Repost: We Need More Patient Problem Solvers

I posted this about a year ago, and what I love is colleagues who say, “Hey did you see this?” I’m just really happy they are finding it their own way and sharing with everybody. That’s a very important reason why some educators blog, tweet, and whatnot.  It’s not about whether they read it then or not, it’s that it creates a culture of sharing and continued reflection and growth. I was very happy to see my colleague post this TEDx talk. (By the way, I’m headed to my first TEDx event in a few minutes. All about inquiry,  innovation, and identity through instruction. I can’t wait. I will share via twitter to try an encourage our faculty into the positive and responsible use of social media even more.) Twitter is how I scored the ticket to the event!

I’m not a big fan of text books. Good tools, perhaps and also convenient. Still, it doesn’t make us better teachers. Furthermore, textbooks in many ways dumb down ideas. In math, textbooks tend to encourage the “one way to get to the right answer” kind of questions. I was great at at decoding textbooks and thus was very successful in high school. But how does someone get an A in a subject like physics and have no clue how the world works? Kids really need to understand how things work rather than learn to manipulate formulas. Students have to come up with problems, reason, and have patience.

His wonderful TED talk explains the problem with math in this country today. Making math real is what it’s about. In second grade, when a child asks, “How many more minutes to recess?” resisting the temptation to tell them and saying instead, “There’s the clock,” provides a real need to learn how to do it.

When you dine with friends and its time to split the bill, it’s amazing how often people pull out their calculators to divide and then calculate the tip. Often they are the same people who knew how to use the quadratic equation at one time. Something is not right in the way math is taught. Hopefully, we as educators learn how to it better.

He’s got a good blog:


Day 1 of Flipping the Classroom

There’s the common expression, “Change is hard. You go first.” Well, I’ve been doing a few firsts this past year or so, partly because I decided not to wait. If I think it’s worth experimenting with, I’ll try it. What I’ve learned is that with a few of these things, I might have been better off talking about it, rather than dive right in. As a result, I may have ruffled a few feathers here and there and had to repair a few work relationships. It was actually a good exercise in growth for me and made me a lot more reflective about what I want to do next.

I started this blog, for example to share what I learned at a conference, but decided to keep it going because I actually enjoy it. Because I had no expectation of anyone else blogging, I was oblivious to the fact that some might feel that they would have to share what they learned via a blog. It’s just my way, and I enjoy it. I also started my own classroom website because I couldn’t wait for our school’s official site to have all the features I wanted. It’s worked for me and my students’ parents and that’s really all it boils down to. There are so many ways to communicate, sometimes the purpose dictates they type.

Well, I’m at it again. After only a couple of weeks since the TED talk “Flipping the Classroom” aired, I unleashed Khan Academy upon my second graders. Honestly, the videos are pretty dry and boring for the most part, but the kids love the exercises, the immediate feedback, and the choice. One child decided for homework tonight to head to the geometry section which asks for the area and circumference of circles. He made a few attempts, got all them wrong and decided he’d come back another time. It was very non-threatening. Today was just the first day, we headed to the media lab so they could learn how to login and logoff. And even though I assigned about 10 to 20 minutes, I noticed that many kids were engaged enough to spend much more time on it. I’m actually more excited about the data that might come back after Spring Break. Why? So much of good math pedagogy is not just helping a child develop a concept, but asking the right questions. Knowing what children have mastered, allows you to target your questions more precisely. Of course good teachers who already know their students well do this, but with the added data, who knows.

One interesting unintended consequence occurred. Many of my students have older siblings. So far, I’ve gotten great feedback from parents, but they wanted to know how their older child could sign in. I told them how and that they could sign me up as their coach if they wished. This is a big experiment. I don’t intend to have students using Kahn Academy in class, but only at home. What I will do, is use the data to help inform the way I teach each child. As Kahn put it in his TED talk, “Flipping the Classroom.”

Kahn Academy approaches math in a very linear, sterile manner, but with some of the basic skills under their belt, they may be able to really grapple with project based learning activities which involve plenty of mathematical problems, creativity, and the beauty of math that doesn’t always get to see the light of day the way our math texts are written. Who knows? This is still day one of doing things a little differently. It may just end up being something faddish, which is something I  usually try to avoid, but when I see some potential in how it can help kids, I’ll dive head first. Sign in for yourself and try some of the later differential questions. Do you even remember how to do them? More importantly, do you know why? I’ll keep you apprised of how my little experiment goes.

When Statistics Mean Something

I just got back from a lecture featuring Stephen Dubner (coauthor of Freakonomics and Superfreakonomics). If you’re not familiar with those books, they try to strip away how we ‘feel’ about a particular topic (for example, teachers cheating on standardized test scores, or the hand hygiene of doctors in hospitals) and they address those sorts of topics “with neither fear nor favor, letting numbers speak the truth.” When we think of economics, we usually think of budgets, currency, the stock markets, etc., but what Dubner and his coauthor Levitt do is look at what some call ‘behavioral economics.’

A few things stood out in his talk. One was the reminder about how we are much more able to perceive traits (good and bad) in others than we are at seeing them in ourselves. Another is how hard it is to change human behavior. Finally, when collecting data, how you collect it is really important. Self-reported data, according to Dubner is usually pretty useless (especially if you ask people on a survey to identify themselves – even as a group). He gave an example of a headline that went something like this: “Recent survey shows that favor of nuclear power has declined.” Hmmm, a survey taken right after a tsunami destroying a nuclear reactor. Dubner mentioned how these surveys/polls are everywhere and those are not the kinds of statistics he is attracted to.  He also warned everyone about using incentives to try to change behavior. They have a tendency to backfire. Dubner also made a case for thinking outside the box. “Be a heretic,” he said, “but remember that most were wrong, many were executed, however, those who were right and lived, changed the world.” In terms of education, this talk reminded me of the importance of keeping our rigorous curriculum balanced between learning basic skills and fostering natural curiosity and creativity. For me, It was interesting to compare his talk with that of David Brooks who I saw last week when he was in town promoting his book. I was fascinated by many of the similarities.

Speaking of numbers, if you haven’t visited the site Gapminder by Swedish Statitician Hans Rosling, you really ought to. Can numbers be fascinating? They certainly can, and he does an incredible job on his interactive website which visualizes the data on world development. He’s done numerous TED talks advocating that one of the ways to stave off world population growth is to create wealth in all nations. His latest talk “The Magical Washing Machine” is only about 6 minutes long and well worth watching till the end when he makes his point. It makes me want to take more statistics courses.


Math is a Fine Art

This weekend, I read the book A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Artform by Paul Lockhart with a Foreword by Keith Devlin. It starts with this quote from Antoine de Saint Exupéry:

“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”

In the foreword, Devlin says that “It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen.” He recommends that every policy-maker, educator and parent of a school-aged child with any responsibilities of teaching mathematics should read this book.

We’ve all seen children humming songs without care to the key that it’s in or how that song might be notated. We’ve also seen children take paints and crayons and experiment with the different media before they are taught about line, color, tone and other aspects of art. Yet, in math, Lockhart says we do not allow children enough time to enjoy and play with math and we ought to do so. According to Lockhart, the current K-12 curriculum in almost any textbook series only teaches kids a series of steps in how to solve a particular type of problem, along with some special notation. As Lockhart puts it, the current system “[destroys] a child’s natural curiosity and love of pattern making.” He claims that math is “simple and beautiful.”

He gives an example  of a triangle in a rectangular box.

How much space do you think it takes up? How do you suppose you can find out?

What Lockhart laments about is that without teachers who understand the beauty of math, we don’t allow children to grapple with this problem long enough before we rush to give them the formula 1/2bh.  If we allow kids to ‘play’ with this puzzle, they may actually discover it themselves.


Children will delight when they discover that by drawing a vertical line from the tallest part of the triangle, they will see that they have created two rectangles, and that the area of each triangle part is half of two smaller rectangles.

Today in class we were working with geo-boards and rubber bands. I teach second grade and the main objective was to create a variety of shapes with right angles to measure area (in square units) and perimeter (in units). When one student, who clearly got the concept was done early, I asked her to play with this puzzle for a while. I built a 3X2 rectangle and a triangle inside it. I asked her to think about how she might find the area of the triangle. After about five minutes, she lit up and with much excitement explained that she could divide the shape into two smaller rectangles and found the area of each one to be half of the rectangle. I asked her to try with a different rectangle and triangle, and her response was instant. What I didn’t give her was the traditional formula. She had basically discovered it on her own without realizing it.

Remember, I teach second grade, so this was exciting for me too. With the other children, some were excited in discovering the area of a rectangle to be the base multiplied by its height. This too was a discovery for these students and the joy of math was evident. I also did not provide a formula for them even after their discovery.

There are many critics to Lockhart’s point of view that it took centuries to arrive at many mathematical theories. He would argue that math which is rich “has been reduced to a sterile set of facts to be memorized and procedures to be followed. They are given the formula: Area of a triangle = 1/2 b h and are “asked to apply it over and over in exercises…By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject…By concentrating on what and leaving out why, mathematics is reduced to an empty shell.”

Finally, here’s one last quote: “Math is not about following directions, it’s about making new directions.”

In the next few weeks, our school is about to go through a selection process for a math curriculum. While I empathize with Lockhart, there also needs to be a balance. I hope that when it comes time to discuss and debate the pros and cons of each math curriculum, we keep an open mind to process, discovery, and relevance, rather than what’s easiest to implement or what is an efficient way in transmitting a set of rules for practice and compliance.

Anyway, the book is an easy read, very compelling, and makes you think. Whether or not you agree completely with the author’s point of view.


CCL – Are Roman Numerals Obsolete

I’m not sure who still uses Roman numerals. I learned them when I was in elementary school. I remember seeing them at the end of movie credits growing up, but that’s long gone. They’re still on the clock tower that houses Big Ben in London, but I suppose one could tell the time on a clock without numbers anyway. I haven’t even worn a watch in the past 5 years. Where else are Roman numerals used. I also thought of using the word sestercentennial in the title of this post, but it’s my 250th post, not the 250th anniversary of my first post.

We’re going to start learning about Ancient Egypt in my class, and I was trying to figure out what exactly the objective would be when we learned about the Ancient Egyptian numerals. It led me to think about when a certain tradition or practice ends and is replaced by something else, in essence, change.

The world is changing rapidly, but is education keeping up? We just had a week off for mid-winter break, and it’s given me time to pause and reflect about a lot of things. This week also gave me the time to figure out how twitter worked, and how it could help me stay apprised and connected of what was going on in the world without cluttering my inbox or remembering to check the various blogs I like. With twitter, I was able to follow two of my colleagues (since they tweeted) among many others I didn’t know who were eager to share what they learned at the NAIS conference last week. Even though I was unable to attend this conference in person as I did last year, being able to follow it gave me the renewed energy and optimism that I had after last year’s event. I don’t know why I hesitated to use twitter. It took thousands of years for Ancient Egyptian numerals to be replaced, but these days, companies that were once on the cutting edge seem to fade before one even figures out how to use it.

The week before our break, our Head of School gave us an article to read titled, “Why a School Doesn’t Run – Or Change – Like a Business.” Written in 2000, many of its points hold true a decade later. The article mentions the difficulty of change for many reasons. The author mentions that while teaching “benefits from regular refreshers and occasional overhauls, it doesn’t demand the kind of continuous updating that, say, law or medicine or high technology do.” A decade ago, I would have agreed with him on this note, but change in education, however slow it may seem to some is inevitable. The difficulties still remain, and school leaders must approach change with clarity, focus, and continuity while respecting educators’ motivation and innovation. The change has to be clear and articulated well. Educators can support change if the ‘why, what, and how’ are addressed. They can support change if it doesn’t mean “do more,” but instead means try doing things differently. Finally, a very important thing he mentions is that educators need to know what won’t change, so they can rely on some continuity.

If the objective of learning about Roman Numerals provides kids with different ways of thinking about numbers, it’s good enough for me and should still be taught. I’m sure our Latin teacher can give me at least X number of other reasons why.

I was going to write about a few articles I read today: George Will had an interesting column about Teach for America. Daniel Pink had a column about detesting the question, “What’s your passion?” Both David Brooks and Paul Krugman have chimed in on what’s going on in Wisconsin as have many others, so I’ll spare boring you with my two cents. You can read the articles by clicking on the links. There’s a lot to write about, but the Oscars are about to start, and I’m sure I’ll have something to else to say soon enough.


Math is Not Linear

Math is often taught in a linear way, and each new year, students revisit topics and build upon them with the ultimate goal at the end of high-school being calculus. Geometry plays a big part, and I remember all kinds of min/max calculus problems involving solid shapes. But what strikes me, is how little the topic of statistics is taught in the K-12 setting. Many who graduate from college are not going to use much of the higher-leveled math in their careers, but they will be exposed to a lot of data – and they should be able to recognize when this data gets misrepresented in the news. When someone reads something that begins, “research says…,” do they understand those numbers and what they mean? Can they recognize a graph when it’s skewed to favor the author’s point of view? Infographics are some of the best visual ways of trying to convey data, but some of them are simply just beautiful art and actually quite misleading. Some, of course, are brilliant.

This week, the New York Times had an article titled Teacher’s Colleges Upset by Plan to Rate Them. The U.S. News and World Report has told many colleges to comply or they may simply get an F rating. Even though many of the colleges (including Columbia, Harvard, and Michigan State) have all stated that the measuring systems are flawed.

This week’s New Yorker, has a piece by Malcolm Gladwell also criticizing the U.S. News and World Report for its general ranking of schools. Honestly, how can you compare apples to oranges? And yet they do. When you factor many of the measurements: endowment, scholarship, tuition, graduation rate, and so on, how do you give each of the measurements equal weighting?

And yet, for some reason, people would rather not think about all the different numbers. They want a nice little number that they can use to order the schools. The U.S. News and World Report admittedly says that the way they weigh its metrics aren’t scientific in any way. Obviously, not a reliable source, yet their publication sales continue to rise.

When measuring teacher effectiveness, the same kinds of things must be considered. How much do you weigh experience, achievement in test scores, degrees earned, anecdotal reputation, etc.? The test scores are a tricky one too. If you’re not comparing the same group of kids, you’re comparing apples to oranges. Furthermore, if you’re comparing one school to other schools, can you aggregate data such as financial aid, diversity, ratio of teachers to students, test scores, and so on. Again, we’re comparing watermelons to tomatoes. Local magazines love to compare schools in the city using data that’s usually dated and not very useful. They also like to come up with an aggregate score and rank schools.

I think some of the metrics measured are legitimate and should be looked at closely, but to give each of those an arbitrary weighting so as to come up with a single number for a ranking is not good math.

These past two weeks in second grade math, the children have been making paper quilts. While not a lot of statistics were involved, the children were employing their knowledge of measurement, area, perimeter, addition, multiplication, fractions, estimation, and problem solving. Not to mention that the quilts also lent themselves nicely to social studies themes like the Underground Railroad as well as integrated nicely with story telling.

So many math text books present fractions in a chapter midway through the year. In reality, fractions are everywhere. They’re in music, quilts, baking, Lego, and so on. Why wait to introduce the concept of ‘half’ midway through the year, when you can use it all year long? I guess I want my students to see the connections between fractions, and measurement, and the operations they need to use to solve problems involving those connections. Fractions aren’t something that exist in a vacuum. They’re part of these children’s world. It’s everywhere – and to wait until chapter 8 (or whatever it is in the book you’re using) is just wrong.

I know I meandered from the topic of statistics using fractions in second grade, but as adults, statistics are all around us. The data is often analyzed by ‘experts’, but we need to be able to do better than take someone else’s word for it or buy into the simplicity of an arbitrary ranking system. Malcolm Gladwell’s article in the New Yorker may seem obvious to many, but many are reading U.S. Weekly News!

Integrating Math and Literature

This month’s Teacher Children Mathematics journal had a great lesson involving the Caldecott winning picture book, Jumanji. Those familiar with the title know that it’s about a pair of children who find a board game and begin playing it by rolling the dice. Things get out of hand and the only way to end the game is by rolling a 12. Then (to a lot of groans), I slammed the book shut and asked the children that we were going to do some math and I would read the end aftewards. I began by asking them  how likely they thought the chance would be that a 12 would be rolled. There was a large range of answers. I then gave each child a pair of dice and asked them to roll it ten times and to record their answers. Finally, we took all their data and filled in a graph on the board. It looked like a nice rolling hill. I then asked the children why they thought this pattern emerged and eventually they started to say that the were more combinations of numbers to add to make 7 whereas there was only one way to get 12. We then read the end of the book and it was a great way for children to experience the concept of probability and how it might affect their lives. Would it be better to build houses on your strip of Monopoly when someone was 1 place away or 6 places away? Why?

Unfortunately, many textbooks are so linear, teaching one concept at a time, they don’t leave room for integration of other math concepts or even literature connections for fun, engaging, lessons like these that ask kids to discover the why behind the math. Many textbook series have ‘literature’ connections by producing their own children’s books. None to my knowledge have measured up to Chris Van Allsburg’s Jumanji – a true classic.