Reading Mathematics Is Different


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These slides are from a webinar on why reading mathematics is challenging for many students and what teachers can do. We will examine 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.

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  • Hello! Welcome to the Reading Mathematics is Different webinar sponsored by the MSP2 social network. Today, we will explore the connection between reading comprehension and mathematics understanding. And introduce four literacy strategies as practical tools to help students learn mathematics and become mathematics readers and problem solvers.
  • My name is Judy Spicer. For almost 13 years I have been a mathematics content specialist working at The Ohio State University. At ORC, our mission is to improve teaching and learning in Ohio. We do this by supporting best practice teaching in Ohio. We maintain a website for K-12 teachers with peer-reviewed lessons aligned to state and national standards in English Language Arts, mathematics, and science. I am delighted to be here today to share insights and materials identified as useful for helping students create mathematical understanding through reading. My picture is a reminder that the mathematics communication between teacher and student is an important component of teaching and learning for understanding. Today there is a media buzz about 21 st century skills and the need for student literacy in all content areas. The just released Time to Act report from the Carnegie Council for Advancing Adolescent Literacy states that "Adolescent literacy in the content areas is pinpointed as "a cornerstone of the current education reform movement, upon which efforts such as the American Recovery and Reinvestment Act must be built." Teachers are challenged to teach mathematics for student understanding, and for students' success on standardized tests. By adding a literacy component to their mathematics classes, teachers can help students achieve both goals. Today we will just scratch the surface about how a focus on improving student literacy skills can improve students’ learning of mathematics. I hope you will come away from our discussion confident that as students' literacy needs are met, good things will happen for students and for their understanding of mathematics.
  • We need to acknowledge that, even though people are born with an innate number sense, people too often believe that they are not able to learn or remember basic mathematics. In order to better understand student difficulties with learning mathematics, let's consider mathematics as a language to be learned. We will focus on how to help students learn the vocabulary of the mathematics language and become readers of mathematics. What does it mean to read mathematics?
  • The key phrase here is "make sense." For mathematics to be important to students, mathematics text, meaning all the printed words and symbols, must make sense! Cognitive science tells us that if something makes sense it sends a signal to the brain that the information is worth tagging for long-term storage! The Time to Act reports: Reading in the content area is where student growth in reading occurs in the later grades. The ability to comprehend written texts is not a static or fixed ability, but rather one involves a dynamic relationship between the demands of texts and the prior knowledge and goals of readers. Retrieved 9/21/09 It is important to note, that t he major difference between reading in grades K-5 and reading in grades 6-12 is the transition from learning to read to reading to learn. These comments require a shift in thinking about literacy and mathematics content. The day of the textbook only being useful for the problem sets should be over.
  • Abstract mathematics concepts, such as numbers, variables, and operations, make visualizing meaning difficult at every level of mathematics. Mathematical symbols are often pictorial with no phonetic clues to their meaning. Learning the meaning of symbols is like sight reading. Symbols must be translated into English, connected to a concept, and finally applied. Word order in problems—frequently there is no thesis statement and the question is often stated last. Students must read through a problem multiple times to understand what is being given and what is being asked. Mathematics textbook design: often written above grade level dense and terse in style Math Texts contain more concepts per sentence and paragraph than any other type of text!!! designed with a complex structure that does not lend itself to reading Awareness of these obstacles will go a long way to supporting student learning.
  • Here we have words for math objects, math operations, and symbols. Only in math class are relatively simple words like combination, pi, and range filled with a depth of meaning not appreciated in English class. Teaching specialized vocabulary will always be a content teachers responsibility because of the special language involved in the field. Mathematics teachers must become alert for language pitfalls that contribute to the misunderstanding of mathematics ideas. Consider words having different meanings in mathematics and common usage such as prime or mean, similar sounding words with distinctly different meanings sign and sine, or multiple words for the same concept such as exponent and power. Does mathematics sound like a foreign language to students? We need to grow awareness of what students are hearing even when the teacher is speaking. Think about the words. When we take the area "of" a triangle, we mean what the students think of as "inside" the triangle. Understanding even small words make a big difference when speaking and reading in mathematics class: Enunciate small, but significant, words precisely: of and off Be aware of possible word confusions "a" can mean "any" in mathematics Emphasize the correct use of little words: are, can, on, who, find, ten, tens, and, or, each Studies indicate that a knowledge of mathematics vocabulary directly affect achievement in arithmetic, particularly problem solving. Heidema 21 Students need to know what we are talking about. Reading in any content area presents special problems because if you don ’ t know content you will have a difficult time understanding the texts, and if you don ’ t understand the texts you are unlikely to learn content. Retrieved
  • Being able to read is an equity issue in all subjects, including mathematics. The National Assessment of Educational Progress (NAEP) has three levels of reading proficiency: basic, proficient and advanced. NAEP scores for 17 year-olds consistently show the same pattern: a majority of students achieve the basic level of reading skills, and at this basic level there are no significant differences based on race/ethnicity or SES. At proficient levels, the scores show stark differences aligned with race/ethnicity and SES. At the most advanced level, less than 10 percent of 17 year olds, regardless of race/ethnicity or SES, are able to comprehend complex texts. The NAEP data, and its consistency across years, suggests that the problems of adolescent literacy involve a range of readers, from those with the most basic skill needs to those who have developed general comprehension strategies, but not the specialized strategies, vocabulary and knowledge base required for understanding complex discipline specific texts. As math teachers, where do we start?
  • Helping students be successful reader of mathematics text involves three interactive elements: Supporting the reader with literary strategies to learn new concepts and vocabulary. We should remember that the ability to comprehend written texts is not a static or fixed ability. Establishing a climate or classroom environment that is appropriate and supportive in ways that include making the connection between effort and achievement. This climate will include discussion, questioning, and helping the student see the usefulness and function of math in the real world. Regularly using Text Features to understand vocabulary and decode the symbols. In this article, a middle school teacher describes how she tries to implement the NCTM standards to engage students in problem solving, do mental mathematics, write about their thinking, and use technology in the mathematics classroom. There are helpful hints about classroom reading practices to interest students in their textbook. In one, each student opens their text, reads for one minute, notes the page number, closes the book, and writes what they remember. The student compares his or her notes to what is actually on the page — according to the author, students are always amazed. The article includes textbook reading activities for whole class, small groups, and partners and an Individual Textbook Scavenger Hunt for the first week of school or anytime a teacher wants to emphasize the importance of reading the textbook With an awareness of these three interactive elements, teachers can use literacy strategies to support student mathematics learning.
  • Literacy strategies support reading which is a constructive process in which readers react to text, using prior knowledge and experience to make connections, generate hypotheses, and make sense of what they read. (Heidema, p.1)
  • " Before " strategies activate students' prior knowledge and set a purpose for reading. " During " strategies help students make connections, monitor their understanding, generate questions, and stay focused. " After " strategies provide students an opportunity to summarize, question, reflect, discuss, and respond to text. ( Retrieved 9/21/09 excellent list of classroom strategies.)
  • When learning abstract mathematical words, just looking up and writing down meanings will not be enough for students to develop conceptual understanding. The use of next four vocabulary learning strategies will be valuable tools for students.
  • This Three-Column Entry is a form of semantic feature analysis designed to help the student understand meaning for a vocabulary word. It can be used before reading or during reading, with reference sources or not, and is an example of writing that can be found in a vocabulary journal.
  • Venn diagrams— a strategy that is familiar to math teachers! A way to sort and organize information of any kind from two or three sources. Venn diagrams Sorting and organizing information Venn diagram templates for comparing and contrasting Seven Strategies to Teach Text Comprehension ORC #6738 provides a useful summary of reading strategies.
  • Frayer Model is an example of a graphic organizer that can be used to organize mathematics meaning and concepts. Use the Frayer Model to develop students' understanding of the words or concepts. Students fold a piece of paper into four parts or quadrants. In the first quadrant, the student defines the term in their own words or use a given definition; in the second quadrant they list facts known about the word; in the third quadrant they list examples of the term; and in the fourth quadrant they list non-examples. Higher order thinking is applied as the student synthesizes and applies the word or concept for the examples and non-examples.
  • In this modified Frayer Model, the student builds understanding of the meaning of perpendicular and the symbol.
  • Semantic Feature Analysis Grid is another good strategy for creating understanding of mathematical terms. This is a handy tool for comparing features of mathematical objects. Generally in semantic feature analysis, the left-side column contains the names of members of the category to which the target concept belongs. The top row contains names of features of members of the category. Students can add terms to the columns or rows. Students fill out a grid to help them compare a term with other terms that fall in the same category.  When the grid is completed, students have a visual reminder of how certain terms relate to each other.
  • Another idea for using a Semantic Feature Analysis is for groups of students or whole classes to discuss whether each item is an example of each concept, marking + for positive examples,– for negative examples, and ? for items which might be examples under certain circumstances. Perfect number is defined as a positive integer which is the sum of its proper positive divisors Deficient number is a number such that the sum of all proper divisors of the number (divisors other than the number itself) is less than the number. Example: the number 21. Its divisors are 1, 3, 7 and 21, whose sum is 32. Because 32 is less than 2 × 21, the number 21 is deficient. Its deficiency is (2 × 21) − 32 = 10. Abundant number is a number such that the proper divisors of the number (the divisors except the number itself) sum to more than the number. example: the number 24. Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number. 24 is abundant. Its abundance is 60 − 2 × 24 = 12. Square number , sometimes also called a perfect square , is an integer that can be written as the square of some other integer; Triangular number is the sum of the n natural numbers from 1 to n .
  • For literacy strategies to become an effective part of a student’s “toolbox,” teachers must provide instruction on how and when to use them.  When providing instruction, consider the following teaching suggestions:>>>>> During the whole-class activity, solicit and compare various responses on how the strategy can work.
  • This is just one of many strategies associated with Polya’s four-step problem solving model.  There are many other problem solving formats. They all should involve multiple reading of the problem. Read aloud or individually and reread, form mental images, and understand problem. There is no algorithmic way to be sure that students "can do" word problems, but students can be taught to approach problem solving with special emphasis on reading and rereading the problem or the mathematics text. Manipulatives and Movement ORC# 11868 gives examples of this strategy (page 7). Problem Solving Standard Format: Step1. Understanding the Problem Vocabulary Paraphrasing the Problem Identifying the Statement Part and Question Part of the Problem Step 2 . Analyzing the Problem ·         Identifying the Statement Data ·         Identifying the Question Data ·         Identifying the Relationship Between the Statement and Question Data ·         Translating the Word Sentence into a Number Sentence Step 3 . Solution Plan Stating the Sequence of Operations Formulating the Equation(s)  Step 4 . Solving the Problem ·         Solving the Equation(s) ·         Finding the Answer to the Problem Step 5 . Answering the Problem Inserting the Numerical Answer into the Problem Sentence Formulating the Answer Sentence.   Retrieved from
  • Retrieved For literacy strategies to become an effective part of a student’s “toolbox,” teachers must provide instruction on how and when to use them.  When providing instruction, consider the following teaching suggestions: Introduce one strategy at a time, and let students apply it several times while you observe what they are doing and where they may need help. Model and explain the use of a strategy in an activity that lets students see how and why to use it. Practice a strategy as a whole class before asking students to use it independently.  During the whole-class activity, solicit and compare various responses on how the strategy can work.
  • Modification of the KWL (What do I K now do I know? What do I W ant to learn? What did I L earn?) Reading comprehension strategy for problem solving.
  • Roulette Problem Solving ORC #11861 gives students a chance to collaborate and communicate while problem solving. Small group strategy that requires students talk about mathematics Emphasizes mathematical discussion and communication to support problem solving. The site also contains hints for test taking and solving word problems
  • Word Sleuth p. 113 Mathematical Literacy , Thompson, et al p. 113 Mathematical Literacy, Heinemann, 2008
  • We need to help students become active readers. Reading should be the active construction of meaning by learners and an important step to becoming self-directed, autonomous learners. Understanding visual and analytic texts is a skill that students need in class and in their everyday lives. Integrating literacy strategies into math instruction will help students to make explicit connections between those strategies and content learning. Effective practices include those in which teachers: Coordinate nonfiction reading, including textbook reading, with more experiential learning activities. To aid comprehension, students need some background knowledge about the content prior to reading. Use nonfiction texts to reinforce and cement concepts introduced through lessons and activities. Use prereading or text preview strategies to orient students to the text. Before reading or making reading assignments, "walk" students through the text, pointing out the way the text is organized, the focus of the content, and troublesome words. Help students to establish a purpose for reading and to make connections to other concepts. Model the use of literacy strategies by reading content-specific texts. Not all text can be approached in the same manner. Students must be able to match their literacy strategies with the demands of the texts. Think-alouds, explicit strategy lessons, and guided instruction are ways to demonstrate the application of literacy strategies. From Bullet 4 leads to
  • This brings me to the end of my presentation, and a confession. As noted on my introductory slide I was a late reader, but once I began to really read, thanks to Nancy Drew, life changed! I trust that by using literary strategies to improve math learning, students lives can be changed too! Thank you for your time and best wishes for every success in your classroom!
  • Reading Mathematics Is Different

    1. 1. Reading Mathematics is Different Presenter Judy Spicer Ohio Resource Center September 30, 2009
    2. 2. <ul><li>Judy Spicer </li></ul><ul><li>Mathematics Teacher (Since forever!) </li></ul><ul><li>Mathematics Content Specialist at </li></ul><ul><ul><li>Eisenhower National Clearinghouse (ENC) (1997-2006) </li></ul></ul><ul><ul><li>Ohio Resource Center (ORC) (2006-present) </li></ul></ul><ul><li>Reader beginning in the 4 th grade! </li></ul>No, I am not a pilot!
    3. 3. <ul><li>People are born with innate number sense. </li></ul><ul><li>Mathematics is part of everyday life . </li></ul><ul><li>Mathematics is more then mastering formulas and procedures. </li></ul><ul><li>Mathematics involves communication . </li></ul>As mathematics educators, we believe
    4. 4. <ul><li>“ Reading mathematics” means the ability to </li></ul><ul><li>make sense of everything that is on a page </li></ul><ul><li>whether the page is a worksheet, spreadsheet, overhead transparency, computer screen, or a page in a textbook or journal—in other words, any resource that students might use to learn and apply mathematics. </li></ul><ul><li> Teaching Reading in Mathematics , ASCD, 2002 </li></ul>
    5. 5. Obstacles to Reading Math <ul><li>Abstract concepts </li></ul><ul><li>Mathematical symbols </li></ul><ul><li>Word order </li></ul><ul><li>Mathematics textbooks </li></ul><ul><ul><ul><li>Literacy Strategies for Improving Mathematics Instruction , ASCD, 2005 </li></ul></ul></ul>
    6. 6. Figure 1.1. Confusing Terms, Formats, and Symbols in Mathematics                                                                                                                                                         Literacy Strategies for Improving Mathematics Instruction , ASCD, 2005 Literacy Strategies for Improving Mathematics Instruction , ASCD, 2005
    7. 7. Rewards of Reading Mathematics <ul><li>Motivating : The ability to read and understand the words and symbols makes mathematics accessible to students. </li></ul><ul><li>Empowering : Students are empowered because when reading they will engage with and make sense of mathematics. </li></ul><ul><li>Conducive to a student-centered learning : Students who are readers can be autonomous, self-directed learners. </li></ul>
    8. 8. Interactive Elements of Reading Reader Reading and Writing to Learn in Mathematics: Strategies to Improve Problem Solving (ORC#: 12736) Classroom Environment Unlocking the Mystery of Mathematics: Give Vocabulary Instruction a Chance (ORC#: 11864) Text Features Getting to Know Your Middle Grades Mathematics Textbook (ORC#: 9282) Teaching Reading in Mathematics , ASCD, 2002
    9. 9. Effective Literacy Instruction Provides <ul><li>Multiple exposures to words and symbols through conversation, visual displays, readings, etc. </li></ul><ul><li>Authentic opportunities to use words and symbols in classrooms and to make appropriate connections for use in other contexts </li></ul><ul><li>Diverse texts , including literary and factual pieces, poetry, and visual materials, from which students draw words and phrases to study more carefully </li></ul><ul><li>Retrieved 9/21/09 </li></ul>
    10. 10. Strategies for Reading <ul><li>Before reading : preview, activate prior knowledge, evaluate reason for reading </li></ul><ul><li>While reading : check understanding by paraphrasing, seek contextual clues for unfamiliar words, integrate information with prior knowledge </li></ul><ul><li>After reading : summarize, evaluate ideas and understanding, look for a broader perspective </li></ul><ul><li> Retrieved 9/21/09 </li></ul>Literacy Strategies for Improving Mathematics Instruction , ASCD, 2005
    11. 11. Vocabulary Learning Strategies <ul><ul><li>Provides authentic connections to previous knowledge </li></ul></ul><ul><ul><li>Places vocabulary words and symbols in a meaningful context </li></ul></ul><ul><li>Makes vocabulary words and symbols visible </li></ul><ul><li>Examples: Concept maps, semantic feature analysis charts, word maps, and Venn diagrams allow students to learn word meanings and analyze structures. </li></ul>
    12. 12. Three-Column Entry Word It usually means … In mathematics it means … Symbol Math meaning … Other meanings … . (decimal point) separates units and tenths digits separates integer part (left) from fractional part (right) indicates end of sentence can mean to multiply when in different position relative to numbers
    13. 13. Venn Diagram for Prime <ul><ul><ul><ul><ul><li>Prime </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>English Mathematics </li></ul></ul></ul></ul></ul>Something that is first and special Mathematical Literacy , Heinemann, 2008 Something that is of first quality, prime meats or prime number Prime numbers are the first, most basic numbers and are special. Every number is prime or can be factored into primes See Sorting Polygons ORC# 1163 for a math vocabulary lesson linking geometry to probability and statistics.
    14. 14. An equation is a mathematical statement that shows that two expressions are equal. Facts/Characteristics 3x – 2 = 4x + 7 (linear equation) ab = ba (an identity) F = 1.8C + 32 (a formula) 5 + 6 = 11 (a number statement) P = 2l + 2w (a formula) x = 3 (statement of value) Non-examples Frayer Model 2x + 3y (expression) 3 (number) perimeter (word) x < y (inequality) = 4.2 (has no left side) Equation -always has exactly one equal sign - the left side is equivalent to the right side - some equations have 0, 1, 2 or more solutions - some equations contain just numbers - some equations are algebraic models of relationships with corresponding graphical and numerical models (e.g., tables) Examples Definition Retrieved 9/24/09
    15. 15. My definition: Perpendicular to Forms a 90  angle Illustration: Example(s) of Use: AB  CD Related words and symbols: Symbol :  Modified Frayer Model two legs of triangle form a right angle & are perpendicular
    16. 17. <ul><li>Category: Whole Numbers </li></ul>Semantic Feature Analysis Teaching Reading in Mathematics , ASCD, 2002 Features/ Properties prime perfect deficient abundant square triangular Terms 2 x x 3 x x x 4 x x 5 x x 6 x x 7 x x
    17. 18. Using Literacy Strategies <ul><li>Introduce one strategy at a time. </li></ul><ul><li>Model and explain the use of a strategy in an activity that lets students see how and why to use it. </li></ul><ul><li>Let students apply it several times while you observe what they are doing and where they may need help. </li></ul><ul><li>Practice a strategy as a whole class before asking students to use it independently.  </li></ul>
    18. 19. Problem Solving and Reading <ul><li>Five-Step Format </li></ul><ul><ul><li>Restate the problem (understand the problem) </li></ul></ul><ul><ul><li>Analyze the problem (find needed data) </li></ul></ul><ul><ul><li>Solution plan </li></ul></ul><ul><ul><li>Find the answer </li></ul></ul><ul><ul><li>Check. Is the answer reasonable? </li></ul></ul><ul><li>See Adapting Literacy Strategies to Improve Student Performance on Constructed-Response Items ORC #11602 </li></ul><ul><li>See Roulette Problem Solving activity p. 7 ORC #11861 to give students a chance to collaborate and communicate while problem solving. </li></ul>
    19. 20. Retrieved 9/2/09
    20. 21. Problem Solving <ul><li>K-N-W-S and KWC </li></ul><ul><ul><ul><li>What facts do I KNOW ? Which info do I NOT need? WHAT does the problem ask me to find? What STRATEGY /operation/tools can I use to solve the problem? </li></ul></ul></ul><ul><ul><ul><li>What do I KNOW ? What do I WANT to find out? Are there special CONDITIONS that I have to look out for? </li></ul></ul></ul><ul><li>See Adapting Literacy Strategies to Improve Student Performance on Constructed-Response Items ORC #11602 </li></ul>
    21. 22. Problem Solving <ul><li>SQRQCQ </li></ul><ul><ul><li>Survey : Read the problem quickly for general understanding. </li></ul></ul><ul><ul><li>Question : Ask what information the problem requires. </li></ul></ul><ul><ul><li>Read : Reread the problem for information needed to find a solution. </li></ul></ul><ul><ul><li>Question : Ask what operations, in what order, should be performed. </li></ul></ul><ul><ul><li>Compute : Do the computation/construct a solution. </li></ul></ul><ul><ul><li>Question : Ask if the solution process seems correct and the answer reasonable. Requires rereading the problem again. </li></ul></ul><ul><li> </li></ul><ul><li>See Roulette Problem Solving activity in ORC #11861 to give students a chance to collaborate and communicate while problem solving. </li></ul><ul><li>Literacy Strategies for Improving Mathematics Instruction , ASCD, 2005 </li></ul><ul><li>Literacy Strategies for Improving Mathematics Instruction , ASCD, 2005 </li></ul>
    22. 23. Communicating Math Beyond the Classroom <ul><li>Make &quot;explain a solution&quot; to someone at home part of homework assignments. </li></ul><ul><li>Assign a &quot;Word Sleuth Team&quot; to research the word origins of a chapter's vocabulary. Be sure to check out The Lex Files </li></ul><ul><li>Encourage students to find and report real world uses of mathematics </li></ul>Mathematical Literacy, Heinemann, 2008
    23. 24. Reading for Understanding <ul><li>Coordinate nonfiction reading, including textbook reading, with experiential learning activities . </li></ul><ul><li>Use prereading or text preview strategies to orient students to the text. </li></ul><ul><li>Model the use of literacy strategies with think-alouds, explicit strategy lessons, and guided instruction. </li></ul><ul><li>Teach vocabulary with attention to helping students learn word and symbol meanings and relationships. </li></ul><ul><li>Provide graphic organizers for note taking and synthesis of important information. Use writing frames, T-charts, and column notes to guide students through texts. </li></ul>
    24. 25. Resources <ul><li>In the NSDL Wiki </li></ul><ul><li>See MiddleSchoolPortal/Reading and Writing Mathematics </li></ul><ul><li>See MiddleSchoolPortal/Quantitative Literacy </li></ul><ul><li>Go to the Ohio Resource Center for high-quality, peer- </li></ul><ul><li>reviewed lessons aligned to Ohio and national standards, and </li></ul><ul><li>professional support materials. </li></ul><ul><li>Find all resources mentioned in this presentation at </li></ul><ul><li>Resources for Reading Mathematics Webinar </li></ul>