In my classroom, I’d probably prefer a more narrow focus — a specific concept over a general math activity. For homework, have students take a photo of parallel lines and a transversal. In class, ask What do you notice?™ Christopher Danielson’s students — future elementary teachers — were asked to photograph a composed unit, which led to a lovely classroom discussion. Dan Meyer kicks the find a positive & negative slope challenge up a notch by holding a steepest stairs competition.
How can technology be best used as a tool for learning mathematics? Calculators can assist with computations when learning other mathematics. iPads can help students communicate their learning. I’m asking about something else. I’m asking about the use of technology to develop new mathematical understandings.
Last week, Marc and I explored this question with about twenty math department heads. First, teachers were given time to explore several dynagraphs. In this version, the equation of each function was hidden. This became the problem to solve.
Following this activity, we wanted to discuss the question above. One approach would be to present several different examples and evaluate each, sharing our criteria. Not very effective. It’s our evaluation, our criteria.
Inspired by this
xkcd
we came up with the following:
After generating a list of possible uses, teachers were asked to plot them in the plane. For example:
Quadrant I: The dynagraphs investigation was placed in the first quadrant (active-understanding). The NCTM Illuminations Pan Balance applet can also be placed here. In this quadrant, learners build depth of conceptual understanding, be it of function relationships or algebraic thinking, through problem solving. Learners encounter many, if not all, of the seven mathematical processes identified in the curriculum. They communicate mathematical ideas, make connections among mathematical concepts and to past experiences, reason and justify their mathematical thinking, and use visualization to make sense of mathematics.
Quadrant II: An alternate version of dynagraphs was placed in the second quadrant (passive-understanding). The equation is no longer hidden, thereby replacing problem solving with observation. My GeoGebra material also fits here. You know the type: drag a slider (or, worse, watch the teacher drag the slider); what do you notice?
Quadrant III: Ah, yes, Khan Academy. Enough said? Probably not. Activity is limited to pressing pause and rewind. The “intuition” video comes later, if at all. In the third quadrant (passive-knowledge), learners consume content.
Quadrant IV: In the fourth quadrant (active-knowledge), you will find Math Blaster, an iPad app in which students practice math facts (+, −, ×, ÷) through gameplay.
Many interesting comments were made by the group. Some highlights:
“It’s about teaching, not technology.”
Activities can slide from the first quadrant. Who’s doing the math? If it’s the teacher, then we’ve moved to the left. Is the focus on “how-to’s” or essential understandings? If it’s the former, we’ve shifted down. The same holds true for uses of manipulatives.
“We can’t always be in the first quadrant.”
Fair enough. This activity provides one answer to the opening question. Quadrant I is the ideal. Is there value in quadrants two through four? I think so. A demonstration can be helpful. For example, this applet can help learners make sense of A = πr². So, too, can this low-tech activity. Is there a place for grapefruit? There may be. But this can’t be where we live. FWIW, it’s not just that KA occupies this space. It’s that it goes about it so badly. If you must have a grapefruit…
“Your axes are wrong.”
At least one teacher suggested that the x-axis be labelled “active learner.” In Math Blaster, children are active in the sense that they are blasting through razor sharp blockades and speeding past the stars on their HyperCycles. A bit of a stretch to call this active learning. Other possibilities for each axis were suggested: student-centred/teacher-centred, conceptual/procedural, process/content, etc. We fully expected this. The intent of this activity was to generate discussion. The imperfection of our labelling of the axes only added to the conversation. The question “How can technology be best used as a tool for learning mathematics?” became “How do students best learn mathematics?”
The workshop was a 4½-hour mixture of problem-solving, show & tell, discussion, and self-directed exploration. This was no ‘sit and git’ workshop (slide 6).
Because teachers at the session came from several different schools, I started with a get to know each other icebreaker. Using the information on their name tags (slide 1), newly introduced teachers created a Venn diagram (slides 8-10) that reflected some aspect of their group.
Presenting on the work of others can be a little odd. Hat tips were given to the mathblogosphere in general (slides 12 & 13) as well as to individual bloggers.
In these groups, teachers solved three problems: stacking cups (slides 14 & 15), LEGO optimization (slides 19 & 20), and visual patterns (slide 23). I connected these problems to related lesson ideas (slides 17 & 21) and teacher-created classroom resource websites (slides 18 & 21).
Next, teachers took part in a couple of activities that could be easily translated to different topics: Pictionary (slide 24) and matching cards (slides 26 & 27). These activities address two of the seven WNCP mathematical processes: visualization (slide 25) and communication (slide 28).
In addition to lesson ideas and teaching strategies, I wanted to draw attention to the mathblogosphere as a place to find conversations (slide 29). Participants chose to read and discuss one of four listed blog posts (slide 30), forming new groups.
Launching off the mathtwitterblogosphere site, teachers were given time to get started using Google Reader, explore on their own, and share their discoveries.
I hope this is helpful to those of you planning presentations on this topic.
Person A (thinks he) has a lot to say. He likes to talk. He needs to talk. You can’t ignore Person A. He won’t let you.
Person B also has a lot to say. Maybe. He doesn’t like to talk. Besides, Person B also has a lot of marking. He brings it to staff meetings and pro-d workshops.
I’ve been both Person A and Person B. The following activity, “The Interview Matrix”, allows me to be neither. I first participated in “The Interview Matrix” in a session facilitated by Jordan Tinney.
The Process
Participant are divided into groups of four. Group members number themselves from 1 to 4.
Over six five-minute rounds (see below), each member interviews and is interviewed by the other three group members. For example, in Round 1, Person 1 interviews Person 2 about Question 1; in Round 2, Person 1 is interviewed by Person 4 about Question 4. In the seventh round, each person writes a summary of the responses to his/her question.
Participants are reorganized into four groups according to question number. Those who asked the same question gather together to share their findings.
Finally, everyone hears the summaries of the four questions from each of the four groups.
The Questions
I chose four themes rather than four questions. Question 1, for example, is actually made up of three questions. It is not necessary for participants to answer all three. Hopefully, at least one of them is of interest to the person being interviewed.
Summary
The word community is used in 3 of the 4 questions. It was interesting that groups interpreted this differently– community of teachers on staff, community of learners in the classroom, community of parents. This was deliberate on my part.
I’ve used this activity with three different groups and each time the participants have enjoyed it. Some math team mentors have taken this activity back to their schools to generate discussions.
One word of caution… at last, everyone gets to hear what that guy with the stack of marking has to say.
In a recent conversation with a group of math teachers, one colleague began a statement about the role of math teachers with this: “I think we can all agree…”
One problem… we did not all agree. “Actually…” I began my reply.
His statement was something like this: “…our primary role/responsibility is to make math easier for students by efficiently providing them with clear and concise explanations.”
There was a time in my career when I might have agreed with him. In fact, I probably spent the first ten years of my career striving to get better at exactly that. And, over time, my explanations did get better. I took pride in my ability to deliver content in bite-sized easy to digest pieces. This ability defined me as a teacher.
Simultaneously, I was growing more uncomfortable with this. I felt like I was teaching punctuation when, really, I wanted to be teaching literature. If I wanted my students to think mathematically, persevere in solving problems, appreciate mathematics, etc. my belief about my role had to change. In short, I had to “be less helpful.” I had to let go of what I had worked so hard to accomplish.
Back to that conversation at the school… here we were discussing the effectiveness of a particular problem-based lesson while holding opposing beliefs about what it means to teach.
“FOSSILS!” –Lewis Black
Lately, I’ve been thinking about ways to bring forward these beliefs. I created an activity and tried it out over the last two days with two groups of math team mentors and administrators. The gist of it:
Place each belief statement where you think it belongs on the truthiness continuum.
If necessary, rewrite each statement so that it can be placed on the far right.
Teachers enjoyed the activity and I enjoyed eavesdropping on some thoughtful conversations. Each belief statement was inspired by actual comments that I have heard in the last two years. For what it’s worth, two of the statements (I won’t tell you which two) were taken directly from the WNCP Mathematics K-7 Integrated Resource Package and educators placed these statements, unedited, on the far right.
Below are some examples of how teachers rewrote statements so that they felt right– from the gut.
There are three types of people: mathy people and non-mathy people.
became All students are capable of learning mathematics. Mathematical thinkers are created, not born.
(Okay, I served up a softball.)
The most effective way to have students learn basic facts is by building brain muscle memory through timed drills and lots of practice. became The LEAST effective way to have students learn basic facts is by building brain muscle memory through timed drills and lots of practice.
Ha! BTW, other groups focused on the importance of having flexible strategies.
And check this out:
The primary role/responsibility of the teacher is to make the learning of mathematics easier for students by efficiently providing clear and concise explanations. became The primary role/responsibility of the teacher is to provide opportunities for students themselves to make sense of mathematics, to scaffold when necessary, and to help students make connections to the big ideas.
Of course, it’s not enough to believe. There’s also the challenge of putting these beliefs into practice. But that, I think we can all agree, is a different conversation.
On Friday I attended a pro-d presentation in which Peter Liljedahl shared his numeracy tasks. Peter’s tasks get students comfortable with ambiguity, get them writing about math, and get them to stop mimicking the teacher.
Early in the session, I wondered how these tasks could address pre-calculus learning outcomes. Later, Peter answered this for me when the conversation turned to finding time. “Why are we afraid to give up what isn’t working?” he asked.
“Who was your math teacher last year?”
“Uh… you were, Mr. Hunter.”
Despite learning (covering?) things like factoring trinomials or writing equations of lines in Math 10, sometimes my Math 11 students would act like they were seeing these things for the first time. So, why am I holding on to this? Why can’t I make time for numeracy tasks?
Peter works with teachers to design numeracy tasks that require the mathematics that students already have in place. This rules out grade level learning outcomes. He joked about trying to steer students towards a particular method of solving a problem – “Students are very good at smelling a word problem.”
While these tasks may not address grade level learning outcomes, they can be used to address the main goals of mathematics education described in our curriculum. Communication, perseverance, risk taking, motivation, engagement, and problem solving – all of these were listed by teachers as necessary to be able to do these tasks and all of these help define numeracy.
As an added bonus, helping students develop these skills will make teaching and learning grade level outcomes that much easier.
I look forward to trying out Peter’s tasks and developing new tasks with Surrey teachers.
Chris Hunter 