Illuminating the Beauty, Humanity, Intrigue in Mathematics

Illuminated Mathematics
Illuminated Mathematics Logo

I recently blogged from the 2011 US Innovative Education Forum (IEF) sponsored by Microsoft Partners in Learning. This is part of a series of IEF guest posts. For more, click my IEF tag. ~  Peter

More than 700 teachers, school leaders, education leaders, and government officials from more than 70 countries attended this year’s 2011 Partners in Learning Global Forum – an action-packed week of education workshops, inspiring networking events, awards, and announcements by Microsoft. Eighteen recipients of the Global Forum Educator Awards were announced at the event. This year’s winners were selected from more than 115 projects, narrowed from more than 200,000 applicants.

The winners in ”Knowledge Building and Critical Thinking” were High Tech High’s Margaret Noble and David Stahnke. “Illuminated Mathematics” is a curated multimedia exhibition produced by the 12th grade class of 2011. Students in Margaret Noble’s digital art class and David Stahnke’s math class were asked to find the beauty, humanity and intrigue behind math in history, philosophy and the applied arts. The goal was to promote math awareness through art, media and design. The event was hosted at the Sushi Performance and Visual Art Center on December 16th, 2010. Projects developed into an array of math abstractions and celebrations in the mediums of sound, video, animation, photography and interactive installation.

Illuminated Mathematics: Website | Project intro | Research Topics | Final Rubrics

Exhibition 2

~ A guest post written by Dave Stahnke ~ High Tech High Media Arts ~

“Everyone, open your books to chapter 7 section 2 as we will be learning how to factor degree 3 polynomials.”

I can imagine this statement being said, in some fashion, within the vast majority of high school math classrooms across our seemingly broken educational system. Almost all of us have at some point taught something that was completely irrelevant to the lives of our students. And we knew it!

Nobody has ever come up to me on the street and asked for help with factoring, or called me late at night, unable to sleep, because they were curious as to why the square root of two is an irrational number.

The fact is that nobody has ever come up to me on the street and asked for help with factoring, or called me late at night, unable to sleep, because they were curious as to why the square root of two is an irrational number. It is unfortunate that this doesn’t happen, but I would be kidding myself if I thought these were genuine student concerns within the realm of what we call “life.” I think it is time for us as teachers to be honest about what we teach, and to question why every student needs to know the entire breadth of standards associated with a particular subject.

Deep vs. Wide

There was a study published recently in Science Education (2009) that made a comparison between teachers who “sprinted” to cover all of the standards with teachers who slowed down and went deeper into the material. The students who “sprinted” ended up scoring higher on the standardized test due to covering more material. But the students who learned through the slower, in-depth approach earned higher grades in their college classes.

Like any great symphony, mathematics represents a pinnacle of human creativity. We teach math to enrich the lives of our students in a way akin to reading poetry or composing music

Is our goal to have students performing better on standardized tests or to be prepared for what they are going to encounter in college and life? The ideal would be that they would be prepared for both. So the questions become, what do we want to leave the students with? How are we going to prepare them for the real world? What do we want them to learn about themselves? And how do we do it? To clear the air, I don’t believe that students are taking my calculus class because they need help doubling a recipe or balancing their checkbook. I believe it is because we want to expose students to the poetry of numbers, to have a new outlook on how to solve problems, to be able to think outside of the box, and to see how the unbreakable human spirit has conquered problems that once mystified the greatest of thinkers. Like any great symphony, mathematics represents a pinnacle of human creativity. We teach math to enrich the lives of our students in a way akin to reading poetry or composing music.

Bringing Math to Life

This year I wanted to do something big that would change the perception of mathematics for my students and the surrounding community… It was time for math to become art and art to become math.

This year I wanted to do something big that would change the perception of mathematics for my students and the surrounding community. My goal was to create a math exhibition that would allow students to showcase their depth of understanding in a creative way. I wanted nothing to do with the poster-board type of science fair displays. I wanted math to come alive through the work of my students. It was time for math to become art and art to become math.

In order to pull this off it was clear that I was going to need help. After all, having the students for only an hour a day seemed to be great limitation to this type of creativity. I enlisted the help of Margret Noble, a sound artist, multi-media teacher, colleague, and friend. I also got help from as many math/physics friends as I could. I contacted about thirty people. Fifteen were willing to act as mentors, spending time meeting with one or more groups of students and/or corresponding through e-mail. All of the mentors were physics Ph.D. students, or had their PhD and were working in labs or as engineers. The students found the mentors to be a great resource. As one student said, “I got a lot of positive feedback from adults. They helped me understand a very complicated topic in a very simple way.”

Exhibition 1

Student Voice and Choice

Margaret and I envisioned mixing multimedia with mathematics by having students create video, sound, photography, and mixed media installations that explored math-related topics. We started the project by creating a list of 50 topics for the students to pick from, though they were not restricted to the list. Once the students had selected a topic we had them brainstorm possible creative ways of expressing it (i.e. their product). Each student also completed a research paper on their topic and gave a power point pre-production oral presentation to explain their topic to the rest of the class.

Along the way, students participated in four in-class critiques of their products, with opportunities to revise after each one. For each critique, students displayed their work on the large screen and the rest of the class would give kind, specific, and helpful feedback. These peer critiques were key to ensuring that students produced beautiful products. As they pushed each other’s creativity and offered new ideas, students’ projects evolved into a variety of forms:

A video with animated fractals, another on chaos theory, an artistic representation of tessellations, a flash video on relativity, music produced using Pythagorean scales, photography that displayed entropy, Pi and mental illness in mathematics, a beautiful silent film which used cryptography to crack a love letter, photography and video of the golden ratio, a video/sound installation on algorithmic compositions using Markov chains, a Leonardo da Vinci model airplane explaining the physics of flight, a comical rap on the life of Pythagoras, and many more.

A student who has struggled with math in the past noted that these peer critiques were instrumental in helping students reach their goals:

During the first two critiques I was a little scared because I didn’t think that our project was good enough and had thoughts in my head saying it could be better. But after the second critique I caught fire. I had many more ideas for our project and I was motivated to make it better. On our last critique a lot of good things were said about our project and it felt good knowing that we were that much closer to having a completed senior project.

Student choice also played a critical role. Contrary to what one might assume, having students choose their own topics to explore created some of the most rigorous and authentic student work I have ever seen. Not only did the students have choice in what they were learning, they also chose how they wanted to display it. Furthermore, as the project work progressed, I realized that once the students’ buy in was there, the usual achievement gap between students almost entirely disappeared. This same student found that this project gave him something to be proud of:

I honestly am proud of my project, because our animation came a long way from what we had in the beginning. A lot of hours were put in, learning Adobe After Effects, perfecting the animation, making the concept of infinite monkey theorem as simple as possible, and staying during lunch and after school so we could finish up and meet the deadlines.

Exhibition

exhibition 3

Students exhibited their final work on a Thursday evening at Sushi Contemporary Performance and Visual Arts, a gallery and performance space in downtown San Diego. The venue had professional lighting and ample wall space for multiple projections. It took us two days to set up the exhibition, hanging photos, placing installations, and installing projectors throughout the space. When the lights were turned down and the student work was illuminated it seemed almost magical. Prior to the exhibition, we had reached out to CNBC (video), Voice of San Diego, and City Beat Magazine to help promote the show. The most common phrase I heard that evening from the parents, media, and other visitors was “I can’t believe that high school students did this!”

As an educator, this experience proved to me that mathematics can not only be enjoyable for students, it can be downright memorable. This was possible through giving student choice and by letting them explore math through their own creative personalities. In the words of my teaching partner, Margaret Noble, “This project worked because math moved from the abstract realm into the tangible. Numbers and concepts became people, culture, history and philosophy that students could illuminate to the public.”

Or, as one student said, “It definitely widened my view of math. At first I thought math was only useful to scientists and mathematicians, but this project showed me that math is everywhere.” What more could a math teacher want?

Reference
Schwartz, M., Sadler, P., Sonnert, G. & Tai, R. (September, 2009). Depth versus breadth: How content coverage in high school science courses relates to later success in college science coursework. Science Education, 93, 5, 798-826.

Image credits: Margaret Noble and David Stahnke

Solve This Problem, You’ll Learn the Skills Along the Way

Wisconsin STEM Summit I’m in the Wisconsin Dells today to deliver a four-hour training session for CESA 6. It’s entitled “21st Century Skills in Action: Project Based Learning in the STEM Classroom.”  We’ll be using a Turning Point ARS and lots of activities so that participants experience the why, what, and how of PBL in the STEM curriculum.

Students explore their world with an expectation of choice and control that redefines traditional notions of learning and literacy. Educators are discovering that they can motivate students with a PBL approach that engages their students with the opportunity to behave like STEM professionals while solving real-world problems.

I was pleased to read an interesting piece in the NY Times on yesterday’s flight. “Computer Studies Made Cool, on Film and Now on Campus” (6/11/11). While the focus is on the growing popularity of computer science, it make a strong case for the project based approach to learning. 

The new curriculums emphasize the breadth of careers that use computer science, as diverse as finance and linguistics, and the practical results of engineering, like iPhone apps, Pixar films and robots, a world away from the more theory-oriented curriculums of the past.

The old-fashioned way of computer science is, ‘We’re going to teach you a bunch of stuff that is fundamental and will be long-lasting but we won’t tell you how it’s applied,’ ” said Michael Zyda, director of the University of Southern California’s GamePipe Laboratory, a new games program in the computer science major. With the rejuvenated classes, freshman enrollment in computer science at the university grew to 120 last year, from 25 in 2006. …

To hook students, Yale computer science professors are offering freshman seminars with no prerequisites, like one on computer graphics, in which students learn the technical underpinnings of a Pixar movie.
“Historically this department has been very theory-oriented, but in the last few years, we’re broadening the curriculum,” said Julie Dorsey, a professor.

She also started a new major, computing and the arts, which combines computer science with art, theater or music to teach students how to scan and restore paintings or design theater sets.

Professors stress that concentrating on the practical applications of computer science does not mean teaching vocational skills like programming languages, which change rapidly. Instead, it means guiding students to tackle real-world problems and learn skills and theorems along the way.

“Once people are kind of subversively exposed to it, it’s not someone telling you, ‘You should program because you can be an engineer and do this in the future,’ ” said Ms. Fong, the Yale student. “It’s, ‘Solve this problem, build this thing and make this robot go from Point A to Point B,’ and you gain the skill set associated with it.” With other students, she has already founded a Web start-up, the Closer Grocer, which delivers groceries to dorms.

Don’t Teach Them Facts – Let Student Discover Patterns

4794114114_dd895561bf Develop a classification system – analyze patterns, create a schema, evaluate where specific elements belong. Sounds like a very sophisticated exercise. Not really, young toddlers do it all the time – sorting out their toys and household stuff into groups of their own design. They may not be able to explain their thinking, but hand them another item and watch them purposely place it into one of their groups. They have designed a system.

Humans experience the world in patterns, continually trying to answer the question – what is this? Remembering where we’ve encountered things before and assessing new items for their similarities and differences. Someone once asked Picasso if it was difficult to draw a face. His reply, “it’s difficult not to draw one.” We see “faces” everywhere.

Filling out a Venn diagram isn’t analysis – it’s information filing.

It’s unfortunate that student don’t get to use their innate perceptual skills more often in the classroom. Instead of discovering patterns on their own, students are “taught” to memorize patterns developed by someone else. Rather than do the messy work of having to figure out what’s going on and how to group what they see – students are saddled with graphic organizers which take all the thinking out of the exercise. Filling out a Venn diagram isn’t analysis – it’s information filing. Instead of being given a variety of math problems to solve that require different problem-solving strategies, students are taught a specific  process then given ten versions of the same problem to solve for homework. No pattern recognition required here – all they have to do is simply keep applying the same procedures to new data sets. Isn’t that what spreadsheets are for?

Continue reading “Don’t Teach Them Facts – Let Student Discover Patterns”

Eight Look 4s When Observing a Classroom: What the Teacher Teaches – What the Students Learn

School desk I rarely quote at length from a blog or news article, but I think this time I’ll break my rule. I first met Mel Riddile a few years ago when we co-presented at a conference. Since then we stay in touch via Twitter and by following each others’ blogs. Mel blogs on policy and practice for NAASP at The Principal Difference and tweets at @PrincipalDiff

His recent blog post “Tests: Will they improve learning?” is a thoughtful response to the recent Science Journal study that concluded that “practicing retrieval produces greater gains in meaningful learning than elaborative studying.”

Mel does a good job of putting the research into the context of the classroom, but the segment I wanted to quote is his closing section –  ”Look 4s for School Leaders.” It’s a succinct guide for principals, instructional leaders and can be used as reflective prompts by teachers.  Put these in your toolkit and don’t forget they are all critical aspects to Common Core mastery.

(Note: They’re also a great companion to my post “Observing a Classroom? Watch the Students, Not the Teacher“) 

Look 4s for School Leaders

    • Closure and Learning – The focus of instruction is not what teacher teaches but what the students learn. The close of every lesson should focus on what the learner has learned not what the teacher has taught. The question is how does the teacher know that the students have learned and mastered the lesson unless there is some type of formative assessment–quiz, test, or activity.
    • Remembering – The only evidence of learning is remembering. When observing a lesson ask yourself how does the teacher know that students will remember what they just learned?
    • Checks for Understanding – Teachers should pause frequently during a lesson to check for understanding. How frequently? As a rule of thumb, teachers should check students understanding approximately every fifteen minutes, which approximates the attention span of the average adolescent. According to the Science study, one of the most effective checks for understanding is the quiz used as a formative assessment. Teachers can pause and ask students to write a summary or take a brief quiz on what they just learned. Immediately re-teaching a concept to a classmate may also be used to test practice retrieval.
    • Timing is critical - When it comes to recall, tomorrow is too late. Teachers need to check for student understanding before students leave the classroom each day.
    • Feedback – “Feedback is the breakfast of champions.” Unless students practice recall (retrieval) and get immediate feedback they will not remember.
    • Defined Instructional Practices – Some students absolutely need a highly structured classroom room environment characterized by identifiable instructional practices, smaller units of instruction, more frequent assessments, coupled with frequent and immediate feedback. However, students who can function equally as well in low or highly structured classrooms are not penalized in any way by the use of structure. In other words, when in doubt, use a more structured approach.
    • Formative Assessments – How often should students be assessed? How frequently students are assessed or asked to practice retrieval depends on their familiarity with the content and the student’s level of mastery. When students are introduced to new content or when they are struggling with a particular concept, they should be assessed more frequently. For example, the skills of proficient and advanced readers need only be assessed annually, while students reading at the basic level or below basic need to be assessed regularly. Frequent assessments mean more feedback. A quiz or summary essay at the close of a lesson will do more for student recall than extensive homework assignments.
    • Mapping – Instructional strategies like “concept mapping” are effective, but they work better if they are used as part of “practice retrieval.” The act of creating a “concept map” in and of itself does not improve learning unless the student makes use of the map as a part of the “practice retrieval” process. Teachers should show students how to use the concept maps to review for a test and not assume that the students know how to do so.

Image credit flickr/mecredis

Forget the Graphic Organizers, Does Taking Tests Help You Learn?

Learning-through-testing This should stir things up!

A New York Times story "Test-Taking Cements Knowledge Better Than Studying, Researchers Say" (January 21, 2011) reports…

 

Graph: NY Times

 

"Taking a test is not just a passive mechanism for assessing how much people know, according to new research. It actually helps people learn, and it works better than a number of other studying techniques.

The research, published online Thursday in the journal Science, found that students who read a passage, then took a test asking them to recall what they had read, retained about 50 percent more of the information a week later than students who used two other methods.

One of those methods – repeatedly studying the material – is familiar to legions of students who cram before exams. The other – having students draw detailed diagrams documenting what they are learning – is prized by many teachers because it forces students to make connections among facts.

These other methods not only are popular, the researchers reported; they also seem to give students the illusion that they know material better than they do."  More