My Common Core Problem Based Curriculum Maps) has crafted a game plan for proceeding with reinventing how math can be taught in schools. He has collected activities and projects from his math blogging colleagues and organized them into a curriculum. When I was still at Stevens/CIESE I did a similar approach with the 6th Everyday Math curriculum since the teachers in the Elizabeth, NJ school district wanted more technology based activities that were not part of EDM. This kind of work requires support from the teachers, pedagogical change coaches (like I was there) and administration support to work. Unfortunately, my effort in Elizabeth has sat dormant since I retired from CIESE in 2007.
What we need is a curriculum (a modern textbook, if you like) that is written from a student's perspective. Most textbooks are written so that adults reviewing the books will find all the required checkboxes checked before they adopt. Why can't textbooks be written as engaging stories that students would buy into? I suspect very few kids would choose the books that are currently coming out of the textbook mills if they had a choice. Engaging stories should drive curriculum. Then students would actually want to do the activities in the book instead of being force fed by the teachers because they are "good for you."
Geoff Krall to his credit has listed links to interesting stories, but they are still on the sidelines in the curriculum game. Some day it will happen. The current technology makes it possible and the math blogging community is putting examples out there every day. However, the devotion to the Royal Road to Calculus continues to interfere with student engagement and genuine learning.
Thursday, November 14, 2013
1. Choose a number. Call it X
2. Add 11. Now you have X + 11
3. Multiply by 6. Result is: 6X + 66
4. Subtract 3. The result is: 6X - 63
5. Divide by 3. The result is: 2X + 21
6. Add 5. The result is: 2X + 26
7. Divide by 2. The result is: X + 13
8. Subtract the original number that you chose. The result is: X + 13 - X = 13
You are jinxed. Problem solved. Case closed. But what if we were teaching typical 6th graders then a more interesting twist on this story is to not assume the obvious (that it always works) and see whether these students could find a number that foils the Jinx Puzzle. Since testing numbers manually becomes quickly tedious a spreadsheet calculator can help with testing a wide range of numbers. There’s just one problem. If you use a spreadsheet you can get a result that actually foils the Jinx Puzzle! Try something like 3.0 x 10 to the 16th power as your number.
This is caused by the fact that the spreadsheet will round off numbers after a long string of numbers is entered or will send a bogus number because we have gone passed its capability to stay accurate.
Dan Meyer in a his 101 questions activity shows how Google Calc fails to handle a subtraction problem by printing 0 instead of the correct answer of 1 just because it doesn’t play well with very large numbers.
In an episode of The Simpson's called The Wizard of Evergreen Terrace, Homer appears to write a valid solution to defeat Fermat's Last Theorem which states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. But given that Fermat's last theorem is proven, is Homer's attempt a real counter example that Fermat and others didn't see? Look at the equation (above) that Homer used using the Desmos calculator to check his work. Looks like the real deal doesn't it?
The Simpsons and their Mathematical Secrets" calls Homer’s attempt at a solution a near miss. If you use a calculator that’s good to 10 places like Desmos, it seems to work. But the devil is in longer approximations. It’s close, but no cigar. And we need exactitude to proof the theorem. Andrew Wiles who proved Fermat’s Theorem in 1995 has nothing to worry about from Homer Simpson.
Sunday, October 20, 2013
Headlining the regional NCTM meeting this week in Las Vegas, Nevada are four educators whose work is transforming curriculum design and delivery and changing the way students think about mathematics. Board member Jon Wray, Karim Ani, Dan Meyer and Eric Westendorf. They are going to be answering the question: What should effective and innovative math instruction look like, and how can teachers create ideal learning experiences for all students?
Wednesday, October 23, 2013: 5:30 PM-7:00 PM
Amazon F (Rio Hotel) - Las Vegas
If that session is the standard for the entire conference, then you're in for a treat if you are attending. Unfortunately, the best I can do is participate virtually via Twitter. Hopefully, there will be lots of blogs generated and videos of presentations.
See a listing of all other technology themed sessions in Las Vegas.
I'll share my take on the conference later this week.
Sunday, October 6, 2013
In my vision of math 2.0 teachers as bloggers are as common place as chalk and blackboard used to be. There is much to learn from the young (and not so young) math warriors who are exploring the new frontier of teachers collaborating on how to make math come alive, personable and of course useful in the lives of their students. Unfortunately to date very few math teachers know about how teachers are learning and improving their craft online with other like minded math educators. Well, a group of these pioneers wanted to do something about it and make it resonate not only in the blogosphere, but everywhere there are communities that help young people learn math.
Though their website comes with a formidable handle mathtwitterblogosphere (MTBoS pronounced mitt-boss) don't let that stop you from joining.
Dan Meyer wrote: "File this [MTBoS] as Reason #437 I'm proud to be a part of this enormous professional community. link"
Today (October 6th) begins 8 weeks of "Exploring the Mathtwitterblogosphere." Join up and follow this crash course.
Also on Tuesday nights you can join the Global Math Department for weekly presentations about math for teachers by teachers.
This week (Tuesday, October 8th 9pm) Karim Kai Ani (of Mathalicious.com fame) will be leading the conversation. You are all invited!
Monday, September 23, 2013
"In April, 1986 at the annual meeting of the National Council of Teachers of Mathematics a group of math educators who were interested in Logo, a computer programming language, met informally and decided to start an organization which eventually became known as the Council for Logo in Mathematics Education or CLIME for short. These educators taught math at all levels and were from all parts of the United States and Canada. What had brought them together was a belief that Logo could make a significant difference in the quality of mathematics education.
Now there are many other resources - including Cuisenaire Rods, geoboards, rulers and compasses, to name just a few - that math teachers use to help them teach. But to my knowledge no one has ever started an international organization for the purpose of promoting their use. What, then, makes Logo so special?
One way to answer this question is to say that Logo, unlike other resources, comes with its own philosophy of education. This philosophy was introduced to the world by Seymour Papert in a book called Mindstorms: Children, Computers, and Powerful Ideas (Basic Books, 1980). In it he said that children seem to be innately gifted learners who acquire a vast quantity of knowledge long before they go to school. What blocks them from learning is not the inherent difficulty of the ideas, but the failure of the surrounding culture or environment to provide the resources that would make the ideas simple or concrete. In other worlds, one reason why math is difficult to learn is because the culture outside the classroom does not provide the materials or experiences that would support the students' classroom lessons."
That was 1988. Since then CLIME has evolved from the Council for Logo to the Council for Technology in math education. This change was made to acknowledge the development of exciting new environments (like Geometer's Sketchpad) which made me realize that Logo was not the only game in town, but that there were other software environments that encourage this kind of dynamic learning that Logo made possible. With the advent of the handheld tablets and smart phones powerful new math apps are being developed. (For example see Keith Devlin's latest entry Wuzzit Trouble.)
It's been a tough time to be an educator with all the hoopla about the common core and its subsequent imprisonment of laptops and handhelds for testing which will keep the technology away from creative teachers who could use them in dynamic ways. So its easy to see the technology glass as half empty.
In his book "Logo Theory and Practice" (1989) Dennis Harper quotes me as saying that "a Logo environment is more of a spirit rather than a thing-something that can only be satisfying if experienced, rather than just languaged." (p.26)
And I still believe that today. I recently wrote to one of CLIME's steering committee/board members Robert Berkman that we need to look at the glass as half full and keep a positive attitude because the Internet does have a long tail (and tales) and there are literally hundreds (thousands?) of places, people and environments where adults are helping our young people "construct modern knowledge" something another one of CLIME's board members Gary Stager promotes in his annual Vermont workshops and in his new book "Invent to Learn."
I've been privileged over the years from the guidance of these and other remarkable people who have been good friends of CLIME and members of the CLIME steering committee over the last 25 years. I want to publicly thank them for their help. (You will be hearing from some of them in future CLIME blog posts.)
See a short summary CLIME's journey since 1986.
Tuesday, September 3, 2013
Whether you like the idea of the Common Core State Standards for math (CCSSM) or not, it is here to stay at least until version 2.0 addresses the eventual problem of the scores not going up. Why am I so sure this will be the case? Because CCSSM doesn't ensure a curriculum that actually helps students understand and learn the topics any better than before. For example, learning fractions. In a recent book Keith Devlin describes how easily it is to get confused when doing fractions. (See my blog.) Schools have to use curriculum that match the standards. That's great. But how can they do it well? Since proportionality is such an important concept and understanding it is so critical a carefully crafted set of activities is needed to prevent misconceptions.
Most school districts will probably choose a textbook program that is correlated with the standards. Problem: Most textbooks do a decent job in correlating, but not in motivating students to learn the math. This summer Dan Meyer had a Makeover Monday blog where teachers submitted typical problems from textbooks and Dan's community offered suggestions as to how to improve them. While reading these blogs I became convinced that we need a makeover of textbooks in general i.e. come up with a more dynamic model for lessons that textbooks could adopt. Currently Dan Meyer and Karim Ani (Mathalicious.org) are creating and encouraging dynamic learning adventures that are interesting to kids and help with deeper understanding.
The giants of the textbook world (Pearson, McGraw HIll, etc.) are trying to modernize their curriculums but they have too much at stake in maintaining the status quo since most teachers and administrators prefer what they are familiar with and find the so called "alternative" models too risky or controversial for district approval. (Larry Cuban describes this phenomenon as dynamic conservatism). Otherwise districts would reject most if not all the mediocre curriculums that are now being published.)
Should Algebra be optional?
Recent articles in Harpers and the New York Times have argued for making the Algebra 1 and Algebra 2 sequence optional especially for kids who struggle with math.
Michael thayer writes:
In an ideal world, kids would sort themselves in this way based on their interests.
Kids in track #1 ("calculus track"): These are the kids who love math, who love the challenge of it, and who see the abstractions of algebra and analysis as pursuits worthy of study.
Kids in track #2 ("statistics track"): These are the kids who recognize the importance and practicality of math, and who see utility for it in their futures.
Kids in track #3 ("one and done"): These are the kids who have had a good experience with math, who have seen the forest for the trees, but do not wish to go deeper as their interests lie elsewhere.
I would also include in track* 3 those students who didn't have a good experience in math and do not have any interest in continuing in math since they would rather use the time to study something else.
My Thoughts on the Path 3 Course
Offer a one year course for students who definitely don't want to do the formal Algebra 1 or 2 path for whatever reason, but still want to go to a "good" college. Their are over 4,000 accredited 2 and 4 year colleges in the US. Getting into a college is usually not a problem, just paying for it is. (Shame on those colleges that afflict serious debt on our students.) I'm sure there are plenty of colleges out there that would accept students who have (as Mike pointed out in his blog) excellent work habits, overall knowledge base, and interpersonal and time management skills who didn't take Algebra 1 and 2 but rather this richer one year 9th grade math course; something like "Mathematics a Human Endeavor - A Book for Those Who Think They Don't Like the Subject" by Harold Jacobs. He wrote his last revision of the book in 1994 and the book is still in demand particularly in homeschooling environments. Anyone out there want to work on an open source version of the kind of one year alternative curriculum that is in the same spirit as Jacobs had in mind? (Here's something I did with his Chapter 3 - Functions and their Graphs.)
Maybe we can do it collectively as an open source project. I volunteer to be a conduit for creating this course! Are you interested? (If so, you might get familiar with Jacobs book to see what I have in mind.)
*Tracking is not the right word for this, because it implies rigidity. These should be paths that students can opt to start on, but can switch to a different path at any time or chart their own course.
Thursday, August 8, 2013
Michel Paul's <firstname.lastname@example.org> recent email post to email@example.com is worth reading. This is Math 2.0 thinking at its best.
Michel Paul writes:
"Computer science is the new mathematics."
Michel Paul writes:
Keith Devlin wrote the following to describe the nature of modern (20th century onward) mathematics to current undergraduates transitioning from high school (19th century and before):
"Prior to the nineteenth century, mathematicians were used to the fact that a formula such as y = x^2 + 3x - 5 specifies a function that produces a new number y from any given number x. Then the revolutionary Dirichlet came along and said to forget the formula and concentrate on what the function does in terms of input-output behavior. A function according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them."- Keith Devlin
Wow! Please read that carefully. Though he wrote this to help today's students transition from the 19th century to modern mathematical thinking, it also describes the kind of thinking one learns to do in computer science! Functions in computer science operate on objects of various types, not just numbers.
Traditionalists lost th
e battle professionally in the mathematical revolution that occurred
a century agobut won in education. Meanwhile, computer science went ahead and got created from the insights of that revolution and turned into the world we now live in. The result? Most K-12 math students and their teachers, us, are unaware of the nature of the mathematical thinking that went on in the 20th century while the technology that surrounds us was built from it!
The ultimate irony - we use 21st century technology, made possible by 20th century math and physics, to teach students how to do 19th century mathematics that they will never use!
think this makes it clearthat the study of programming
canprovide a way for math students to encounter and develop some intuition regarding concepts underlying modern mathematics that the traditional high school curriculum does not provide.
"What I cannot create, I do not understand."
"What I cannot create, I do not understand."
- Richard Feynman===================================
"Computer science is the new mathematics."
- Dr. Christos Papadimitriou