What Good Mathematicians Do: Check Your Work

A few weeks ago, I hung a large poster in my room titled,"What Good Mathematicians Do". The poster, by McDonald Publishing Co., has eleven things listed on it. This week I asked students to sit facing me with their back to the poster. I asked them to tell me any of the things they had noticed or could guess were on the poster.

The most common answer in all of the groups was, "Check your work!" In fact, for most groups, this was the only answer they could initially come up with. I facillitated a few activities and discussion around the poster and was able to drive home a few more points, but I found it interesting that the "Check your work" answer stood out so prominently.

I guess I shouldn't have been surprised. At every grade level, teachers remind children to check their work. It has been drilled into them, even though it seems that many choose to ignore this advice. "Check your work!" is a common admonition for just about every math task from kindergarten on up.

What if math teachers were just as diligent and consistent at teaching the other aspects of being a good mathematician? What if "Know how to explain your work," and "Ask questions," were called out as frequently as "Check your work"? I think we would have higher achieving students. If they were just as familliar with the other success strategies, students would probably be more capable of checking their work, and ultimately more successful.

The Significance of Valentine's Day to Math Teachers

For a math teacher, Valentine's Day has another meaning. After we finish with the party, candies, cards and romantic dinners, the calendar reminds us of another upcoming event. Valentine's Day, February 14, means that we have just one month until the mathematician's holiday: Pi Day, March 14.

Teachers begin getting ready for this event early. There are a variety of activities and teaching opportunities leading up to this special day. Granted this year's Pi Day is on a weekend, but it can be celebrated a day early just as we are having Valentine's parties today. Of course the significance of Pi Day being on the 14th day of the 3rd month is to help children remember that the value of pi is 3.14.

I like to read the book Sir Cumference and the Dragon of Pi by Cindy Neuschwander. (ISBN-13:978-1-57091-164-4) It is a fairy tale that establishes in a creative way the definition of pi. Other titles in this series include:
Sir Cumference and the First Round Table
Sir Cumference and the Isle of Immeter
Sir Cumference and the Great Knight of Angleland
Sir Cumference and the Sword Cone

Main characters in these books include Sir Cumference, his wife Lady Di of Ameter, and their son Radius who is half as tall as his mother. Children in fifth and sixth grades enjoy these books, but may not fully understand all the concepts. I have read them to seventh and eighth grade classes who enjoyed them as well.

So after all of the mushy lovey stuff of Valentine's Day, it's time to haul out the Pi Day materials and let the real fun begin!

The Queen of Mean

The mean is one of those measures of central tendency (along with median, mode and range) that students must learn and use to analyze data. I use a simulation presented as a small group demonstration, to illustrate what the mean is and how it is found.

I start with 5 or six sheets of different colored construction paper spread out on a table. I tell the students that these represent houses. I then use unifix cubes or something similar to represent people/families that live in each house. The cubes that live in a house are all the same color (ie the brown cubes are referred to as the Brown family.) Each house has a different number of people living there.

Then the Queen (sometimes it is me and sometimes I just refer to her) makes a decree that there must be the same number of people living in each house. So people must be moved to fulfill this mean-spirited decree which breaks families apart, and the Queen is dubbed "the Queen of Mean" (I then embellish and repeat this part of the story about how mean she is.)Students can work to figure out how many must be in each house. If they have no clue how to do this, I suggest taking all the people and spreading them around one at a time. (This is the same as adding all of the people together, then dividing by how many houses there are--the steps for finding the mean.) I emphasize several times to put all the people together, then divide them out.

After several simulations with different numbers of people, students get the idea of how to find the mean. I then take the cubes away and ask if they can do it without using cubes. I pose another problem and let them go through the steps of solving it. From this point on, whenever a student doesn't rmember how to find the mean, I say, "Do you remember the Queen of Mean?"

"Oh, yeah!" they respond and I don't have to say anything else.

Pulling Teeth and Getting Blood from Turnips

The task before me is like pulling teeth and getting blood from turnips! OK--well maybe it's not quite that bad. I have been working today with students on writing answers to math problems; not just the kind of math problem where you give the answer in one or two words, but those that say "Show your work." and "Explain your thinking." These are the kind of problems we frequently find on standardized tests.

We start by making an effort to understand the problem. This usually means reading it over a couple of times at least. We then define the specific question(s) that we must have an answer to. We look at what information is already given to us in the problem and decide what operations/methods will be used to solve the problem. All this is mapped out on scratch paper, not yet ready for the final draft.

If we get to the point where we get a correct answer, we feel pretty good about ourselves and are tempted to communicate that particular answer with just two or three words. But there I am with my turnip juicer and tooth pullers insisting that everyone must now write in great detail how they arrived at their answer. Then we look back to make sure it isn't just details and explanation, but that a clear statement answers the direct questions that were asked.

Many students find this to be a hard process. Writing and language tasks may not come easily to them. In their minds, they consider tasks like this to be appropriate for language arts classes, but not math. They want to give up. And there I am with my dental tools and juicer as well as any incentives I can come up with.

Slowly but surely, I start to see some improvement. I have modeled how to answer these questions what seems like a bazillion times. And I see my students beginning to write just one more detail or begin to show their work. The process of organizing their thoughts into words to explain how a problem is solved is starting to show through.

Model, think aloud, model some more. Practice, practice, practice. I hope and pray that it works!

Ready-made Powerpoint Presentations

I have been using Pete's Power Point Sation to find powerpoint presentations that I can use with my students. It is found at: http://www.pppst.com/. There are powerpoints for just about all subjects. I can save these powerpoints in my own "instructional presentations" folder and modify (if desired) and use them in my classes. Most of the presentations last just a few minutes. There are some jeopardy games which I downloaded and use as a template by changing the categaories, questions and answers.

This site is a tremendous timesaver that enhances the quality of my educational program. My own introduction to a class one day, can be followed up the next day with a short presentation that reviews what I went over, but in a little diffferent format. Students respond well to this medium, and math concepts are reinorced.

The Vocabulary of Word Problems: When Math Teachers Must Teach Reading

Some students struggle with math because they struggle with reading. And the trend in mathematics education lately, is to include a lot of applications which involve more reading than ever. Gone are the days where students would do math problems in isolation, so those who didn't read well could still do well in math. Math books contain more written words and language-based tasks than ever. So math teachers find themselves teaching reading skills, especially as they pertain to understanding word problems. This is nothing new; good math teachers have always taught math vocabulary and applications.

For word problems, the basis seems to be vocabulary. Most of us have lists of catch phrases that signify certain operations. For example, "how many more?" indicates subtraction. But we cannot rely totally on vocabulary and catch phrases. Sometimes the situation described in word problems describes a mathematical concept without any signal words or catch phrases.

I am currently trying to get my students to recognize when to use a proportion to solve a problem: when there are three pieces of data and two labels. They all can solve proportions quite well, but sometimes have a hard time knowing when to use one. Having a general guideline has helped them tremendously.

With students in Title I classes, I have found it necessary to slow down and require the students to read (and reread) problems. I have noticed that sometimes math teachers skip having students read instructions, but paraphrase them instead. This can sometimes cause confusion which leads to incorrect work. When a student asks for help with a problem, a common response I have is "Read the problem to me."

Every now and then, it is helpful to remind myself that I am not only a math teacher; I must teach reading as well.