Monday, November 21, 2011

Who Has?

Sidewalk card game by Lewis Hine
Who has played the Who Has game? I have. It’s a really good one, because it keeps all students on their toes.

Basically, you print up a bunch of index cards and hand them out to the class. If you’ve got 30 cards and 22 students, give extras to some of your stronger students. Choose a student to read their card. The game will go around the room until you are back to the first card.

Who Has? can be played with cards for any topic or subject area. There is a nice collection of Who Has? cards for different math activities on the excellent Mathwire website.

I’m going to specifically recommend two of them: More or Less and Place Value. I’ve encountered many students, even in the upper elementary grades, who do not fully understand place value. I think these Who Has? decks might help. Let me know if you try them.

Sunday, November 20, 2011

Finding Balance

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.

Thursday, November 10, 2011

The Powers of Ten

Lee Orlando, fifth grade teacher, kindly contributed this article.

Our Bridges November calendar focuses on decimals and "base-ten fractions," and we've been having some good discussions around the nature of decimal numbers.

Yesterday, we were ordering some numbers that included whole numbers and mixed numbers (expressed as decimals)... one student confidently announced:  "3 is greater than 33.45 because any decimal is smaller than a whole number."   The salient feature for this student was the decimal point, and she was working under the misconception that whenever you see a decimal point, the number is automatically smaller than any whole number.  Just that comment alone kept the students talking and debating for quite a while!

I also did a quick formative assessment: Instead of giving students decimal numbers already arranged appropriately (in a column) in order to add/subtract, I simply gave them the numbers and had them arrange them in order to add (there were four numbers). Perhaps you can guess what many students did. They applied their understanding of place value of whole numbers (ones, tens, hundreds, etc, lining up the numbers from right to left) and totally disregarded the decimal point.  Here were all these decimal numbers, neatly lined up as if they were whole numbers, with the decimal points totally misaligned. These were students who have been adding and subtracting decimal numbers in our weekly math computation practice, but when given the numbers separately - not pre-arranged in a column - their lack of conceptual understanding about decimals and their values was completely transparent.

All this has led to lots of discussions about the power of zero which we had already explored in our Great Wall of Base Ten, but which was now coming back in light of decimal numbers.  I have been digging up some cool resources around this, including this awesome video which you may already know: "The Powers of Ten."

Sunday, October 23, 2011

A new way of problem-solving

Lee Orlando, fifth grade teacher extraordinaire, is the guest author of this article.
Does this scene sound familiar? It’s time for math problem-solving, so you gather your students and present the problem at hand. Together, you read through the problem. You may direct students to underline the important information. You may also ask them to restate, in their own words, what the problem is asking them to do and encourage them to think about strategies that might work to solve the problem. Time for your students to “go forth and solve.” A few may do so, but, soon enough, a sea of hands is waving throughout the classroom: “I don’t know what to do!” or “I still don’t get it!”

Beth Hulbert and Marge Petit have developed a method of posing math problems that first immerses students in the problem’s context. The idea is, if students can fully visualize what is happening in the scenario and understand how the different math elements of the problem are related, they are much better prepared to solve the question posed. To do this, students first investigate answers to questions about the scenario they themselves have generated. Then, and only then, are they presented with the question that accompanies the task.

Here are the steps:

1. Introduce the scenario - not the question. Beth suggests that, when possible, the scenario is introduced kind of in the same vein as telling a story - in other words, with some feeling of authenticity to grab students' attention.

2. The scenario is  posted and students are invited to think about what kinds of math questions would fit this particular scenario. [I typed the scenario on paper and had them write their questions right in the meeting area.] After giving them a minute to think about some possible questions, the teacher calls on students to share some questions. She records these on the chart paper.

3. Students are invited to pick one of the questions on the chart paper (or one of their own, if that seems okay) and solve it. Beth said that this is where you can guide and differentiate. For example, for those quick-thinkers, you could direct them to solve other questions on the chart once they are finished with their first one. If you have kids you know need some extra support, you could direct them to a student question that would best prepare them for solving the actual question.

4. After 10 or so minutes hunkering down with student-generated questions, call the group back together, lead a quick share-out if you wish, and introduce the problem's original question (as presented in your math curriculum or whatever). By now, students have messed around with the components of the scenario, the numbers now have a lot more meaning, and they've made sense of the context. They are much better equipped to solve the question the teacher has for them and are much less likely to return to their seats and say "I don't get it!"

Sunday, October 2, 2011

Math and Image

Lately, I’ve been thinking about the role of images in mathematics. When it was time to create fliers for upcoming elementary school Math Nights, I decided to forgo the usual clip art. I used some mathematical images, including Sierpinski’s Triangle and a fractal tree like the one shown here. I’m happy with the way the fliers turned out.

Professor Tim Whiteford blogged about these fliers and more in a post entitled “Sierpinski and the Joys of Learning Math”. Says Tim, “...communicating the aesthetic component of math, [is] a critically important element if we are ever going to help students enjoy math for what it is, the science of pattern. Imagine learning to read and write without poetry, fiction, literature and creative writing? Imagine if the only thing we learned in language arts was the ability to read directions and write formal descriptions? Imagine if reading and writing were reduced to a purely utilitarian function?”

Teachers often use more than words and numerals to teach mathematics. They incorporate image and structure by encouraging students to work with manipulatives, make towers, draw tesselations, and create patterns. In light of this good work, should we continue to explore creative ways of teaching math? I think so.

Matthew Peterson discusses how language can get in the way of math in his TED talk called “Teaching Math Without Words”. In this 8 minute video we get a glimpse of a software program designed to show math with pictures. I found the software visuals intriguing, as are other apps and interactive math websites (i.e. a Tetris-like game called Factor-tris). As I watched the video, I found myself wanting to defend the role of language and dialogue in mathematics education but Matthew beat me to it. He shared a poignant story about a student with autism finding richer language as a result of his work with the math program.

A great companion to Matthew Peterson is Temple Grandin’s “The World Needs All Kinds of Minds”. If you haven’t read her book, Thinking in Pictures, you can get the gist in this 16 minute TED Talk. Temple mentions math, saying “You see, the autistic mind tends to be a specialist mind. Good at one thing, bad at something else. And where I was bad was algebra. And I was never allowed to take geometry or trig. Gigantic mistake. I'm finding a lot of kids who need to skip algebra, go right to geometry and trig.” Temple is one of my heroes. I love what she has to say about her visual thinking and I love the embroidery on her quirky western shirts.