Vermont Fest 2014 and a Snake

The sign of a good conference is when you come away from it questioning everything about your work.

I like this snake drawing from the New York Public Library more than the three terrible photos I took of people in banquet rooms at the Killington Grand. I would never consider publishing a blog post without a picture.

At Vermont Fest this year, I found myself wanting to make a list of a bunch of things all students should get the chance to do in school.

Skype/hangout/video interact with people in other parts of the world

Type an equal sign into a spreadsheet

Make a personalized, LED-lit badge to explore how light works

Have maker/genius/whatever hour/day/week to explore something interesting to them

Publish their content (writing/photos/videos) on a blog and comment on the work of others

Learn about how computers work by writing some code

Be asked the questions How are you feeling? What do you think? and What do you need?

Use Google Research Tools to find and cite things.

Sit in chairs listening to a teacher talk a lot less.

Some great presenters I encountered include Chris Lehmann (@chrislehmann), Bonnie Birdsall (@bonniebird), Dan French (@danfrench), Adam Provost (@batman44), Lucie deLaBruere (@techsavvygirl), Charlie Macfadyen (@csmacf), Tricia Hinkle (@VTScienceFair), Matt Dunne (@@mattdunnevt). Check them out.

Here are a couple of links.

Using Google Sheets in the Classroom

Tricia Hinkle’s site

Bonnie Birdsall’s writing tech site

The Vermont Fest site with all kinds of other links to presenters

Relational thinking about numbers

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 + 2

14 + 1 = 13 + 2

19 + 4 = 19 + 3

4 + 4 + 4 = 4 + 4 + 3 + 1

4 + 4 + 4 = 8 + 4

23 + 5 = 25 + 3

47 + 62 = 62 + 47

10 + 10 + 1 = 1 + 20

5 + 5 = 5 + 5 - 2

30 - 2 = 20 - 2

10 - 9 = 9 - 8

We 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.

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.