Math Vocabulary: The Need for Intentional Instruction

Another valuable insight that came from my classroom observation was that my students were not strong enough readers to take what they read and translate it into mathematical operations. They needed more work on vocabulary. The administrator directed me to a website which has a table of math vocabulary words that translate to mathematical operations. I made a chart with this information, then made a sorting activity by putting the words on separate cards along with cards for the operations of addition, subtraction, multiplication, division, and equals. I used this activity to focus on teaching the mathematical association each word had. Instead of just pointing it out as we did the word problems, I made math vocabulary the focus of a teaching mini-unit. We did this activity regularly for several days and then sporadically in the weeks that followed.

Today, as we approached word problems and translating them into algebraic expressions, I had the students use a highlighter to mark vocabulary words that signaled specific mathematical operations. They were then able to determine the algebraic expression that was needed because they knew what operation to use based on the vocabulary words. I realized this had indeed been a successful undertaking when I saw the students answering the assigned tasks independently with correct answers.

The valuable insight that I gained here is that I needed to stop and do some intentional instruction focused specifically on what I wanted them to learn. The overall task of solving word problems needed to be set aside for a time to work on the individual skills needed to complete the overall task. Sure, as we had done word problems, I had pointed out what various words meant and how they would translate into math. But these kids needed to have intense training along with the message that they needed to know how to do this. And it paid off in increased learning and skills for my students.

Book Review: What Successful Math Teachers Do

This year when I made out my annual goals, one of them was to read the book What Successful Math Teachers Do, Grades 6-12.It is written by Alfred S. Posamentier and Daniel Jaye and was published by Corwin Press in 2006. (ISBN 1-4129-1619-4) I had seen an advertisement for it and was interested. In the meeting where I discussed my goals with the principal, he directed me to Barbara in the library and she ordered it for the school. Two weeks later, the book arrived and I have kept it on my night stand, looking at it frequently. I wanted to buy my own copy and ended up doing just that, because it is a valuable reference for math teachers, and because my lunch that contained beets dripped on it a little bit.

I understand that there is a counterpart to this book for math teachers of grades K-5 also.

After observing countless mathematics classrooms, the authors compiled a list of 79 strategies that are present in math classrooms where there is a high rate of student achievement. These are listed and explained in the book. After a strategy is given, it is followed by sections titled, "What the Research Says," "Teaching to the NCTM Standards," "Classroom Applications," and "Precautions and Possible Pitfalls."

The strategies are further organized into chapters titled, "Managing Your Classroom," "Enhancing Teaching Techniques," "Facillitating Student Learning," "Assessing Student Progress," "Teaching Problem Solving,"and "Considering Social Aspects in Teaching Mathematics."

I will admit that this is not the kind of book I can sit down and read cover to cover. For me, it is easiest read by reading one or two strategies at a time. Because I have been teaching for quite a few years now, I spent less time in the classroom management and assessment sections and focused a bit more in the problem solving and teaching techniques sections.

This is a book that I would love to discuss with other math teachers in a group setting. I will admit that in my mind, I was able to dismiss a few of the the strategies as less important than some of the others and would enjoy some discussion on them. Just three of the strategies that caught my eye were:

#37 Teachers can help students learn to ask better questions.

#59 Have students study written model solutions to problems while learning and practicing problem solving.

#66 Help students learn without relying on teacher-centered approaches. Give them carefully chosen sequences of worked-out examples and problems to share.


I am hoping that someone will want to be in a group to discuss this book--I think the strategies in it are interesting and valuable, not only to new teachers, but to seasoned veterans as well.

Supporting the Spiral Even in Middle and High School

I was recently observed by one of the building administrators. Over the years, I have been observed quite a bit, but I can't say that the countless hours put into observing my teaching and then the written summary have ever resulted in something that actually helped me with my teaching and subsequently student learning. Most years, I got the impression that administrators sat in my classroom to reassure themselves that I was really teaching the students, and then they could go back to their offices happy and content. Many years, I have not been formally observed, but when I am, they note that I am creative and capable etc. Years ago, as a new teacher, when I was yearning for some valuable and helpful insights, the best anyone did was, "The letters on your bulletin board are a bit crooked." Needless to say, for me observations have just been something that happen from time to time with little value or effect.

But this year is a refreshing breath of fresh air. On this observation in addition to the numerous commendation comments, there were two observations that I could sink my teeth into and put into practice. I was actually excited to get this feedback because it gave me a better sense of direction with what I needed to do and also clarified something I had been thinking about already. One suggestion, I will be dealing with in a subsequent post, but the other reaffirmed my concept of how students learn and how instruction helps them learn.

The observation was done while I was guiding a small group of seventh grade students through some math word problems. (In light of my last post, I hesitate to use the p-word here, but for the sake of clarity,I am.) There was no common theme in the problems, so it would be classified under mixed practice rather than just addition or subtraction etc. My students especially need practice reading problems and figuring out how to solve them, so I try to do exercises like this frequently. I did not write the problems myself and one of the problems required a lot of "guess and check" work. Now "guess and check" is a viable method that we teach for solving math problems, but it can also be downright tedious. This particular problem could have been solved by using algebra, but my students have not had algebra yet and the answer key said students should use the "guess and check" method. So after we went through all the hassel of trying every possible solution and eventually arrived at the correct one, I mentioned to my group that when they got a little older and learned algebra, there would be a simpler and easier way to solve this question. Then we moved on.

The administrator's observation was that at that point after we had the correct answer, I could have modeled how to solve the problem using algebra. The students would have been able to follow some, if not all of the process, and it would have served as an introductory/pre-teaching moment. It also would have piqued their curiosity and helped to lay a foundation for the math instruction that lies ahead. DUH! I know this and it makes sense, but it took an outside observer to point it out.

For a number of years now, elementary math education (as well as other subject areas)has been based on the spiral concept. I won't say that I'm always a big fan of how some curriculums implement the spiral, but I am a big believer in the concept. Whenever possible, you introduce a concept or skill before students are expected to master it. They are given several opportunities to practice the skill so that when mastery is expected, it will be there. Future learning is built upon the prior knowledge.

Perhaps the realization/reaffirmation that the spiral is important for middle and high school students as well as elementary was my philosophical highlight of the week. I am now even more conscious and deliberate in trying to work in a few more moments that support the spiral.

Just a Thought about Problems

Last week, during vacation from school, I had surgery on my foot and my dog died. Earlier this winter, we lost electricity for three days which caused our basement to flood because the sump pump wasn't working, resulting in significant damage. Last summer, I was in a car accident that gave me whiplash that I am just getting over now. In the past few years I have been in a position where I have been responsible for making health care decisions for several family members who were incapacitated. And then I sat with them while they died and subsequently made the funeral arrangements. THESE ARE PROBLEMS!

When students come through my door and I have math tasks for them to perform, why in the world do I refer to them as problems? They are tasks and challenges, but really not problems. Believe me, a number of my students have real problems of their own and don't need any more. I am trying to teach my students sound practices for approaching math and I want them to feel empowered rather than burdened by it. So every now and then I choose words other than "problems" in math. I hope students can view math with a welcoming attitude rather than misery.

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.