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."