# MSP:MiddleSchoolPortal/Reading and Writing Mathematics

### From Middle School Portal

## Reading and Writing Mathematics – Introduction

In the*Principles and Standards for School Mathematics*, the National Council of Teachers of Mathematics states that "students who have opportunities, encouragement, and support for writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically” (NCTM 2000).

Students who learn to read and write mathematics gain essential skills, as Joseph G. R. and Nancy C. Martinez, authors of books on the subject, have pointed out:

- They learn to use language to focus on and work through problems, to communicate ideas coherently and clearly, to organize ideas and structure arguments, to extend their thinking and knowledge to encompass other perspectives and experiences, to understand their own problem-solving and thinking process as well as those of others, and to develop flexibility in representing and interpreting ideas (quoted in Barton and Heidema 2002, pp. iii-iv.).

Far from expecting teachers to stretch their class time to include yet more content, we offer here resources that can enrich math instruction as teachers help their students better understand the content they are already tackling.

For the student, reading the math textbook or handouts or extended response problems presents built-in challenges. The vocabulary of mathematics can be confusing, with some words meaning one thing in a mathematical context and another in everyday settings. Symbols can look alike, and different symbols can represent the same operation (for example, *, x, and • for multiplication). Graphs vary in format, even when representing the same data.

Writing is valued as a way of communication in most school subjects, yet rarely in math. If students can learn to explain their thinking in solving a math problem (using drawings or tables or graphs as well as words), they acquire a means of setting out their work logically and refining their thinking as they communicate their understandings.

Graphical representation is a powerful means of communicating all kinds of data, as evidenced in its use in newspapers, advertisements, campaign literature, economic reports, and more. A handy set of selected online tools for creating histograms, box plots, scatter plots, and other graphs can be found in the section titled Tools for Displaying Data.

Each section of resources contains articles, many by teachers, who share their experiences, rationale, and classroom methods. Each section also offers lesson plans or activities appropriate for middle school students.

Enjoy the challenge of opening your students to mathematical communication!

## Reading in Math Class

**Reading Mathematics is Different**
This recorded webinar and related resources provide insight as to why reading mathematics is challenging for many students and what teachers can do. Also examined is how mathematics symbols, vocabulary, and content presentation can create roadblocks to students’ mathematics understanding. Learn how to address students’ difficulties by approaching mathematics as a language and to use specific strategies to improve mathematics learning.

**Reading in the Mathematics Classroom**
Written by Diana Metsisto, a middle school mathematics coach, this online chapter from the book *Literacy Strategies for Improving Mathematics Instruction* involves both the “why” and the “how” of integrating reading in the teaching of mathematics. She begins with a brief but insightful summary of the theory, then offers a number of concrete classroom strategies.

**Reading and Writing to Learn in Mathematics: Strategies to Improve Problem Solving**
Math educator and author Clare Heidema describes six strategies — K-N-W-S, SQRQCQ, three-level guide, process log, and RAFT — and shows how teachers can use them in the math classroom.

## Vocabulary

**MathLexicon - a collaborative vocabulary authoring toy for extending English with math word parts**

Natural Math network members can never have enough math words! MathLexicon's prefix and suffix database is open for public submission. You can use it to design new exciting math terms. Output examples, with the root "cat":

- concatgon: with opposite of cat angles

- pentacatology: the study of five of cats

- catplex: a cat with many parts

**Unlocking the Mystery of Mathematics: Give Vocabulary Instruction a Chance**
Math teacher Bizzie Cors realized that her students needed to “construct meaning for all vocabulary terms and connect to prior knowledge as well as to new concepts and algorithms.” This led her to create a new process to teach vocabulary development. Described here is what she calls the “sticky-note chain” process; its final product is a graphic organizer complete with sticky notes, connections, and problems created by the students themselves.

**A Maths Dictionary for Kids**
This animated, interactive dictionary for kids explains over 500 common mathematical terms in simple language. Each term is illustrated and often accompanied by an interactive applet that makes the definition visual and immediate.

**Math Words: From Beginning Algebra to Calculus**
This interactive mathematics dictionary offers many terms and formulas appropriate for older middle school students. The illustrations, diagrams, and applets help define the terms every bit as much as the text does. As noted by the authors, over a thousand illustrations and examples are provided!

**Sorting Polygons**
In this classroom activity, students identify and classify polygons according to various attributes (number and lengths of sides, sizes of angles, and so forth). They then sort the polygons in Venn diagrams, according to these attributes. Vocabulary learning centers on recognizing properties of polygons and how they are related.

**Literacy Strategies: Knowledge Rating**
Knowledge Rating is a pre-, during, and post- reading activity in which students analyze vocabulary words from the text before reading. The site provides directions for implementing the strategy and sample charts for recording information. The strategy is recommended for content area reading.

## Reading the Textbook

**Getting to Know Your Middle Grades Mathematics Textbook**
This article by Diane Kahle, an experienced teacher of middle school mathematics, shares general tips, small group and whole class ideas for textbook reading, and a ten-question scavenger hunt to help students learn how to find information in their mathematics textbook.

**ACCESS: Textbook Feature Analysis**
Here you have a series of questions to guide students through understanding their textbooks. Questions about types of text, sidebars, typography, color, symbols and icons, images and graphics, organization, and navigation appear on the left-hand side of the activity sheet, with space for answering on the right. As author Jim Burke explains, its purpose is to teach how the textbook works by showing students how all these elements are organized.

## Using Trade Books to Teach Math

**The Book of the Month and Mathematics: An Integrative Approach**
To illustrate her ideas on how to integrate literature into the teaching of mathematics, math coach Christy Rhoades includes a complete lesson plan for middle school students based on the book *Sweet Clara and the Freedom Quilt*. Follow the link to her lesson on “Finding the Best Path to Freedom for Clara and Jack.”

**How Much Is a Million?**
This lesson focuses students on the concept of 1,000,000. It allows students to see firsthand the sheer size of 1 million and, at the same time, provides them with an introduction to sampling and its use in mathematics. Students will need grains of rice and a balance to figure out the approximate volume and weight of 1,000,000 grains of rice.

**One Grain of Rice**
In this lesson, students take on the role of a villager in a third-world country trying to feed her village. While listening to you read aloud the book *One Grain of Rice* by Demi, students work collaboratively, using algebra, exponential growth, and estimation, to come up with a plan to trick the raja into feeding the village.

Other trade books you might use in teaching math:

**The Measure of the World**, a historical novel by Denis Guedj, tells the true story of Pierre Méchain and Jean-Baptiste Delambre, an astronomer and a mathematician charged with measuring the meridian through Paris to create a reference for defining the meter.

Middle schoolers may be surprised and pleased to find ratios treated as the subject of these next three picture books. You can find the books in school or public libraries. They are also available from online booksellers.

**Cut Down to Size at High Noon**
by Scott Sundby and illustrated by Wayne Geehan

This parody of classic western movies teaches scale and proportion. The story takes place in Cowlick, a town filled with people with intricate western-themed hairstyles that the town's one and only barber creates with the help of scale drawings. Enter a second barber, and the town does not seem big enough for both of them! The story reaches its high point of suspense when the two barbers face off with scissors at high noon. The duel ends in a draw of equally magnificent haircuts, one in the shape of a grasshopper and the other in the shape of a train engine, and the reader learns that scale drawings can be used to scale up as well as down.

**If You Hopped Like a Frog**
by David M. Schwartz and illustrated by James Warhola

Imagine, with the help of ratio and proportion, what you could accomplish if you could hop like a frog or eat like a shrew. You would certainly be a shoo-in for the Guinness World Records. The book first shows what a person could do if he or she could hop proportionately as far as a frog or were proportionately as powerful as an ant. At the back of the book, the author explains each example and poses questions at the end of the explanations.

**If the World Were a Village: A Book about the World's People**
by David J. Smith and illustrated by Shelagh Armstrong

How can you comprehend statistics about a world brimming with more than 6.2 billion people (the population in January 2002)? One answer to understanding large numbers is to create a scale where 100 people represent the total world population and change the other numbers proportionally. In a world of 100 people, how many people (approximately) would come from China? (21) From India? (17) From the United States? (5) In the same way, the book presents statistics about the different languages spoken in the world, age distributions, religions, air and water quality, and much more.

## Writing in Math Class

**A Case for Using Reading and Writing in a Mathematics Classroom**
Speaking from her own experiences as a math teacher, Sarah Kasten tells how -- and why -- she introduced reading and writing in her classroom. She shares how she directed her classes to do 5-minute, impromptu writing assignments, explain their problem-solving process, or even explain a new concept and create their own example problems.

**Using Writing in Mathematics**
This article includes specific suggestions for managing journals, developing prompts for writing, and providing students with feedback on their writing. In addition, the site includes two sample lessons for introducing writing activities in a math classroom.

**Writing in Mathematics**
A brief teacher-to-teacher article on getting started with writing in math class — moving from think-pair-share to a less-known model: think-write-pair-share.

**Bias Sampling**
The purpose of this activity, designing a survey, is “to demonstrate how the results of a poll or other scientific study can be biased by selecting special types of people to respond or by asking only certain questions.” In this well-constructed lesson, students gather opinions on how much homework time is appropriate. Who should they interview? What questions should they ask? The task culminates in a persuasive, hopefully unbiased, report to the school principal.

**Math Out Loud!**
In this one-pager, Robyn Silbey, a school-based math specialist, contends that “speaking and writing in math offer students an opportunity to synthesize their thinking and articulate it for others.” Moreover, she offers practical ideas for carrying out the classroom process.

**59 Writing Prompts for Math Teachers**
Teachers will find these prompts useful for students who are writing in math journals, learning logs, and classroom reflections.

**Math and Communication**
You’ll find solid tips on encouraging and supporting math talk in this brief piece by well-known math teacher Kay Toliver.

**Adapting Literacy Strategies to Improve Student Performance on Constructed-Response Items**
This article discusses ways of adapting various reading strategies to help students improve their answers to extended-response questions on the mathematics portion of high-stakes tests. A practical article directed to teachers.

**Writing in the Mathematics Classroom**
From the book *Literacy Strategies for Improving Mathematics Instruction* (ASCD, 2005), this chapter by Cynthia L. Tuttle includes a wealth of examples from students. She offers descriptions of such processes as written responses to math problems, structured writing guides, and assessment tasks. The chapter is not available online, unfortunately.

## Communicating Through Graphing

**Mathematics as Communication: Graphing Information Collected Over Time**
This lesson plan focuses on interpreting and creating graphs that are functions of time. Two activity sheets focus on graphs of time versus speed; two others look at how many times an event occurred in a specific amount of time. Inventing stories to correspond to the graphs is challenging but fun!

**Bones: Does Drinking Soda Affect Your Health?**
Students read graphs and interpret raw data to determine if there is a correlation between drinking soda and the rate of bone fractures in teenage girls. Students are encouraged to organize data into tables to look for associations, but are cautioned that additional factors many influence the appearance of cause and effect.

**Working Hours: How Much Time Do Teens Spend on the Job?**
To answer this question, students must interpret a bar graph to determine the average number of hours teenagers work per week. Related questions ask students to calculate averages for additional data sets.

**State Data Map**
Information can be represented in many ways. This applet allows the user to represent data about the states using colors. The state with the highest data value is darkest; other states are shaded proportionally. Several sets of data are already entered and available for examination: state population, land area, representatives in Congress, gasoline usage, and more. Users can eliminate the data from any state in order to note the consequences, or enter their own data. A box plot accompanies each map representation, showing the data in a different but corresponding format.

**Grading the Graphics**
Students work as graphics illustrators on a newspaper, interpreting, creating, and analyzing graphic displays. The page opens to the tasks as well as the solutions for each task and, most helpful, a scoring rubric.

**Using NBA Statistics for Box-and-Whisker Plots**
In this lesson, students use information from NBA (National Basketball Association) statistics to make and compare box-and-whisker plots. After reviewing the concepts of minimum, maximum, median, upper quartile and lower quartile, students create three box-and-whisker plots for sets of data on the heights and weights of basketball players. In each case, the students consider the effects of changing one piece of the data, such as eliminating the height of the tallest player. Detailed instructions for the lesson, assessment options, and all materials are included.

**Accessing and Investigating Population Data: National Population Projections**
In this activity, students investigate population projections from 1990-2100 using data from the U.S. Census Bureau web site. Using the five specific population pyramids, students investigate population projection data for the United States over a 110-year period. They examine how the population data is distributed over time and explain what factors might contribute to these trends. An activity sheet and thoughtful questions, included in the lesson plan, guide the class investigation.

**Exploring Linear Data**
In a lesson that connects statistics and linear functions, students construct scatter plots, examine trends, and consider a line of best fit as they graph real-world data. They also investigate the concept of slope as they model linear data in a variety of settings, which range from car repair costs to sports to medicine. Handouts for four activities, spread out over three class periods, are provided.

**Emergency 911! Bay City**
Last week there was an accident at the Waterfront Amusement Park in Bay City, resulting in the death of two teenagers. The owners of the amusement park have charged that if ambulances had responded more quickly, the two teens would have survived. The students’ task is to compare two ambulance companies on the basis of their timely response to Emergency 911 dispatches. Students must use graphs or measures of center and spread to present a persuasive argument for choosing one of the ambulance companies over the other. Included are samples of student work and scoring.

**All Aboard**
Students are given the timetables of two trains, each going along the same route, but one coming and the other going. Carefully crafted questions ask students to create and analyze a graph of position versus time for one train, then to explain mathematically the ways a graph of the other train’s run should resemble the first graph. Finally, they create the second graph and try to explain the unexpected discrepancies between the two graphs. A full solution and scoring rubric are included.

**Graphic Representation in the Mathematics Classroom**
In a well-illustrated professional reading from *Literacy Strategies for Improving Mathematics Instruction* (ASCD, 2005), Loretta Heuer explains students’ difficulties in reading graphics and creating visual representations. Student-generated drawings offer insight into their understanding of such abstractions as fractions and help students self-correct their thoughts. Not available online, unfortunately.

## Tools for Displaying Data

This section offers applets for creating several types of data display. They can be useful for class presentations as well as for offering “pictures” of those long lists of numbers collected as data. Students can engage more easily with the visual displays and make better sense of their data through these interactive exercises. The last two tools offered here help in collecting all that data: a stopwatch and a spinner.

**Create a Graph**
Students will learn how to create area, bar, pie, and line graphs. They are provided with information about what each type of graph shows and what the graphs can be used for, along with an example of each type of graph. They can create the graphs using their own data.

**Bar chart (grades 6-8)**
This virtual manipulative enables students to make a bar chart, three to twelve columns wide and five to twenty rows tall. They can label columns and click on cells to make the chart. A special feature: Students can enter data as quantities, and then by clicking the percentage button, they can instantly see the percentage relationship of the quantities.

**Circle Graph**
This activity allows the user to graph data on a circle graph. Users can use predefined data sets or enter their own data. Especially good for showing examples of this type of display!

**Histogram (grades 6-8)**
This virtual manipulative enables students to construct histograms to summarize data. They can, first, view examples of real-world data, such as the number of minutes between eruptions of Old Faithful, then clear the data and enter their own. Finally, students can switch the display mode and see the same data as a box plot.

** Histogram**
This second applet for creating histograms offers as examples real-world data sets more appropriate for older middle school students. The applet automatically displays the number of elements in the data set, the mean and standard deviation, and a frequency chart along with the histogram for each data set.

**Box plot (grades 6-8)**
This virtual manipulative enables students to construct box plots to summarize data. As students enter data into a table, the applet displays the minimum and maximum data values, the lower and upper quartiles, and the median. The number in the data set, the average, and the standard deviation are also shown.

**Stem-and-Leaf Plotter**
This activity generates a stem-and-leaf plot from data that students enter. After examining the display, students are challenged to give the mean, median, and modes; feedback is immediate and answers given, if requested.

**Scatterplot (grades 6-8)**
To visualize the relationship between two variables, a scatter plot is often used. Here students click on a point on a grid to enter data or add values to a list. The manipulative displays the data points but also a line of best fit and its equation, for those students studying algebra. Besides instructions on using the applet, you’ll find an interesting activity that engages students in comparing their height with their hand span.

**Data picking**
In this interactive game, students create a table of data they collect from the onscreen characters and then select a scatter plot, a histogram, a line graph, or a pie chart that best represents the data. The amount of data increases and the type of data representation changes according to which of three levels of difficulty is selected. Tips for students are available as well as a full explanation of the key instructional ideas underlying the game.

** Stopwatch**
Functions just like a real stopwatch as well as recording set times, accurate to the nearest tenth of a second.

**Adjustable Spinner**
Users can create a game spinner with one to twelve sectors to look at experimental and theoretical probabilities.

## SMARTR: Virtual Learning Experiences for Students

Visit our student site **SMARTR** to find related math-focused virtual learning experiences for your students! The **SMARTR** learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

## Careers

**The FunWorks**
Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

## References

Barton, Mary Lee, and Clare Heidema. 2002. *Teaching reading in mathematics*, 2nd edition. Aurora, CO: McREL (Mid-continent Research for Education and Learning).

National Council of Teachers of Mathematics. 2000. *Principles and standards for school mathematics*. Reston, VA: Author.

## Author and Copyright

Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

Please email any comments to msp@msteacher.org.

Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org.

Copyright June 2009 — The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.