it is to keep track of where I am on a long list of stitches. But my version is even easier to keep track of. Because I understand what I'm doing, I have easy to reference landmarks, and I just don't need that level of detail.

Why is the original so complicated? The previous round was worked in groups of 3. So when you're working groups of 2 into groups of 3 sometimes you land on a stitch in the middle of the group and sometimes you land on a space between the groups. Sometimes that matters, but here it really doesn't, I do the same steps to make my stitch either way and it just makes it easy to lose track of where I am when they specify the difference. Also, you don't start on the corner so you have to say "do this little bit of the side, then a corner, then a whole side, now repeat from the corner, but stop before you finish the last side because you already did a little bit of it at the beginning." My rewrite assumes you're looking at your square as a whole piece and can identify corners as well as deduce where to stop so you end up going around exactly once.

Why am I talking about crochet patterns on a math education blog? I'm sure you've already drawn parallels. It's tempting to give kids precise instructions that help them navigate every little difference they might encounter, that way they're sure to get it exactly right! But at what cost? Providing every little step means we lose track of the big picture. It's so easy to skip a step when we don't understand what it is we're trying to accomplish. And if this problem were a little different (in my case if I wanted to change the design to be a larger square or a rectangle) the hyper detailed instructions are useless and we have to start from scratch. When we're asking kids to follow a pattern, let's be sure to point out the landmarks as we go. Because that's what will help everyone navigate when they're on their own - which is absolutely my goal.

June 2, 2019

Math Play Mats

In March I'd bought some yarn and was looking up crochet patterns. I came across fiddle mats for people with dementia. I wondered whether my friends with toddlers would like such things, and then I wondered if I could make it a mathematical play thing. Turned out I could! A math play mat includes a ten frame, a rekenrek (or is it a pair of rekenreks?) and a pocket full of felt shapes. My beta testers have found lots of great ways to play, and made one request for a change to version 2.

I'm so glad I made these because I got to play math with littles. I learned that I am sorely lacking giggles in my life. I played on the floor of a conference center, at a cafe, and in a living room. So I can attest to the fact that these are great portable toys. Soraya, Jenna, and I played for almost an hour and only stopped because we wanted to go to the park before it rained. She played more, and in different ways the very next day. I love it when the best case scenario you imagine is surpassed by reality!

The things they've played:
  • Match my shape. 
  • Sort all the shapes.
  • Name my shape. (Some are standard shapes but others are flowers or snowflakes. Which leads to - what's the difference between a flower and a snowflake?)
  • Can you find a ____? (Green shape? Square? Green square?)
  • Put things in the pocket and take them out.
  • Move the beads so the top row matches the bottom.
  • Count the beads.
  • Count the shapes.
  • How many ____? (Count, cover, how many are hidden? How many more to fill the squares?)
  • Comparisons (are there more purple or green?)
  • Decorate mom with the shapes.
  • Put the shapes in the ten frame.
  • Make a pattern in the ten frame.
At one point Soraya made a joke that turned into a pattern. She announced she was going to make groups of four. So she put one shape down in the ten frame saying “one.” Then she put another shape down and said “four!” And giggled. This small human who's still working out how the numbers go after 10 knew full well that saying four after one was unexpected and would make us laugh. So of course Jenna and I laughed! And Soraya continued saying one then four until she was laughing so hard she couldn’t talk. It was adorable, I'm grinning just remembering it. And it gets better! When we recovered from laughing, Jenna and I got instructions to make new number patterns. I was supposed to say one, then say fifteen, as I put shapes in the squares. Jenna and I started strategically picking shapes to go with the numbers we were supposed to say (like green for one and yellow for fifteen). Soraya picked up on what we were doing and made lots of rules.

The one complaint: 
Jenn's kiddo, Z, kept asking her to cut off the beads so she could make more patterns. While I won't remove them entirely, because I'm confident eventually Z will find cool things to do with them, I do want to make them less floppy so they won't get in the way. Added bonus: they're currently made by threading beads onto braided yarn which is omg so tedious. My upgrade is to use pipe cleaners which will make my life so much easier! Plus it's easy for adults to take the pipe cleaners off the mat if anyone else demands it and then put them back later.

Can you see the concentration? These people are doing math. (Z, Soraya, and Shelby's kiddo)

I would love to see what happens with these toys at different ages. The difference between an almost 3 year old and an almost 3.5 year old was noticeable (yes, I know, tiny sample size). While I'm no expert in the trajectory of learning to count, I have done enough reading about it to know that the answers to all of these questions are different:
  • Can they count the shapes in a pile? 
  • Can they count the shapes organized in a ten frame? 
  • Can they count the beads by pointing to each one? 
  • Can they count the beads by sliding each one?

Seeing where kids are and how they grow is so fun. Another place for growth was one kiddo could find me a square, and could find me a green shape, but finding a green square was too much. I hadn't thought about how much harder searching for two characteristics simultaneously was until I was gifted a green shape that wasn't a square!

Fave tweet:

So what's next? I'm definitely going to try the pipe cleaner method for the rekenreks, and I'm playing with different methods to get the ten frame on there because sewing isn't my favorite and I'd like that part to look cleaner. If I can come up with a quick way to outline the ten frame and attach the rekenreks then I'd love to make a whole bunch of these and bring them to something like toddler play time at the library. I could also send them to you, dear reader, if you have a kiddo in your life, a classroom full of them, or you're a kid at heart. Because lets be honest sensory play with felt shapes is fun for everyone! I do recommend playing together though, as the potential for giggles is much higher. And we could all use more giggles in our life.

Update: my new version worked out so there's now a website for these things: tinacardone.com/math-play-mat

May 18, 2019


Last February I left teaching and started working for Illustrative Mathematics. It's been an amazing opportunity to think deeply, collaborate with fantastic people, learn new things, and build a product I'm immensely proud of. More units have been released since our initial preview, go check them out! https://curriculum.illustrativemathematics.org/HS/teachers/index.html

One thing I got to learn at IM was how to code images using tikz. Who knew all those "graphing a picture" projects in Desmos would turn out to be so useful? Not only can I generate a stick figure, my code is so fancy it's easy to move the arms around to recreate the entire semaphore alphabet. So cool right?? (This is where you nod and say, "Sure Tina..." while waiting for the point of this post.)

Another thing I learned about while working at a non-profit was the grant process. And this post is a riff off of the RFP, Request for Proposals. Since the HS curriculum will be finalized in June I'm looking for a new job. I have some ideas, but I know that the collective mind of this community will help me dream up things I can't even imagine. The more I learn the more I realize that this world is filled not just with jobs, but entire careers, that I never even thought of. So I'm asking for your help.

As I've mentioned before, during my reading of Dare to Lead I identified my values as community and making a difference. So I'm holding those two ideals close as I consider possible paths. Community means a variety of things, but it includes my family, so I'm restricting myself to jobs I can do remotely or locally. Other than that I'm trying to cast a wide net as I consider options and opportunities.

Current thoughts:
- Follow our curriculum to its next stage by working with one of IM's partners to get our materials professionalized according to their style and added to a new platform.
- Coach or teach at a local high school.
- Open a math and science museum downtown.
- Do a variety of consulting work and write a third edition of Nix the Tricks to help market that.
- Find something short term and apply for a PhD program (I even came up with a topic last night - studying math instruction for students with disabilities, specifically how they so often get stuck doing rote work with tricks or a list of steps).

I'd also love to work for Elizabeth Warren's campaign, but based on the fact that I get daily emails from her asking for money, I'm feeling doubtful that she's looking to pay people with no campaigning experience. However, activism work both within and outside of math education is appealing. That's definitely an area where I don't even know what jobs exist, let alone how to find them.

So, let me know what job openings you hear about, what avenues you think I should research, and your favorite parts of the new IM HS curriculum!