If I Had a Hammer

Peter, Paul & Mary

I just read a great blog post by Tracy Zager. Her blog is called Becoming the Math Teacher You Wish You’d Had.

It is about getting students out of the normal (sometimes boring) routines they’ve become accustomed to in math class.

Tracy describes being in a workshop with Brian Hopkins and doing a bunch of math problems in groups. Her group solved a problem and then there was discussion and learning about the mathematics that best fit that problem. When Brian posed a seemingly similar problem, Tracy assumed they would be applying that same mathematical construct to the new problem. That was not the case, which surprised Tracy.

“...Brian disrupted the predictable, pitter-pat routine of math class...

What I see in schools is we cue kids to know what tool to use. If we’re two weeks into a unit on fractions and we give them a story problem, the kids figure fractions are involved. If the name of the chapter is “Multiplying Two-Digit Numbers” and it’s written on the bottom of the worksheet, the kids are going to assume they should multiply some 2-digit numbers. If we’ve written an objective about linear equations on the board, kids figure the answer is going to involve linear equations. If my new tool is the hammer that divides fractions, I’m going to use that hammer until my teacher tells me it’s time to switch hammers.”

As educators, we are often frustrated by our students’ lack of ability to make sense of and solve problems (the first Common Core math practice standard). Yet, are we giving students experiences in math class that help or hinder their ability to solve problems?

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.

Summer Gems

Here are two articles you shouldn’t miss this summer, and they make a nice pair.

“Why Do Americans Stink at Math?” by Elizabeth Green, July 23, New York Times, and “Most Math Problems Do Not Have a Unique Right Answer” by Keith Devlin, August 1, 2014, Devlin’s Angle.

I’ve come to the conclusion that our old ideas and stereotypes about math and education are hard to shake. It really means reading articles like these as they come out every month or so, in order to truly change our minds and our practice.

To whet your appetite, here are a few quotes:

One of the first math classes he observed gave him such a jolt that he assumed there must have been some kind of mistake. The class looked exactly like his own memories of school. “I thought, Well, that’s only this class,” Takahashi said. But the next class looked like the first, and so did the next and the one after that. The Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them. - from “Why Do Americans Stink at Math?”

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find "the right answer." No one thought for one moment that there could be such a thing. - from “Most Math Problems Do Not Have a Unique Right Answer”

The last summer gem is this youcubed video about using Number Talks in math class. How would you solve 18 x 5 mentally?

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.”

The Importance of Curiosity

One student was curious about tornadoes. Photo from NPR.

I look forward to getting the Marshall Memo via email. This week I found myself reading an Edutopia article entitled “Curious Homework: An Inquiry Project for Students and Parents” by Suzie Boss. It emphasizes the importance of curiosity in life and education, and laments the fact that students don’t always find a way to express their callings and passions in the context of school.

In the article, we learn about a teacher named Scot Hoffman who started his work on curiosity by bringing in folks with interesting careers or passions - a mountaineer, a doctor - and interviewing them in front of the students. Then it was the students’ turn to explore their own sense of curiosity. It would be fun to try this in the Burlington community, let students choose and interview folks, and partner with community members in addition to parents in order to pursue their own interests.

Vermont Math Leadership Council

the view from my porch

Vermont has a new education organization - Vermont Math Leadership Council - thanks to Julie Conrad, Tracy Watterson and others for getting it off the ground. Thanks to me we have a blog with the easy URL vermontmathleadership.org. Thanks to Tim Whiteford, there is a bunch of great posts on the blog! Check it out. We’d like to get more material from more voices on this blog and increase our readership. Anyone can join the organization. Our next meeting is January 29, 2014.

I was about to post something here on my blog about the new Institute of Education Sciences Educator’s Practice Guide called Teaching Math to Young Children. But then I saw that Tim beat me to it by posting about the very same thing on the VMLC blog. So, read his post here.

What I like about this publication is that it helps both rookies and veterans understand how best to work with young children and math. There are clear, specific recommendations that are easy to follow and a selection of easily usable games and ideas that illustrate the concept.

Whereas we often encounter activities or lessons and need to figure out what math we’re really going for, this resource communicates the math goals clearly and first, then hands us the activity to use.

Stereotypes about Math

In our new technological world, employers do not need people who can calculate correctly or fast, they need people who can reason about approaches, estimate and verify results, produce and interpret different powerful representations, and connect with other people’s mathematical ideas.

The Stereotypes About Math That Hold Americans Back, By Jo Boaler. The Atlantic, November 12, 2013. 

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?”

Shrub

Edible Wasatch 2013, photo by David Vogel

My brother, David Vogel, and his partner, Rachel Hodson, are the owners and editors of a magazine called Edible Wasatch. It’s about local food in Utah, but I can’t wait for it to arrive in my mailbox in Vermont each season. It’s always full of gorgeous photos, great articles, and interesting recipes.

Shrub is the name of a syrup made from fruit, sugar and vinegar. I found it in the summer 2013 issue of Edible Wasatch I just received. You can add a small amount of shrub to seltzer water to make tasty, homemade soda kids will love.

Teachers and parents: Try making shrub with your kids. With this activity, you can combine wellness, local food appreciation, culinary skills, and math. What could be better? Then reward yourself by using some of the kids’ leftover shrub to mix up a delicious summer cocktail.

Strawberries have been so perfect around here lately, I think I would make shrub with those. However, you can use any fresh, ripe berries or other fruits. I have raspberries growing at my house now, so I might combine those with some local strawberries.

Here’s the recipe, straight from Edible Wasatch:

Basic Shrub

One part fruit or berries

One part sugar

One part vinegar

Wash and cut up the fruit and put into a bowl or a jar large enough to contain all of your ingredients. Add the sugar and stir or mash it with the fruit until it’s evenly distributed. Cover with cheese cloth or a kitchen towel to protect it from fruit flies and leave it to macerate - at room temperature for a few hours or overnight - until the fruit has broken down and become juicy.

Add the vinegar to the fruit and sugar mixture - apple cider vinegar or any kind of wine vinegar, not distilled vinegar. Stir or shake until the sugar is completely dissolved. Strain off the liquid and discard the solids. Store your shrub in a bottle in the refrigerator and enjoy for up to a year.

See their original article and recipe here.

Ideas for teachers and parents

Let kids do all the work, including deciding which fruits or berries are in season and would work well, picking or buying those, making the shrub, and making the soda. They could even sell it in the neighborhood instead of lemonade and hand out the recipe to customers.

Here’s some math. Ask kids things like...

• What does “part” mean in this recipe?
• How do we know how much of an ingredient to use when “one part” is called for?
• How is this measurement different than other kinds of measurements we’ve seen in recipes (teaspoons, cups, etc.)?
• How should we decide how much shrub to make?
• How much fruit, sugar, and vinegar would we need if we wanted to double the amount next time we make it?
• What would we do if we wanted to try making shrub that was less sweet?
• Have you ever seen another recipe that called for “parts”? How could you figure out how much you need for “parts” when you know the total amount of the recipe you desire?

Other possible questions

• If we use a few teaspoons of shrub to make homemade soda, about how much sugar will one glass of soda contain?
• How does our homemade soda compare to a commercial soda? Amount of sugar? Type of sugar? Other ingredients? Taste? Color? Cost? Environmental and economic impact?
• What does “macerate” mean?
• What other ways could we use shrub besides making soda?
• Why can shrub stay in the fridge for so long without spoiling?
• What questions do you have?

What Google has learned about hiring

A few days ago, the New York Times published an interview by Adam Bryant entitled “In Head-Hunting, Big Data May Not Be Such a Big Deal”. Bryant interviewed Lazslo Bock, a Google executive, about what Google has learned about what works and what doesn’t when it comes to hiring new employees.

Google has the resources to crunch numbers on employee performance or success and match that to hiring criteria. Most of the rest of us don’t, so this is worth reading. Here are a few things that stood out for me, as someone who participates on hiring committees and tends to wonder how good I am at judging applicants.

“Years ago, we did a study to determine whether anyone at Google is particularly good at hiring. We looked at tens of thousands of interviews, and everyone who had done the interviews and what they scored the candidate, and how that person ultimately performed in their job. We found zero relationship.”

“On the hiring side, we found that brainteasers are a complete waste of time...What works well are structured behavioral interviews, where you have a consistent rubric for how you assess people, rather than having each interviewer just make stuff up...[also] where you’re not giving someone a hypothetical, but you’re starting with a question like, “Give me an example of a time when you solved an analytically difficult problem.”

“Twice a year, anybody who has a manager is surveyed on the manager’s qualities...We then share that with the manager, and we track improvement across the whole company. Over the last three years, we’ve significantly improved the quality of people management at Google, measured by how happy people are with their managers...for most [managers], just knowing that information causes them to change their conduct.”

Lastly, this is my favorite bit from the article:

“...G.P.A.’s are worthless as a criteria for hiring, and test scores are worthless...We found that they don’t predict anything...the proportion of people without any college education at Google has increased over time...academic environments are artificial environments. People who succeed there are...conditioned to succeed in that environment. One of my own frustrations when I was in college and grad school is that you knew the professor was looking for a specific answer...You want people who like figuring out stuff where there is no obvious answer.”

These insights have far-reaching implications for teachers, not just members of hiring committees.

Estimation Jars

Today I’m creating Estimation Jars for Math Night at John J. Flynn Elementary School. This is always a highlight at Math Night and it’s fun to watch kids at work on their estimate. I have one jar full of Rolos (there is something appealing about the gold wrapping), a smaller jar containing Skittles, and a very small jar with unpopped popcorn.

This year, I plan to accept all estimates within a reasonable range for each jar and then randomly draw a winner. That’s different than past Math Nights when the person who estimated the number closest to the actual number won the jar. I am hoping this helps promote a more correct understanding of estimation, with the goal being to produce a reasonable estimate, not to land on or closest to the exact number.

It is a good idea to have a few jars containing different sized objects so students can think about how the relative size of the objects affects the number that fit in a space. Ongoing math explorations can be done using estimation jars outside of an event like Math Night. Here is an excellent write-up and video about how classroom teachers can use estimation jars with their students, with a focus on the all-important concepts of doubling and halving.

Salt + Fat

The Extraordinary Science of Addictive Junk Food by Michael Moss, Published Feb. 20, 2013 in the New York Times Sunday Magazine.

I enjoyed this article in last weekend’s New York Times. In it, you will find important information about the workings of the food industry and will have a better understanding of the obesity epidemic. Also, I couldn’t help but notice how mathematics is such an important part of the story. When you get to the part about the food guru Howard Moskowitz, (“I’ve optimized soups, I’ve optimized pizzas. I’ve optimized salad dressings and pickles. In this field, I’m a game changer.”) notice his ability to use statistical analysis to take bucketfuls of taste test data and turn it into usable information. This is why he has become a legend in his field.

Marjorie on Math Munch

There’s a great blog called Math Munch. I mentioned it in a previous post, but I have to revisit it here. I’ve signed up to receive new posts from Math Munch via email, something I don’t usually do. But I’ve found that this is worth the space in my inbox. Each post has a few different topics, everything from cool paper folding projects to high quality videos and online games. You must check it out!

The current Math Munch post includes a piece about a woman named Marjorie Rice who became interested in tesselations and discovered some new ones on her own. These pictures are from her website. What’s great about this story is that a) she’s a woman, b) she didn’t have a math background beyond high school, and c) she was curious, worked at something, and found enjoyment and success.

I won’t go on and on here, but the other two stories with Marjorie’s in this issue of Math Munch are equally worth reading.

The pictures I’ve included here are some of Marjorie’s tesselations.

Radiolab Numbers

Radiolab is so cool. It’s a public radio show you can listen to on your public radio station or online. My dad recently told me about a Radiolab show called Numbers. It originally aired in November 2009. Each show is an hour long and has several segments on a single topic. If you listen to the Numbers show, you’ll first hear a Johnny Cash song, then learn about the innate number sense of infants, and move on to other fascinating topics like Benford’s Law, a surprising observation about the first digits of numbers, and a forensic accountant who uses it to investigate fraud, and on and on. The show is a very human take on numbers. I’m sure you’ll enjoy it.

Is Welding a STEM job?

New York Public Library Digital Gallery

Victor Prussack, Burlington’s Coordinator of Magnet Schools, found this Thomas Friedman article in the New York Times over the weekend and was kind enough to send it to me.

The piece begins with a profile of a sheet metal business owner from Minnesota named Traci Tapani. She can’t find enough skilled welders.

“Many years ago, people learned to weld in a high school shop class or in a family business or farm, and they came up through the ranks and capped out at a certain skill level. They did not know the science behind welding,” so could not meet the new standards of the U.S. military and aerospace industry.

“They could make beautiful welds,” she said, “but they did not understand metallurgy, modern cleaning and brushing techniques” and how different metals and gases, pressures and temperatures had to be combined.

Welding “is a $20-an-hour job with health care, paid vacations and full benefits,” said Tapani, but “you have to have science and math. I can’t think of any job in my sheet metal fabrication company where math is not important. If you work in a manufacturing facility, you use math every day; you need to compute angles and understand what happens to a piece of metal when it’s bent to a certain angle.”

Who knew? Welding is now a STEM job — that is, a job that requires knowledge of science, technology, engineering and math.

I was reading this article to my kids tonight and it occurred to me that even today’s doctors and lawyers - always regarded as professions for the highly educated - have undergone a STEM transplant. Doctors of today have unprecedented levels of information access and global collaboration, and are using online medical records, sophisticated medical imaging technology, gene therapies, remote robotic surgery, and more. Lawyers might have to deal with things that didn’t exist not long ago like various forms of DNA and IT evidence.

If you wanted to be an administrative assistant or a librarian, you used to be able to use a Rolodex and the card catalog. Now these jobs are all about technology.

New York Public Library Digital Gallery

The Struggle

John J. Flynn Elementary Principal Graham Clarke alerted me to a very nice piece on NPR today. It’s called Struggle for Smarts? How Eastern and Western Cultures Tackle Learning.

A researcher was studying students in a Japanese school, and ended up observing a fourth grade math class.

"The teacher was trying to teach the class how to draw three-dimensional cubes on paper," Stigler explains, "and one kid was just totally having trouble with it. His cube looked all cockeyed, so the teacher said to him, 'Why don't you go put yours on the board?' So right there I thought, 'That's interesting! He took the one who can't do it and told him to go and put it on the board.' "

Have a listen. It’s only 8 minutes.

Teachers’ Expectations

When I arrived at work last week, several people were excitedly talking about the NPR piece they’d heard on the radio during their drive in. Here is a link to the Morning Edition show, Teachers’ Expectations Can Influence How Students Perform, which aired September 17.

I recommend listening to the audio, but you can also read the transcript. Back in 1964, Harvard professor Robert Rosenthal began studying how teachers’ expectations influence student achievement.

[Rosenthal] found that expectations affect teachers' moment-to-moment interactions with the children they teach in a thousand almost invisible ways. Teachers give the students that they expect to succeed more time to answer questions, more specific feedback, and more approval: They consistently touch, nod and smile at those kids more.

"It's not magic, it's not mental telepathy," Rosenthal says. "It's very likely these thousands of different ways of treating people in small ways every day."

It is difficult to truly change our beliefs, but there is a way. Recent studies have shown that teachers who actively worked on their teaching through videotape analysis and targeted work with coaches in their classrooms to change their behavior also experienced a significant shift their beliefs about students.

Teach your children well

 

Here’s a nice article from the Wall Street Journal for parents, with an accompanying video. 

This short piece hits the important points when it comes to math and parenting. 

The one thing I would add is my favorite question for my daughter when she asks for help with her algebra homework: “What do you think you should try next?”

Mindset

Charles Coiner, 1961, from Smithsonian American Art Museum

I am a huge fan of Carol Dweck’s work on people’s beliefs about the nature of intelligence. A student’s mindset has a measurable impact on his or her success in math, among other things.

We praised the children in one group for their intelligence, telling them, “Wow, that’s a really good score. You must be smart at this.” We praised the children in the other group for their effort: “Wow, that’s a really good score. You must have worked really hard.” That’s all we did, but the results were dramatic.

Doesn’t that statement make you want to read more? Here is a condensed version of Dweck’s research (you can read the book, Mindset, and find more articles here).

And here is a student-friendly version of the article. Perfect non-fiction reading for fourth and fifth graders!