Writing in Mathematics

Teaching students to communicate mathematically is essential if you wish to raise any standardized test scores. The last few years I have tried to include more justify your response questions into my classroom. Why? When a student can communicate their mathematical ideas clearly many positive things occur - it deepens the meaning of the concepts they are learning, it helps students retain information, and it strengthens their use of proper mathematical vocabulary and proper notation. Of course, students struggle with writing their mathematical explanations. It is important that the classroom teacher not give up on this endeavor. To help the students, model the written communication on a regular basis. I have found studying and analyzing responses to released state test questions and released AP questions have assisted me and my students greatly.

Reading in Mathematics

The opening of another school year is coming fast! Time to contemplate various goals. What area or areas do I want to focus on? Our students take so many important standardized tests, how can I help them be more successful? I believe if our math students did a better job of reading the test questions, they would have a greater chance of answering the question correctly. Many times I hear students say they just didn't understand what the question was asking.

Consequently, we need to help our students develop solid reading skills within the context of mathematics. Active reading strategies of underlining text and circling key words help, but is there more we can do? I recently came across a K-N-W-S strategy / template. Tom Stull, a high school math teacher from Ludlow High School, has developed this guide. Students use the guide to assist them when analyzing word problems.

K - What facts students KNOW
N - What information is NOT NEEDED
W - WHAT the problem is asking them to find
S - What STRATEGY they will use

Tom has quite a few reading strategies that he shares which can be found on the More Reading Strategies in Action - Mathematics High School. For those of you interested in other levels or other areas, the More Reading Strategies in Action website has some great resources available.

The Last of the Complex Diagrams

If you tried the previous problems for the past two posts, I have two more for you. I use both of these problems at some point in time in my classes. They are both excellent for stressing the importance of organizing thoughts and work. I also ask my students for a written explanation of their problem solving strategy.



The triangle question really forces one to be very organized in order to find all 28 triangles. This is a great problem to have students work together on.



The square problem allows one to analyze a pattern. When discussing this problem have students make up a list of the number of individual squares, of squares made up of four smaller squares, of squares made up of nine smaller squares, etc. The students should see a pattern occurring. This will allow you to move into examining patterns and finding rules.

Killing many birds with one stone

If you started to create the "thinking" environment with the previous rectangle problem (by the way there are a total of 9 rectangles), then you want to continue along this line. I would suggest that the next day you have a second problem similar to the previous one for students to work on. I have created a triangle problem.



This problem allows me to accomplish several goals. First, I give it to the students as soon as class begins. This allows me a chance to take care of administrative details like attendance. Second, after they have worked on it for a few minutes, I ask them to check with their neighbor to see if they agree. If they don't agree I ask them to try to decide which answer is correct. This problem is more challenging than the previous, so usually students do not agree on the answer. Consequently, I have them discussing their methods and they are engaged. I will ask them what they think the answer is and I will write these answers on the board. Now, I have the opportunity to accomplish my third goal which is to stress the importance of organizing their thoughts and their work. I will number the individual triangles and then I will make a list of triangles I am combining to form another triangle. After we discuss various strategies and discuss methods of organizing their count, the students usually beg me for another one to try. I tell them we will do another one tomorrow and there will be prizes for those who get it correct. Consequently, I have them looking forward to coming to math class! So, this quick activity from the day before which had the purpose of creating a safe thinking environment has provided me many benefits including the engagement of my students in problem solving!

Create a "thinking" environment immediately

I have heard teachers comment that students can't (and do not) think - all they want is to know what the right answer is. Well, I disagree. I do believe many students are hesitant to share what they are thinking and there is the natural fear of being wrong in front of everyone. After all we do not like to be wrong in front of everyone, why should they? Consequently, it is important to create a safe and comfortable "thinking" environment in your classroom.

I start on day one by setting the expectations and guidelines. Now, I am not a big fan of handing out a page of rules and regulations. I do discuss the importance of respect and then I ask them the following question.



I give the students a few minutes to work on this (depending on the students math background I may need to discuss that a square is a rectangle) and then I write the numbers from 1 to 15 on the board. I ask for a show of hands how many students think there is one rectangle, two rectangles, etc. and I do a tally on the board. When I get to 15, I will ask if anyone has a different answer and will place that on the board, if there is one. This is the first day of school, so students tend to be fairly quiet and there is not much conversation going on. I will then ask if anyone would like to change their answer. I do this as I am sending them a message that it is okay to change your answer. Next I will count the tally marks and usually it does not equal the number of students - there are always a couple of students who hesitate participating in this activity. Consequently, I will announce that there appears to be a few people who hadn't decided yet and did they now want to vote. Finally, I do the problem (count the rectangles) and demonstrate my method. As I am doing this I will hear various comments like "Oh, I forgot that one" or "I didn't see that one". We will then discuss their strategies.

When I finish I point out three very important things:
1) If the students got the problem wrong, do they now understand? I discuss the importance of speaking up, asking questions, and never leaving the classroom confused.
2) It was NO big deal if they were wrong! We are all going to be wrong at some point in time. Everyone was polite and respectful and that is how this class will always run!
3) Non participation is unacceptable. I do stress that I will not embarrass them, but I can't read their minds so they will need to help me by participating and sharing what they are thinking.

Consequently, I have the built the foundation of a safe and comfortable "thinking" environment through an activity. I encourage the students to copy the problem down and have their parents try it.

In my next post I will provide the answer to the question and will discuss how to continue building this safe and comfortable "thinking" environment.

Word Walls in Mathematics

Mathematics is filled with confusing vocabulary! For example, the word median is used with statistical measures (the middle term in a set of data that is in order numerically) and in geometry (the median of a triangle is the line segment from one vertex to the midpoint of the opposite side). Ask students to find the difference and you might want them to subtract two numbers or you might be asking them to tell you how the numbers differ - odd vs even, prime vs composite. Begin a geometry unit and you have new vocabulary coming at the students fast and furious. So, how can we help our students master the vocabulary as quickly as possible. First, as I mentioned in my previous post, use the vocabulary correctly and use it often. Second, try putting a word wall up in your classroom. There are many ways to design a word wall. You might want to start with the vocabulary word, followed by the definition, followed by a visual. You might want to incorporate the visual within the word. Examples of this can be found at The Broward County Public Schools - Exceptional Student Education page. You will need to scroll down to they yellow link - Mathematic Word Wall. You will need Adobe Reader to view the document. I would suggest that when you begin the school year you have a word wall started. As the year progresses have the students take charge of the word wall. The possibilites are endless - be creative and have fun with it!

Goodbye Plug and Chug, Hello Substitute and Evaluate

Mathematics is a very challenging discipline for many people. So, why do textbooks and teachers make it even more confusing by communicating poorly? Our students are capable of understanding the concepts if we would use the proper terminology. For example, why do we tell students to plug and chug? What does it really mean to plug and chug? What we really want the students to do is substitute the value into the expression and evaluate. I know what you are thinking, what is the big deal? After all, students knew that when you told them to plug and chug you meant place the value in for the variable and perform the order of operations. Well, have you thought about what happens when on a standardized test the directions say to substitute and evaluate? Have you noticed that students try to solve algebraic expressions? Why? They do not understand the difference between an expression and an equation. Here is another example. How many books and teachers ask students to reduce a fraction? Doesn't reduce mean to make smaller? Does the fraction actually get smaller? No, the fraction is equivalent! No wonder students have difficulty understanding the concept of a fraction. What we should be asking the students to do is simplify to lowest terms. One last example, how would you read -3? If you said minus three, doesn't minus mean subtraction? Isn't this really negative three or the opposite of three?

Don't feel bad if you are guilty of poorly communicating the mathematics to your students. In many instances we are mimicking what was in the textbook and relying on how we were taught mathematics. I would encourage all math teachers at all levels to think about the terminology they use. If you are in the habit of using mathematical slang or poor terminology, try to break the habit as soon as possible. Discuss this issue with other math teachers at your school. If everyone begins to use proper terminology, students will also use proper terminology and you will see them make great strides in being able to communicate mathematically.