It is always good to hear about female mathematicians. Today, Google featured a graphic with the tag “Ada Lovelace’s 197th birthday”. Investigation revealed that Ada Lovelace was the daughter of the poet, Lord Byron, and lived in the 1800s.
A paragraph in the Washington Post caught my attention.
At the age of 17, Lovelace was among the first to grasp the importance of Babbage’s machines, Google noted. In her correspondence, as reported by New Scientist magazine, Lovelace said that “the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves.” She also noted that the Analytical Engine “does not occupy common ground with mere calculating machines” and had the potential to run complicated programs of its own.
Apparently, Lovelace wrote the first algorithm designed to be run by the Analytical Engine. Some say she should be considered the first computer programmer.
Here is another Lovelace quote to ponder (from Wikipedia:Ada Lovelace)
[The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine...
Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.[58]
There’s a great website called Inside Mathematics. I learned about it at the Common Core Math sessions this year with Bob and Judi Laird, Sandi Stanhope, and Fran Huntoon, of the Vermont Mathematics Initiative.
Inside Mathematics is a treasure trove for teachers. It has problems of the month linked to Common Core standards. The best part about the problems is the fact that they begin with an easy level and progress to more difficult levels related to the same math and situation. It’s kind of like how the Bedtime Math website works.
Here’s one we tried today with Sandi Stanhope: The Wheel Shop. It starts innocently enough with Level A. There are 18 wheels in the shop that sells tricycles. How many tricycles are there in The Wheel Shop?
Then we went on. The Wheel Shop sells other kinds of vehicles. There are bicycles and go-carts in a different room of the shop. Each bicycle has only one seat and each go-cart has only one seat. There are a total of 21 seats and 54 wheels in that room. How many are bicycles and how many are go-carts?
Nick Mack had a lovely way to solve this one today at the workshop which involved doubling 21, then adding 2 wheels to bicycles until he had the right number of wheels. The bikes with the added wheels became the number of go-carts.
Nick said his second graders did a similar problem involving ducks and sheep. They drew a bunch of ducks then added 2 more legs to several of them in order to “turn them into sheep”. He said the sheep that began as ducks were fairly ugly, but the math worked out.
Today Sandi said, "Math should be hard. Let kids figure it out."
Here is the Level C problem:
Three months later some vehicles have sold and new models have been brought into the Wheel Shop. Now, there are a different number of bicycles, tandem bicycles, and tricycles in the shop. There are a total of 135 seats, 118 front handlebars (that steer the bike), and 269 wheels. How many bicycles, tandem bicycles and tricycles are there in the Wheel Shop?
Today Nick started making a spreadsheet for this one and I did something similar tonight. My spreadsheet had columns for handlebars, seats, and wheels and did some multiplication and addition with formulas to get the totals. I guess-and-checked my way as the spreadsheet did the calculation. Setting this up took about 10 minutes, but my husband, Jim, had already gotten the answer and was eating muffins with the kids in the kitchen by the time I figured it out.
Jim had written his solution on an index card. He explained it as follows:
There are 118 total bikes (because there were 118 total front handlebars).
Multiply that by 2 and get 236 (so there must be 236 wheels).
But there are 269 wheels, so I subtracted. 269-236=33. There are 33 tricycles.
There were 135 seats but only 118 steering handlebars, so I subtracted. 135-118=17.
There must be 17 tandems and 33 tricycles. This left 68 regular bicycles because I subtracted 17 and 33 from 118 to get 68.
Enjoy the problems of the day! We did.
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?”
On Sunday, the New York Times published a piece by Andrew Hacker with the provocative title “Is Algebra Necessary?”. Several people have emailed me the link, so it seems like it has garnered some attention. It is worth reading.
While I disagree with most of Hacker’s anti-advanced-math argument, the statistics about the barrier students face when they are not proficient in high school math is important to consider. Failure in math does profoundly affect a young person’s options when it comes to college and career choices. How many people do you know whose math ability was a significant factor in their choice of college major and career? Just the other day, a friend admitted that she’d always dreamed of being a doctor, but she wasn’t good at math so she went into journalism instead.
Rather than lowering the bar for students, we should look at how we can improve mathematics education. Algebra is about abstraction, which is one of the most fundamental ideas across disciplines. When I was a software engineer, much of my day-to-day work was about designing abstract functions with variables to hold unknowns. It was about finding clean, elegant solutions to complex problems through thought, collaboration, trial and error. We conjectured, tested, proved. I couldn’t have learned concrete “job skills” ahead of time. I had to keep learning, inventing and working with big ideas. Great algebra and other math classes are excellent preparation for students, as are great science, literature, writing, and history classes.
Hacker’s article prompted many thoughtful responses. I’ve collected some here along with excerpts for your reading pleasure. I’ll end with a nice video of Paul Lockhart talking about the wonderfulness of math.
The New York Times Letters to the Editor: A National Conversation on Math
Andrew Hacker is right: most students will never need to use algebra. Many will struggle to learn it nonetheless. But the answer is not to let students quit as soon as they begin to struggle.
I myself hated mathematics for many years. Through algebra, geometry and trigonometry, I cursed a system that compelled me to take such “useless” courses. Eventually, I was required to take calculus, the most dreaded of all math courses. I prepared for the worst.
It came as a surprise, then, that I quickly found myself enjoying the class. The reason was that I had finally encountered a talented math teacher with a passion for the subject. His passion proved infectious, and now, a year later, I’m looking to study mathematical biology at an Ivy League university.
It’s an outcome I would have never predicted just a few years ago. It could have never happened if I had been allowed to quit when I first struggled with math.
ZACHARY MILLER
Kitty Hawk, N.C., July 29, 2012
Motherlode: Adventures in Parenting by Jessica Lahey
...I know precisely where I lost my battle with math, the moment I was informed clearly and unequivocally that I simply wasn’t “a math person.” My seventh-grade math teacher, an otherwise lovely man, called each of his students up to his desk one by one in order to write a “1” (for the honors track) or “2” (for the standard track) on the school’s official math placement forms. As I watched from over his hunched and courduroyed shoulder, he wrote a beautiful, decisive and neat “1” on my form.
There it was, in permanent ink. I was good at math.
“Jess, could you come back up here for a minute?” he asked as I floated back to my seat.
He reclaimed my form, and carefully overlaid that beautiful “1” with a dark, clumsy “2,” pressing hard with his black pen in order to make sure the ink obliterated any evidence of his indecision.
And from then on, I wasn’t good at math anymore....
Scientific American: Abandoning Algebra is Not the Answer by Evelyn Lamb
Mathematicians are recruited by hedge funds, consulting firms, and technology companies not because they already know how to balance portfolios, what the best corporate strategies are, or how to optimize user interfaces, but because their mathematics degrees indicate experience and acuity at problem solving. It’s easier for companies to teach someone with a strong mathematics background how to do their specific work than to teach someone who knows the company business how to solve problems. And, like it or not, algebra is one of the first places students start to learn these problem solving skills....
Math education needs to improve, but if illiteracy were on the rise, I don’t think we’d be talking about eliminating reading from the curriculum.
Rob Knop
Algebra is fundamental to nearly all of "higher math". Even if you want to do more than the most basic of things with statistics, you need to know some algebra. To give up on that would be right on par with the giving up on the teaching of history as anything other than memorizing the occasional date, and to give up on the teaching of English literature as anything other than being able to read a short document for simple surface content and to put together a simple declarative sentence. If you want people to be educated beyond elementary school and beyond "job training", then algebra is one of the intellectual foundations of our civilization that simply cannot be neglected.
Harvard University Press Blog: Is Mathematics Not Beautiful?
For all his focus on the pain and fear of mathematics, Hacker has little to say about its beauty. He does suggest that we should treat mathematics as a liberal art, “making it as accessible and welcoming as sculpture or ballet,” but in the service of rationalizing its marginalization rather than encouraging its embrace.
Today
I asked three kindergarteners to figure out whether a bunch of number
sentences were true or false. I was partially inspired by an excellent
book called Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Carpenter, Franke and Levi.Here is what I gave them:11 + 3 = 12 + 214 + 1 = 13 + 219 + 4 = 19 + 34 + 4 + 4 = 4 + 4 + 3 + 14 + 4 + 4 = 8 + 423 + 5 = 25 + 347 + 62 = 62 + 4710 + 10 + 1 = 1 + 205 + 5 = 5 + 5 - 230 - 2 = 20 - 210 - 9 = 9 - 8We
didn’t get to the last 3 on the list yet, but not because of a lack of
enthusiasm. The kindergarteners loved this activity and can’t wait to
have a chance to finish it. They worked independently to decide whether
they thought each one was true or false. Then I simply asked them “How
did you know?”. If they decided very quickly - quicker than they could
have done a calculation for each side of the equal sign - I asked in amazement if they would explain their thinking. I also asked them to
prove their answer was correct. This work really got them thinking, not
only about the meaning of the equal sign but also about relationships
between numbers and properties of addition and subtraction.
How do we teach students about the equal sign in math?
Professor
Tim Whiteford brought this up in a meeting recently. Says Tim:
“Traditionally we have used language like ‘three plus four makes seven’
or ‘three and four are seven’. We now know that both these forms of
language actually develop in children a misconception about what is
happening in this piece of procedural knowledge. Children tend to think
that the equals sign makes things happen.” (see Tim’s full blog post on
the equal sign)
I
remember having this misconception as a child, and children in the U.S.
continue to struggle with it today. I was looking at the 3rd grade
NECAP released items last year and noticed lots of students got this
question wrong: 1+4+?=6+14. (Many students incorrectly chose 1, which
makes sense because 1+4+1=6.)
Researchers
at Texas A&M University found that 70% of U.S. middle
school students lack understanding about the equal sign. Students in
other countries like Korea and China do not have the same
misconceptions. When students begin algebra in middle school,
understanding the equal sign is critical for their success. (full article here)
On
the bright side, this seems like a relatively easy thing to fix. I
visited a second grade class the other day and watched the students
excitedly working with a number balance scale.
Their teacher used this tool to help them develop their concept of
equality as a relationship, as opposed to an operation. If you don’t
have a number balance scale, here is a very nice virtual pan balance scale from NCTM Illuminations, and a virtual number balance scale.
We can also mix up the way we write equations. I could decide to write 7=9-2 instead of 9-2=7.
At
what age do students need to learn the correct meaning of the equal
sign? Why wait? This is a Mathematics Common
Core State Standard for first grade: 1.OA.7.
Understand the meaning of the equal sign, and determine if equations
involving addition and subtraction are true or false. For example, which
of the following equations are true and which are false? 6 = 6, 7 = 8 –
1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
