Talking Math with My Kids

I’ve gotten so many great ideas from Twitter that I wouldn’t know where to begin describing them all. One of my newest favorite ideas comes from Christopher Danielson and his “Talking Math with Your Kids” project. As he points out,

Parents know that we need to read 20 minutes a day with our kids.

In the same vein, it seems clear that we should make exposing our kids to mathematics a daily goal. At our house, our kids have always been around a lot of conversations about mathematics, but until recently I hadn’t been making a conscious effort to engage with them mathematically. (I have a 3-year-old son and a 1-year-old daughter.) It’s been fun to see where the 3-year-old is in his mathematical development. Here are some things we’ve talked about recently:

  • While buying school supplies: My son’s class required three boxes of tissues and my daughter’s class required two boxes of tissues. I explained this to him. On one hand we held up three fingers and on the other hand we held up two fingers. I asked him, “How many boxes of tissues do we need to buy?” His initial response was, “Three-two!” Then I asked him to count my fingers: “One, two, three, four, FIVE! We need FIVE boxes!” We went on to talk about that three plus two equals five (3+2=5), and then he let me count his fingers and I counted that two plus three equals five (2+3=5) as well.
  • Before watching TV: After picking both kids up from school, we have snack time and the 3-year-old can watch a few minutes of a Mom-approved TV show. (Usually it’s some PBS cartoon; for a long time, his favorite has been Dinosaur Train.) When I asked him how many minutes of Dinosaur Train do you want to watch today? he thought for a long while. I could see he was really trying to think of a very large number. He then excitedly yelled, “TEN!” We clapped and agreed he could watch ten minutes of TV during snack time.
  • On the way to school today: He asked if I was going to go to work today and I told him yes. Then I asked him if he knew I was a teacher, too, just like his teacher at school? After some conversations about whether or not I took a school-bus to my school (I don’t), he asked where my school was and if it was very far away. I told him it was twenty minutes away. Although he can count to twenty, I don’t think he has a sense of what twenty looks like, or how big it really is. He then asked lots of questions about my 20 minute distance:”Is it more than six minutes?” Yes.
    “Is it more than seven minutes?” Yes.
    … “Is it more than eleven minutes???” Yes.

    Then we were at his school and I told him, “It’s even more than nineteen minutes.” He said, “Oooh. So it IS more than six minutes.”

  • Practicing Counting: He’s been learning whole numbers larger than twenty at school recently. We were practicing counting together, and he said: “…twenty-seven, twenty-eight, twenty-nine, twenty-TEN!” We laughed and told him that after twenty-nine comes thirty, and his face let us know this did not make sense and he was not happy. If it goes eight, nine, ten, why does it not go twenty-eight, twenty-nine, twenty-ten? This seems like a really valid concern.

    I used to know a lot more French than I know now. Our conversation made me wonder what he will think in a few years when I can explain to him about soixante-dix (they use sixty-ten for “70”) and even quatre-vingt-dix-sept (four-twenty-ten-seven for “97”).

Escaping the Lectureculture

For years now I’ve been a reader of Robert Talbert‘s column Casting Out Nines hosted by The Chronicle of Higher Education. Last week he wrote a post (“Is lecture really the thing that needs fixing?“) that gave me a lot to chew on. Here’s where I find myself today:

  1. Lectureculture is a set of machinery that self-replicates and it has political, social, psychological, instructional, and institutional components. It is pervasive and I find it in the world all around me, and some of the cultural natives don’t even recognize its existence.
  2. When I run a course, my #1 goal is to help learners move from being introduced to a concept to understanding and displaying mastery of the concept. Lecture is not the most effective way to help learners*.
  3. If I do nothing but lecture in my classes, I am helping sustain lectureculture and I am not helping my learners toward mastery the best I can, in violation of my #1 goal.

My plan of action: I’m teaching “Calculus II” again this semester. Although I’m using a standards-based approach, I must fess up that last semester nearly all of our class time was devoted to either lecture or assessment.

I am a lectureculture native and it is hard for me to let go. But I have come up with two ways I want to add non-lecture content delivery this semester (that don’t involve me tossing out all of my old materials).

First, I plan to continue last semester’s “Madness Mondays.” On those days, I introduced my students to ideas not necessarily tied to our course. I wanted to pick topics that I thought would inspire curiosity or happy befuddlement in my students, so they would walk away wanting to know more about what they had heard. (Examples: The Cantor set. Hilbert’s Hotel. Countably infinite vs uncountably infinite). I hoped to approach these ideas using a type of moderated discussion, letting the students ask questions to each other and talk about what was perplexing, interesting, fascinating, confusing, etc.

Second, I was really inspired by a recent video by Jo Boaler about “Number Talks” and I plan to try doing a weekly “Number Talk” (or something like it) with my calculus students.

My husband asked me why I wasn’t combining these things under one umbrella. To me, they hit two different–but equally important–goals for my course that can’t be found directly on our syllabus. They are

  1. I want my students to develop an appreciation for mathematics outside of what will show up on their next exam. I want them to be exposed to the kinds of questions mathematicians ask. I want them to practice the difficult skill of speaking with others about mathematical ideas.
  2. I want my students to become more fluent in numeration. I want my students to practice looking at the same problem from multiple perspectives. I want my students to see mathematics as a creative endeavor and get away from the idea that what mathematicians do is “apply a standard algorithm, proceed the same way, get the right answer.”

[Many of my digital colleagues seem to use some type of presentation requirement in their courses to get at item (1.) above. While I think that having students present math problems, solutions, ideas, etc. to each other would help develop this skill, and other skills too, I remember how terrified I was as an undergraduate at the thought of standing up in front of people and I don’t think I could impose those feelings on anyone in my classroom.]

Hopefully I will come up with other ways to push back against lectureculture in my classroom.

*As I was writing this post, the following MOOC announcement appeared in my Twitter feed & seemed quite apropos:


Reboot of my list of standards

I’m about to start my second semester of using a standards-based approach in Calculus II. One of the things I wanted to change was my list of standards. Last semester, I ended up with about sixteen standards. When thinking about improvements for this semester, I wanted to pull apart my standards in a different way and I wanted to have more of them. Also, another big goal I have is to offer a broader picture of what calculus is really about. I’ve decided to re-categorize my (now) thirty standards under some Big Questions. Here’s what I have so far:

  • What background skills are important before we begin?
  • What kinds of applied problems can we solve using integration?
  • What techniques can we use to evaluate integrals?
  • How can we add infinitely many things together?
  • When and how can polynomials be used to approximate functions?
  • How can we model phenomena if we know how they change over time?
  • What can we say about the motion of objects moving in more than one dimension?*

Here’s a Dropbox link to my current standards list: m220-f2014-standards.pdf (Apologies if this link isn’t stable; this is a working document undergoing continual changes)

* Thanks to Joshua Bowman for help with this last one!