One of my favorite conversations to have with my students (Grades 9-11) has quickly become centered around the concept of zero.
It first arose naturally last semester, when a Grade 9 student blurted out: “But 0’s not a number, it’s the absence of number!” (I’ve forgotten the context of this remark.) I remember a few other students laughing at this, and I know that this was unfortunately a negative formative experience for her, because she wrote about it in her end-of-year reflection. And yet, when I prodded further at a later point in time, I found that many other students, if not most, held the same belief. We had a whole-group conversation about it, talked about how 0 behaves in a similar way as other numbers in many ways (you can add it, subtract it, multiply it…), but still, I think a lot of students held on to this conception. This was especially clear when the students were solving equations and came to a problem such as this: Continue reading
Exactly where [a lesson] moves depends on such complex factors as the structures of those present, the context, and what has been anticipated. It may move toward more formulated understandings, if such formulation is relevant to the play space or if it becomes part of a further exploration. It may simply move to other sorts of activities. This, of course, is not to say that we should just allow whatever might happen to happen, thus abandoning our responsibilities as teachers. Rather, it is to say that we cannot make others think the way we think or know what we know, but we can create those openings where we can interactively and jointly move toward deeper understandings of a shared situation.
(Davis, 1996, p. 238-39)
My Grade 9 students are currently working on recognizing, analyzing, graphing, and solving problems involving linear relations. Linear relations lend themselves so naturally to describing patterns à la www.visualpatterns.org, and this is precisely how we got our toes wet in the topic: For several days, my students had been analyzing, extending, and (productively) arguing about a variety of linear and non-linear patterns. The intention of these first few lessons was to have students develop (or, in some cases, refine) an understanding of constant and non-constant change and to connect patterns in pictures to patterns in tables of values.
As the students began to connect ideas, I looked to develop an activity that gave students an opportunity to apply the generalizations emerging from the phenomena that we were playing with. Continue reading
Here’s something you already knew: Some days are really hard. Some days weigh heavy on your shoulders as you leave your classroom, feeling defeated. If I’m honest, some days feel like this*:
What’s your point, Ilona?
I’m not sure. I guess I just needed to look at those emotions square in the face. Tomorrow, I will try again, hopefully a little wiser. And the next day. And the day after that.
*(Sam assured us that he feels much better now.)
“What you do speaks so loud that I cannot hear what you say.” ― Ralph Waldo Emerson
Since the first day of school, I’ve been working hard to try to establish a classroom culture where students feel comfortable taking risks, asking questions, sharing and building on each others’ fully-formed and partial ideas, and acknowledging and correcting their mistakes; where all students feel that their contributions and questions are valuable and worthy of consideration. I have tried to do so by pointing out (less often than I should) when a student or a group is exemplifying one of these norms, by waiting (again, less often than I should) after questions and contributions to give more students the time they need formulate and share their ideas, by giving tasks that are accessible to a wide range of students and can be tackled with a variety of strategies, by eliciting and celebrating different solution paths, by highlighting different kinds of mathematical smartness (h/t Ilana Horn)…
And then, I proceeded to potentially sabotage it all with an inexcusable split-second decision. Continue reading
In our division, classes on the first day of school are only 15 minutes long. By the time students settle in and introductions are made, there is hardly enough time to wrestle and play with an interesting math problem. I saved that for the second day. Instead of going through the syllabus, however, I gave the time over to my students to reflect on the following questions:
- When I see the word ‘mathematics,’ I think of…
- A good experience with mathematics was when…
- A bad experience with mathematics was when…
- This semester, I expect to…
(The students completed the prompts in their new math journals, which I will hopefully get a chance to write about once the routine is more firmly established.) Continue reading
I’ve been substitute teaching for about a month now, which has been a roller-coaster ride (the fun kind). On a few of those days, I was left a lesson or activity to facilitate, but most days I’m not so lucky. Understandably, most teachers prefer to leave substitutes with a work period (alright, let’s call it what it is – glorified babysitting). However, I really enjoy engaging with students, especially when math is involved, and so I usually can’t resist showing the students a mathematical “magic” trick that I leave as a challenge for them to figure out during the period. I now have a small collection of tried and true “tricks” that I like to pull out at the beginning of class, but there’s one in particular that kills every. single. time, no matter the age group (although it will likely need to be adapted for grades below 6; I haven’t tried it). Continue reading
Last month, Nat Banting described a fantastic task on his website called the Dice Auction. You should really just read the original post, but I will summarize it as best as I can here.
The premise is that you are invited to an auction, and given a budget of $10 [I changed the budget to $15 for my students to encourage a bit more risk taking]. Everyone at the auction has the same budget. The participants are all bidding on certain events that may occur when two 6-sided dice are rolled (e.g., both numbers are greater or equal to 5; a single 2 is rolled; both numbers are odd; etc.). After all the events have been auctioned off to the highest bidders, the two dice are rolled 20 times. Each time the event that you purchased occurs, you collect a prize. Bidding always begins at $1 and goes up in increments of $1. You cannot bid against yourself. The order of the events up for auction is known beforehand. If you choose not to spend (some, or all of) your money, the auctioneer will sell you prizes at a cost of $2 per prize after the bidding has ended. Your task is to get as many prizes as possible.