Category: Strategies

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Learn Mathematical Thinking From the Wrong Answer

I recently saw this video clip from an old Abbott and Costello film (thanks to my Twitter network). It reminds us that math isn’t simply about learning a computational process, or getting the right answer. It’s pretty clear that Lou Costello has learned the wrong algorithm, and he defends his approach it with great determination.  See the same mathematical thinking by Ma and Pa Kettle. 

We learn math skills so that we can apply mathematical thinking to the problem solving we will need in our lives. Thus, much can be learned from the procedures we use to generate both the “correct” and the “incorrect” answer. Sharing our thinking with others allows us to negotiate a deeper understanding of algorithms and their application in the real world.

The video clip neatly “illustrates” a teaching strategy from Teach Like Your Hair’s On Fire by Rafe Esquith. The book details Esquith’s fifth grade teaching methods in a rough LA neighborhood. Esquith shows us what students can learn from the wrong answer and his process can be easily applied across the curriculum. While it makes for an excellent test taking strategy, its real power is that gives students an engaging perspective to think more deeply about teaching and learning.

Esquith writes,  “Let’s say I’m teaching addition. Just before I give the kids their own problems, I put one more problem on the board:

63 + 28 = 

Rafe: All right, everybody. Let’s pretend this is a question on your Stanford 9 test, which as we all know will determine your future happiness, success, and the amount of money you will have in the bank. (Giggling from the kids) Who can tell me the answer?

All: 91.

Rafe: Very good Let’s place that 91 by the letter C Would someone like to tell me what will go by the letter A?

Isel: 35.

Rafe: Fantastic! Why 35, Isel?

Isel: That’s for the kid who subtracts instead of adds.

Rafe: Exactly. Who has a wrong answer for B?

Kevin: 81. That’s for the kid who forgets to carry the 1.

Rafe: Right again. Do 1 have a very sharp detective who can come up with an answer for D?

Paul: How about 811? That’s for the kid who adds everything but doesn’t carry anything.

In Room 56, the kids come to learn that multiple-choice questions are carefully designed. It is rarely a matter of one correct answer and three randomly chosen incorrect ones. The people who create the questions are experts at anticipating where students will go wrong. When a kid makes a mistake somewhere in the course of doing a problem and then sees his (incorrect) answer listed as a potential solution, he assumes he must be correct. My kids love to play detective. They enjoy spotting-and sidestepping-potential traps.

When students in Room 56 take a multiple-choice math test with twenty problems, they see it as an eighty-problem test. Their job is to discover twenty correct answers and sixty incorrect ones. It is hysterical to listen to the sounds of the class when the students take a standardized math test. The most common sound is a quiet giggle of recognition. The kids love to outsmart the test and can’t help laughing as they discover one trap after another.”

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Inspire Your Students To Tell A Story With Prezi

I've been using Prezi in my workshops and have written a few posts illustrating it's features – brainstorming with Prezi and embedding Prezi in your blog.

Enjoy "Create a Prezi" – a fine video that demonstrates the process in under 5 minutes. (No, I didn't create it!) It will inspire your students and give them a chance to break out of the lineal / PowerPoint style. (Thanks for the tip – Twitter/ashleyproud)

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Innovative Teaching is to Sustainable Farming as Test Prep is to _____?

Recently I spoke at a project-based learning conference in Wisconsin. I had been reading Michael Pollan's "The Omnivore's Dilemma,” so I had farming on my mind as I drove from the Milwaukee airport to Janesville WI past vast cornfields punctuated by enormous grain silos.

Pollan observes that high-yield corn is a product of genetically identical plants that can be densely planted without fear of any stalks monopolizing resources. As corn dominated the midwestern landscape, the region became an agricultural monoculture of expansive corporate cornfields – pushing out other crops and more diverse family farms. Cheap corn created the "Concentrated Animal Feeding Operation," where never-ending truckloads of feed are used to fatten cattle in the least time possible. "Big" corn and cattle production are artificially supported by vast, but unsustainable, industrial inputs of fossil fuels, petro-chemicals, and an elaborate transportation system.

And somewhere on the drive to Janesville, I got thinking that Pollan's indictment of corporate agriculture might be extended to some aspects of education. The testing regime is turning our kids into a high-yield, uniform commodity. Rows and rows of competent, standardized students, that can be delivered according to employers' specifications for a "skilled workforce.” Children “force fed” in test prep programs in efforts to quickly “fatten” the scores to meet AYP. Like the cornfields and feedlots that are disconnected from local ecosystems, the movement toward national educational standards erodes at local control and innovation.

Fortunately when I got to the conference I saw another side of contemporary education – innovative teachers. It was like walking into a sustainable farmers' market.

The conference was held at the TAGOS Leadership Academy and hosted by Project-Based Learning Systems, the developer of Project Foundry, a web-based management tool for innovative learning environments. Teachers had come from across the country – Chula Vista CA to Waterville ME. Like sustainable farms, their schools were deeply rooted in their communities, each closely tied to its unique local social ecology. Their programs fostered interdisciplinary learning, like the symbiotic polyculture of a farm based on a rotational interplay of crops and animals.

PF-plans The PBL approach is based on the notion that rather than simply apply bodies of knowledge to problems, the exploration of problems can generate new bodies of knowledge. Teachers didn't attend the conference to simply “sit and get,” they were there to share. After my introductory talk and a planning session using my audience response system, the teachers self-organized into a series of peer-teaching sessions that took them through most the rest of the conference. 

The next day I headed home feeling upbeat. I had met many fine teachers and instructional leaders who reminded me of why I went into education. Most of all, I thought about the scores of teachers across the country, working in innovative schools (or perhaps subversively innovating in traditional schools), committed to raising a “crop” that can sustain itself through a life time of learning.

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Motivating Students – A Make and Take Workshop for Teachers

This week I head back to Edison School of Engineering & Manufacturing, to conduct a two day follow up workshop.  Our previous work together identified four target areas: 

  1. Motivating students
  2. Making learning relevant
  3. Student-centered learning strategies
  4. Effective use of technology

We are going to start by modeling a “Brainstorm, Group, Label” activity (See Tool 13).  It will also  set our agenda for the “make and take activities” of the workshop. Day two begins by modeling a “Fishbowl Discussion Group” on the topic of effective next steps. We’ll use the ideas generated in the fishbowl to design a Strategic Planning Grid (below) to prioritize their staff development for the fall.

Plan-grid

Most of our time over the two days will be spent assisting teachers in designing specific lessons. I’ve assembled some Literacy Strategies that teachers can use as starting points for modify their existing lessons. 

“Non-readers”  for students who lack decoding skills.

“Word-callers” for students who can decode, but lack comprehension skills.

“Turned-off readers” for students who have the decoding and comprehension skills, but lack motivation or engagement.

I’ll also be showcasing some web tools that are very engaging for students.

Wordle (text analysis) 

Many Eyes (data and text visualizations)

Prezi  (presentation tool) 

Dipity  (timeline builder)

Bubbl.us (brainstorming)

Flickr Tag Related Tag Browser (image tag analysis)

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Classroom Discussion Techniques that Work – Try This Hollywood Classroom Walkthrough

Recently I blogged about the teacher-centric information flow in the traditional classroom. See: Engage Student Discussion: Use the Social Network in Your Classroom  If you would like to see my point illustrated, you can do a quick "Hollywood classroom walkthrough" with this clip from "Stand and Deliver." Before you play the video, create a diagram with eight small circles labeled teacher and student responders 1-7. As you watch the video, keep track of the sender/ receiver in each exchange of information with lines and arrows. Once you have finished with the diagram, reflect on a few broader questions:
1. Were the students comfortable offering their answers?
2. What feedback did the teacher give after each student answer?
3. Did the students get any closer to a valid answer as each, in turn, ventured a response?

Go back and look at your information flow diagram. You'll notice that every answer was directed to the teacher. After the first six  answers the teacher found a clever way to say "your wrong," without explaining why. Students made a series of guesses at a correct solution without any evidence that they learned anything from the prior responses. Finally a student shows up at the door with correct answer and it's not even clear that she heard any of the earlier answers. 

Some might admire the comfortable climate of this classroom – after all, students were very willing to risk a response. Ironically the only one making fun of them was the teacher (a practice more suited to Hollywood than a real classroom.) Others might consider this an example of rather Socratic approach – but I don't see the teacher posing any new questions to expand student thinking. When you strip away all the clever (inappropriate?) repartee you are left with a very teacher-centric discussion – with students guessing at a correct answer. 

This approach reminds me of an illustration I saw in "Math Is Language Too: Talking and Writing in the Mathematics Classroom" by Phyllis Whitin. It's a drawing done by Justin, a second grader, writing and drawing about his relationship with math.

Justin-math
Like Justin, the students in "Stand and Deliver" don't see math as a topic for peer discussion or reflection. Rather, they "do the math" for their teacher. While these two examples focus on math, this dynamic could be true of many whole group discussion across the curriculum. I admit to being equally guilty of a dominating classroom discussion as a rookie social studies teacher. "Class, what were three results of the War of 1812? … Anyone? … Anyone??"
After years of facing this type of discussion, students learn that their comments are of provisional value until "approved" by the teacher. Over time, students stop listening to each other and only focus on what the teacher says or validates – "will that be up on a test?" When students are put in small group discussion, they rapidly get off subject. With no teacher to validate their comments, they naturally gravitate to other subjects where peer comments are valued – "what are you doing this weekend?" 
In my workshops I train teachers in discussion techniques that foster student reflection and interaction. The strategies are focused on getting the teacher out of the role of information gatekeeper and encouraging student-centered dialogue. 
Want to encourage students to redirect their thinking and reflection away from the teacher and toward one another? Try a research-based discussion technique like the Fishbowl-discussion 68 KB PDF