As I blogged in my Apollo 13 video post, Watch Problem Based Learning in Action "While our students have been conditioned to 'learn the basics - then solve the problem,' that's not how life always works."
Here's a great 4-minute video by Dan Meyer that gives three examples of how to bring real-life problem scenarios into the math classroom. To paraphrase Dan, "In these examples student have to first ask the question - what information do I need to solve this problem? The textbook usually gives you that information. But here students build the problem and decide what matters. The question that's usually buried at the bottom - it's the last thing in the textbook problem - now becomes the first thing in the student's mind. I want to make that question "irresistible" to the student, so they have to know the answer." For more great ideas on how "math makes sense of the world" - go to Dan's blog dy/dan
During this summer program students entering eighth grade were coached by an intern in ways to investigate and talk about the math in their lives. Here's the 4 strategies the students used:
1. Look for math in real life - Nic ponders the permutations in picking out his clothes. 2. Frame your experiences as word problems - Shanice eagerly monitors price changes in a coat she wants to buy. (Spoiler alert: she gets it!) 3. Try out different ways to solve problems. Nik crafts a way to determine his baseball batting average. 4. Explain and share your thinking. Shaniece describes what they do when one them gets stuck on a problem.
Watch the video to hear what they discovered in their own words. "I see math when I'm walking down the street.... I see math in myself."
Intel is hosting an education digital town hall at the Newseum that will explore new ways to "cultivate tomorrow's thinkers and entrepreneurs to sustain economic and educational success." (December 7 at 8:45 a.m. - 11:45 EST) Participants include Education Secretary Arne Duncan; Angel Gurria, the Secretary General of the Organization for Economic Co-operation and Development; Rob Atkinson with ITIF; and Tom Friedman of the New York Times.
Let's see how the Duncan sidesteps the issue of testing and innovation - while US students spend endless hours honing their test taking skills, the demand for routine skills has disappeared from the workplace. Anyone know of a meaningful and rewarding career that looks like filling out a worksheet? Maybe Friedman will be willing to tackle the stifling impact of testing on creativity thinking among our students. For my thoughts on the subject, see my post "As NCLB Narrows the Curriculum, Creativity Declines"
"Education for Innovation" a live digital town hall
You can submit questions you would like the moderators, PBS NewsHour’s Gwen Ifill and Hari Sreenivasan, to discuss with the speakers. Then, vote the questions you like best to the top. Click here
You can join the for the live, interactive webcast on Tuesday, December 7 at 8:45 a.m. - 11:45 EST or join the conversation at Twitter/InnovationEcon use the hashtag #Ed4Innovation
More on the Program for International Student Assessment (PISA)
PISA is an assessment (begun in 2000) that focuses on 15-year-olds' capabilities in reading literacy, mathematics literacy, and science literacy. PISA studied students in 41 countries and assessed how well prepared students are for life beyond the classroom by focusing on the application of knowledge and skills to problems with a real-life context. For a detailed example of how PISA assesses sequencing skills see my post "Why Don't We Teach Sequencing Skills?"
Response to sample question This short response question is situated in a daily life context. The student has to interpret and solve the problem which uses two different representation modes: language, including numbers, and graphical. This question also has redundant information (i.e., the depth is 400 cm) which can be confusing for students, but this is not unusual in real-world problem solving. The actual procedure needed is a simple division. As this is a basic operation with numbers (252 divided by 14) the question belongs to the reproduction competency cluster. All the required information is presented in a recognizable situation and the students can extract the relevant information from this. The question has a difficulty of 421 score points (Level 2 out of 6).
First some background ... my original post used a video clip from "Stand and Deliver" to map the information flow in the traditional classroom. I also used the illustration below (from "Math Is Language Too: Talking and Writing in the Mathematics Classroom" by Phyllis Whitin) to demonstrate how students learn to "do the math" for their teacher, rather than see math as an opportunity for peer discussion, problem solving or reflection.
Here's the comment to my post that I received from "Pjack." I'm glad to see that at least one student is reflecting on his progress as a learner. (for more on student reflection see my post on the Reflective Student)
"The way math is taught is can be somewhat disheartening in many cases, as illustrated by that kid's drawing. As a high school student, and one who isn't that great with numbers (art kid here), one of my favorite classes I've ever taken, of all the most unlikely things, was summer school physics. The teacher did a brief lecture, gave us some formulas for how to calculate this and that, put us in groups of our choosing, had us figure out one problem per group in a collaborative fashion, and then present the answer to the class, whether it was right or wrong. The class would then give constructive feedback, and ask us questions, which we would work as a class to answer. The teacher sat at his desk the entire time, willing to offer help to those that asked but otherwise removed. The thing he repeated was, "What you put in to it you get out of it." Needless to say, it was an interesting experience, and one of the first times I did math collaboratively. Sadly, many of the students (soph/juniors in high school) made comments like, "He doesn't teach!" or were generally terrified of this responsibility. Really goes to show how little we feel prepared to take control over our own learning, at times. I notice this sort of teacher-dependency in some amount in almost every class."
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."
Recently I blogged about the teacher-centric information flow in the traditional classroom. See: Engage Student Discussion: Use the Social Network in Your ClassroomIf 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.
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
Today I listened to NPR's Scott Simon and Keith Devlin of Stanford University, answer the question: Why Do We Need to Learn Algebra? (NPR Weekend Edition Saturday~February 28, 2009). Devlin described how spreadsheets have become essential to managing everything from your finances to your fantasy football team. And of course, spreadsheet are basically collections of algebraic formulas. You can follow this link to the NPR story, comments and audio file. Teachers might use Devlin's comments as a springboard for getting students to think and discuss the application of algebraic thinking in their lives.
This is essential, since algebra is emerging as an academic gate keeper. I'm not a math teacher, but I suspect it stems in part from the fact that many students lack basic computation skills. But more importantly, students have to be able to transition from concrete lower order thinking skills (arithmetic) to higher-level and more abstract thinking (algebra and beyond).
As Doug Reeves has noted, "The single highest failure rate in high school is Algebra I. After pregnancy, it’s the leading indicator of high school dropout. The leading indicator of success in Algebra I is English 8. The Algebra 1 test is a reading test with numbers.” District Administrator, April ‘05
If Reeves is correct, then this is as much a literacy problem as a math problem. Teachers of all content areas can pitch in to support the higher order skills (analysis, evaluation and creating) that will help students with more advance mathematical thinking.
Student Reflection on Classroom Discussion and Problem Solving
I recently received an insightful comment to my post "Classroom Discussion Techniques that Work - Try This Hollywood Classroom Walkthrough" I thought it was worth reprinting the observation as a separate post.
First some background ... my original post used a video clip from "Stand and Deliver" to map the information flow in the traditional classroom. I also used the illustration below (from "Math Is Language Too: Talking and Writing in the Mathematics Classroom" by Phyllis Whitin) to demonstrate how students learn to "do the math" for their teacher, rather than see math as an opportunity for peer discussion, problem solving or reflection.
Here's the comment to my post that I received from "Pjack." I'm glad to see that at least one student is reflecting on his progress as a learner. (for more on student reflection see my post on the Reflective Student)