Redefining Math (still)

I find myself thinking a lot about redefining math, and posting things on this blog about that topic. Here, again, is a quote from a brilliant mathematician that forces us to rethink our idea of what “math” means.

I must include a paragraph from this article verbatim. Terry Tao, math prodigy.

That spring day in his office, reflecting on his career so far, Tao told me that his view of mathematics has utterly changed since childhood. “When I was growing up, I knew I wanted to be a mathematician, but I had no idea what that entailed,” he said in a lilting Australian accent. “I sort of imagined a committee would hand me problems to solve or something.” But it turned out that the work of real mathematicians bears little resemblance to the manipulations and memorization of the math student. Even those who experience great success through their college years may turn out not to have what it takes. The ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning and, perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians. Tao now believes that his younger self, the prodigy who wowed the math world, wasn’t truly doing math at all. “It’s as if your only experience with music were practicing scales or learning music theory,” he said, looking into light pouring from his window. “I didn’t learn the deeper meaning of the subject until much later.” (p. 46, 7.26.15 NYTMag)

Read more here, and add this to the “How do we redefine mathematics in school?” pile.

How to relate, by Alan Alda

I am watching reruns of M*A*S*H right now as I reflect on seeing the one and only Alan Alda in person at UVM’s Davis Center. I love watching M*A*S*H, as I feel it has held up really well over time and is a perfect mix of funny and tragic. The character Hawkeye Pierce, played by Alda, stands out even though the other cast members are excellent.

Alan Alda was also brilliant in Woody Allen movies, like Crimes and Misdemeanors (“If it bends, it’s funny. If it breaks, it’s not funny.”) Then I began seeing him getting involved in science with Scientific American and other shows that ran on PBS channels. Although today was a snow day and the roads were pretty bad, I and the intrepid Nina Madore weren’t going to miss the opportunity to see him live at UVM. I figured nobody would be there, but the giant Silver Maple Ballroom was packed.

I expected Alan to talk about specific science concepts, but, instead, he talked about being an effective communicator. He got rid of the podium used by those who introduced him in order to be more accessible to the audience. Actively demonstrating the theme of his talk, Alan looked carefully out at the faces in the audience and spoke in a warm, conversational manner without reading any notes. He shared the alarming statistic that 95% of Americans are not considered “science literate”. Then he went on to make the case for scientists learning to be better at relating to laypeople and working to make their research accessible.

Alan shared several ideas that scientists can use to think about improving communication. He talked about using emotion to help lodge ideas in the minds of the audience and using suspense and drama as a hook. At one point, a courageous audience member took the stage to demonstrate by walking across the stage with a very full glass of water. Another audience member was enlisted to demonstrate “the curse of knowledge” by tapping out a song. The curse of knowledge is what happens when you know something so well that you forget what it is like not to know it. In order to be a good communicator/teacher, one must break the curse and use empathy to understand the audience.

Alan’s talk was excellent. I wish more colleagues had been able to attend and that we could participate in some of the workshops he was going to be leading at UVM. His foundation, The Alan Alda Center for Communicating Science has a website worth visiting. UVM is going to begin a similar program and partner with him and his colleagues. In the meantime, elementary and middle school teachers should take a look at The Flame Challenge. Teachers of fifth and sixth graders can register here to let their students be the judges of a contest to see which scientist can best explain the answer to a science question. This year’s question is “What is sleep?”.

Alan has important messages for teachers and scientists and really anyone with a heartbeat. It’s worth visiting his website, reading more, and watching videos. Alan, thank you for coming to Vermont. I look forward to seeing UVM’s efforts in this area.

STEM and Equity in the Burlington School District

I am enough of an artist to draw freely upon my imagination. Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.

― Albert Einstein

Recently, a budget draft included funding of a STEM position at Flynn Elementary. Here is why adding and/or strengthening STEM education at Flynn and in all Burlington schools is the right thing to do.

STEM is cutting edge education.

STEM stands for Science, Technology, Engineering and Mathematics. At its best STEM education consists of hands-on, student-centered design challenges and project-based learning that cultivates the imagination.

Einstein’s remarks about the importance of imagination have never been more apt.

The world is changing so rapidly that no one can anticipate what, specifically, students will need to know. We must equip them for success in a in a technology-driven world that demands continual learning. Students should be offered broad and varied opportunities to take charge of their own education, collaborate, communicate, design and synthesize.

STEM is diversity and equity.

Quality STEM education is enemy of the achievement gap. Women and people of color are still underrepresented in STEM classes in high schools, STEM majors in college, and STEM fields in the workforce. Women and people of color are equally capable of excelling in this work and they provide much-needed perspectives. We have to ask why the demographics of these fields continue to be skewed. The fact is that we still live in a world full of biases and systemic discrimination. Education is a critical part of the solution. Stereotypes, biases and barriers don’t go away without concerted effort to counteract them.

The importance of high quality STEM education goes beyond giving students opportunities in STEM fields. It makes school a place where more students are able to be successful, period. Hands-on, student-centered design challenges, a hallmark of STEM, tend to work better for all students than traditional pedagogy. Students who are from low income families, are non-white, are new Americans and/or English language learners are at greater risk for under performance in school. STEM helps these students. Education for equity means providing engaging, challenging work for all students.

As a K-5 Math Coach in the Burlington School District, I have personally witnessed the power of STEM education. Individuals who are less successful in traditional learning situations shine when engaged in high quality STEM learning opportunities. I’ve watched students who typically exhibit challenging behaviors and low academic performance become energized about computer programming, exploring fractals in art and nature, catapult building, geodesic dome construction, and on and on. It is hard not to become an advocate of this type of education after watching marginalized students get opportunities to demonstrate their brilliance.

STEM is for Burlington

It is wise to fund STEM education in Burlington. By investing in STEM, we are focusing on what really works for achievement and equity. This is a forward-thinking opportunity to empower our students to be the leaders, thinkers, and makers of a changing world. Imagine that.

Geodesic Dome Building, Part I

Ada Leaphart, Integrated Arts Academy Art Teacher Extraordinaire, and I have embarked on an adventure. We’ve got all the fourth graders at IAA building geodesic domes out of newspaper.

I really don’t know how this is going to end up, so that is why I have named this Part I. I hope to report back with more news as the project progresses.

Ada and I decided to keep things loose. We didn’t want to figure everything out for the students by creating our own dome first, learning all the lessons, and then presenting a tidy scenario.

Instead, we have never attempted to create a geodesic dome out of newspaper or anything else, for that matter. Sometimes STEM+Art (I, unlike others, am not wanting to call that STEAM) should be messy and students should have the fun of making mistakes and doing all the figuring.

Before the students got started building, Ada showed a photo of a geodesic dome and asked students what they noticed. We got some math+art conversation going from that, as students noticed many geometric shapes, like triangles, pentagons, hexagons, and trapezoids. Then we talked about old Bucky Fuller (every class asked if he was still living) and how he really wanted to make the world a better place for everyone by using an efficient structure like a dome for shelter.

Kids couldn’t wait to get started. Here are the instructions we are using from PBS.

It turns out, you can roll newspaper in a loose, floppy, weak way or you can roll it in a very tight, very strong roll. Students shared successful and unsuccessful techniques. Among the successful techniques invented by students are 1) Asking someone else to help you tape the roll, 2) asking someone else for help, period, 3) twisting the roll when it is finished to make it even tighter, and 4) using a pencil to act as the center of the roll, then shaking it loose once the roll is finished.

We ended up with enough rolls to make one or two domes. 65 usable rolls are needed. Next time we will need to establish a Quality Control group to assess, select, and count the rolls we’ve got. Ada and I aren’t sure how the whole thing is going to go. At the end of it all I would like to set the dome on fire in the playground. I am not sure we will get permission for that. Oh well, we’re going to roll with it.

Shelburne Farms Mini Maker Faire

I went to the Maker Faire at Shelburne Farms today for the first time.

This is what I did.

I made a magic wand with an LED light that turns on and off by touching a wire to a battery on the wand. I first had to figure out how to connect the LED to some wires, then run them to the battery correctly. After that part was working, I added sparkly silver ribbon to the stem of the wand and encased the LED light in crumply clear plastic tape for light refraction purposes. Voila! I am ready to put spells on people. Joanna Elliott, Flynn Elementary parent and teacher, was the wizard behind this project. See her fabulous art blog.

I made a puzzle book, a square flexagon (a previously unknown-to-me relative of the hexaflexagon) for comic-book type story-telling, and a mini book that could contain anything from math facts to the secrets of the universe. A matchbook size mini book can be made and then kept in an actual matchbox. Book Arts Guild of Vermont people helped me do this. Students might want to make these after reading the Red Clover Book entitled The Matchbox Diary by Paul Fleischman. This is an activity for any budget.

I spoke with Richa, who is going to assist with a course called Intro to Relational Databases at Girl Develop It Burlington. There are classes and meet-ups. I want to go.

Champlain College Emergent Media Center folks explained what they are working on. Their new Maker Lab that had its grand opening party last night.

I saw a robot-building challenge and presentation by Joe Chase and his team of students from Essex High School. Joe is my neighbor and it was great to see him up there advocating for more design and engineering work in schools. My daughter took his robotics class a few years ago and loved it.

I saw and did many other cool things, including experimenting with magnets with Frank White from CreateItLab and speaking with the effervescent Michael Metz of Generator, Lucy deLaBruere, Courtney Asaro, and Graham Clarke, both of Flynn Elementary School in Burlington.

What a great day! Takeaways included an Arduino Robot Kit made by YourDuino.com and the knowledge that so many people are working on creating engaging opportunities for people of all ages in the Burlington area.

Who Gets to Graduate?

On Sundays I love to play tennis and read the New York Times. There was a great article this past Sunday. Author Paul Tough shares some sobering statistics about who is graduating with four year degrees, who isn’t, and why.

You must read the whole article. Go and do that now. My big takeaways: remedial classes and groups don’t work and students in some demographic groups (low-income, non-white, parents who didn’t graduate from college) are at risk but can be helped. It isn’t really new information. It is just more evidence supporting certain ideas.  I want to know how we can apply this to K-12 public schools.

Every college freshman — rich or poor, white or minority, first-generation or legacy — experiences academic setbacks and awkward moments when they feel they don’t belong. But white students and wealthy students and students with college-graduate parents tend not to take those moments too seriously or too personally...It is only students facing the particular fears and anxieties and experiences of exclusion that come with being a minority — whether by race or by class — who are susceptible to this problem. Those students often misinterpret temporary setbacks as a permanent indication that they can’t succeed or don’t belong…

“What I like about these interventions is that the kids themselves make all the tough choices,” ... “They deserve all the credit. We as interveners don’t. And that’s the best way to intervene. Ultimately a person has within themselves some kind of capital, some kind of asset, like knowledge or confidence. And if we can help bring that out, they then carry that asset with them to the next difficulty in life.”

Helping students doesn't mean showing them how

A few days ago, I was in a professional development meeting with paraeducators at C.P. Smith Elementary School. We were discussing how best to help students in math. We had a great conversation about it and talked about the idea of asking instead of telling. Our focus was #1 and #3 of the eight Common Core Math Practice Standards. I’d printed some questions that I have seen on several different websites, so I am not sure of the original author. Here they are:

Make sense of problems and persevere in solving them

• How would you describe the problem in your own words?
• How would you describe what you are trying to find?
• What do you notice about...?
• What information is given in the problem?
• Describe the relationship between the quantities.
• Describe what you have already tried. What might you change?
• Talk me through the steps you’ve used to this point.
• What steps in the process are you most confident about?
• What are some other strategies you might try?
• What are some other problems that are similar to this one?
• How might you use one of your previous problems to help you begin?
• How else might you organize...represent... show...?

Construct viable arguments and critique the reasoning of others.

• What mathematical evidence would support your solution?
• How can we be sure that...? / How could you prove that...?
• Will it still work if...?
• What were you considering when...?
• How did you decide to try that strategy?
• How did you test whether your approach worked?
• How did you decide what the problem was asking you to find? (What was unknown?)
• Did you try a method that did not work? Why didn’t it work? Would it ever work? Why or why not?
• What is the same and what is different about...?
• How could you demonstrate a counter-example?

After the paraeducator meeting, I came across an Edutopia article entitled, “Takeaways from Math Methods: How will you teach effectively?” It is about pre-service teachers who have taken a math course. This jumped out at me.

Helping Students Doesn't Mean Showing Them How

Before admitting [preservice teachers], we interview each one and ask, "Why do you want to be a teacher?" The most common response is, "I want to help students," a sentiment that PSTs describe later as "giving good explanations" or "making it simpler" -- notions of helping which are underdeveloped.

A synthesis of research in mathematics education by James Hiebert and Douglas Grouws identified two teacher actions that impact conceptual understanding. One is to engage students in productive struggle. Merely telling students how or making things simpler does not actually help them understand as much as providing challenging tasks and time to "dig in" to a problem.

To help them redefine how to help students learn, I encourage [teachers] to embrace their sense of accomplishment when they solve a challenging task, recognize the pride they feel when they share a unique way to solve a problem, and reflect on what such feelings might mean for a student in their own classroom.

The way this piece is written states an important idea in a concise, clear way. I hope to keep pushing myself and others to ask, watch, and listen instead of show and tell.

NYT: How to Fall in Love With Math

From WolframMathWorld

From yesterday’s New York Times Opinion Pages piece by mathematics professor Manil Suri.

Despite what most people suppose, many profound mathematical ideas don’t require advanced skills to appreciate. One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems.

Think of it this way: you can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music. Math also deserves to be enjoyed for its own sake, without being constantly subjected to the question, “When will I use this?”

Making Mistakes

Here's a TED Radio Hour show featured on NPR entitled "Making Mistakes". Excellent, and very relevant to mathematics education and life in general.

St. Louis STEM Reader

I attended the NSTA STEM Conference in St. Louis last week with several colleagues. It was a very worthwhile trip.

Top Quotes

“Engineering naturally integrates mathematics, science, social studies and language literacy.” - Tamara Moore

“Tell kids they are doing engineering. Use that word. Students, especially girls, tend to go to medical school or major in science in college if they enjoy STEM. Very few choose engineering because they don't know what that means.” - Liz Parry

“If you are not iterating, it is not engineering.” - Ann McMahon

“Is it really STEM or is it just John Dewey?” - Tara Bell

Top Takeaways

1. High-quality STEM education is more about the how than the what. We can’t teach students everything they will need to know. We can help them learn how to learn.

2. The habits of mind related to engineering can be a thread running through all content areas. Empathy, collaboration, failure, iteration, perseverance, social justice, activism, leadership.

3. STEM has the potential to make a real difference for equity and diversity. It has been shown to be an effective way to engage struggling or at risk students. Currently, success in STEM subjects in school and in careers is very skewed toward certain demographic groups, and something needs to be done to change that.

Favorite Presenters

Tara Bell, STEM coach at Washington STEM Academy in Champaign, IL

Bob Goodman (New Jersey Center for Teaching and Learning: Ridgewood, NJ) Demystifying Science with the Progressive Science Initiative (PSI) Equity requires that all high school students learn physics, chemistry, and biology. PSI is a free open-source program that is helping schools achieve that goal.

Liz Parry, Coordinator, STEM Partnership Development, The Engineering Place, College of Engineering, NC State University

Ann P. McMahon Ex-Aerospace engineer. Strategies for integrating STEM with social and emotional learning. Ann recently gave a TEDx talked titled Engineering Empathy (use password tedxgladstone).

Beth Bender, Principal of Gateway STEM High School, St. Louis. A public magnet school, 85% FRL, 55% African american, increasing ELL. There is a lack of awareness about engineering, and it is mostly male. Engineering students give demonstrations for other students during lunch; recruiting for the high school is done with hands-on tasks, live puppies.

Tamara Moore, University of Minnesota. STEM Education Center. Said she would soon be running a pilot program of K-6 STEM integrated units that include a heavy children’s literature component.

Resources

Family Engineering (book and website recommended by Liz Parry)

TeachEngineering

Ideo and the Design Thinking for Educators toolkit

New Jersey Center for Teaching and Learning

eGFI

Gateway STEM High School

The Deep Dive

EnLIST Entrepreneurial Leadership in STEM Teaching and Learning

University of Minnesota STEM Education Center

TryEngineering

National Center for Universal Design on Learning

Sugata Mitra: School in the Cloud

Sugata Mitra recently won the TED Prize for 2013. His wish is to build a virtual school where children learn from each other.

Well, I bumped into this whole thing completely by accident. I used to teach people how to write computer programs in New Delhi, 14 years ago. And right next to where I used to work, there was a slum. And I used to think, how on Earth are those kids ever going to learn to write computer programs? Or should they not? At the same time, we also had lots of parents, rich people, who had computers, and who used to tell me, "You know, my son, I think he's gifted, because he does wonderful things with computers. And my daughter -- oh, surely she is extra-intelligent." And so on. So I suddenly figured that, how come all the rich people are having these extraordinarily gifted children? (Laughter) What did the poor do wrong? I made a hole in the boundary wall of the slum next to my office, and stuck a computer inside it just to see what would happen if I gave a computer to children who never would have one, didn't know any English, didn't know what the Internet was. - Sugata Mitra

The Future of STEM Education

Hear Professor Roni Ellington’s inspirational talk from TEDxBaltimore, 2013.