It All Adds Up: Language in the Mathematics Classroom

1. It All Adds Up:Language in the Mathematics Classroom Douglas Fisher & Nancy Frey San Diego State University www.fisherandfrey.com .

2. Increasing Specialization of Literacy Shanahan & Shanahan, 2008

3. Math Reading Goal: arrive at “truth” Importance of “close reading” an intensive consideration of every word in the text Rereading a major strategy Heavy emphasis on error detection Precision of understanding essential Conclusions subject to public argument Shanahan & Shanahan, 2008

4. Looking inside the math text… • Close reading required for precision • Arriving at the truth • Mastering technical vocabulary • Rereading for error detection

5. Mathematics Text 1.1 Introduction to Linear Equations A linear equation in n unknowns x1, xx…, xn is an equation of the form a1x1 + a2x2 +…+ anxn = b, where a1, a2,…,an, b are given real numbers For example, with x and y instead of x1 and x2, the linear equation 2x + 3y = 6describes the line passing through the points (3, 0) and (0, 2). Similarly, with x, y and z instead of x1, x2 and x3 the linear equation 2x + 3y + 4z = 12 describes the plan passing through the points (6, 0, 0), (0, 4, 0), (0, 0, 3). A system of m linear equations in n unknowns x1, x2, …, xn is a family of linear equations Shanahan & Shanahan, 2008

6. TEACHER RESPONSIBILITY “I do it” Focus Lesson Guided Instruction “We do it” “You do it together” Collaborative “You do it alone” Independent STUDENT RESPONSIBILITY A Structure for Instruction that Works

7. In some classrooms … TEACHER RESPONSIBILITY “I do it” Focus Lesson “You do it alone” Independent STUDENT RESPONSIBILITY

8. In some classrooms … TEACHER RESPONSIBILITY “You do it alone” Independent STUDENT RESPONSIBILITY

9. And in some classrooms … TEACHER RESPONSIBILITY “I do it” Focus Lesson Guided Instruction “We do it” “You do it alone” Independent STUDENT RESPONSIBILITY

10. TEACHER RESPONSIBILITY “I do it” Focus Lesson Guided Instruction “We do it” “You do it together” Collaborative “You do it alone” Independent STUDENT RESPONSIBILITY A Structure for Instruction that Works

11. Modeling in Math • Background knowledge (e.g., When I see a triangle, I remember that the angles have to add to 180.) • Relevant versus irrelevant information (e.g., I’ve read this problem twice and I know that there is information included that I don’t need.) • Selecting a function (e.g., The problem says ‘increased by’ so I know that I’ll have to add.) • Setting up the problem (e.g., The first thing that I will do is … because …) • Estimating answers (e.g., I predict that the product will be about 150 because I see that there are 10 times the number.) • Determining reasonableness of an answer (e.g., I’m not done yet as I have to check to see if my answer is makes sense.) • Fisher, D., Frey, N., & Anderson, H. (in press). Thinking and comprehending in mathematics classroooms.

12. The sum of one-fifth p and 38 is as much as twice p. Okay, I’ve read the problem twice and I have a sense of what they’re asking me. I see the term sum, so I know that I’m going to be adding. I know this because sum is one of the signal words that are used in math problems [for a list of signal words see figure 2]. I also know that when terms are combined, like one-fifth p, they are related because they make a phrase ‘one-fifth of p’ so I’ll write that 1/5 p. The next part says and 38, so I know that I’ll be adding 38 to the equation. Now my equation reads 1/5p + 38. But I know that’s not really an equation. I know from my experience that there has to be an equal sign someplace to make it an equation. Oh, they say as much as which is just a fancier way of saying equal to. So, I’ll add the equal sign to my equation: 1/5p + 38 = . And the last part is twicep. And there it is again, one of those combined phrases like one-fifth p, but this time twice p. So I’ll put that on the other side of the equation: 1/5p + 38= 2p. That’s all they’re asking me to do. For this item, I just need to set up the equation. But I know that I can solve for p and I like solutions. I know that you can solve for p as well. Can you do so on your dry erase boards?

13. Vocabulary Development • Word walls on which teachers post 5-10 words on a wall space that is easily visible from anywhere in the room. The purpose of the word wall is to remind teachers to look for ways to bring words they want students to own back into the conversation so that students get many and varied experiences with the words.

14. Vocabulary Development • Word cards in which students analyze a word for its meaning, what it doesn’t mean and create a visual reminder.

15. Vocabulary Development • Word sorts in which students arrange a list of words by their features. Word sorts can be open (students are not provided with categories) or closed (students are provided categories in which to sort).

16. Vocabulary Development • Word games in which students play with words and their meanings. For example, this might involve a bingo game of sorts where students write words from the class in various squares and then the teacher randomly draws definitions until someone gets bingo. We also like games such as Jeopardy, Who Wants to be a Millionaire, or $25,000 Pyramid as they allow students to review words while having a bit of fun. A great website that provides information about vocabulary games is: • jc-schools.net/tutorials/vocab/ppt-vocab.html

17. Productive Group Work • Individual Accountability • Interaction • Use of academic language