5
Reading, Writing, and Speaking in Mathematics
As teachers of mathematics, we sometimes forget that the words and phrases that are familiar to us are foreign to our students. Students need to master this language if they are to read, under­ stand, and discuss mathematical ideas.
-Thompson & Rubenstein
Thompson and Rubenstein were referring to the issue of language and mathematics learning in mainstream classrooms. They state vocabulary or the fluent use of terminology is a necessary condition for overall mathematics achievement. If the learning of mathematics is highly dependent on its language and the teaching of math vocabulary is crucial, it is doubly so for ELLs.
English language learners in secondary schools should have many opportunities to communicate their mathematical ideas and questions. Talking and writing about their mathematical thinking helps ELLs build word knowledge and oral expression and clarify their thinking. Discussions with the teacher or peers are also useful monitoring tools for teachers. Through listening and recording student conversations and peer problem solving, teachers can monitor individual student progress. Mathematics is no longer viewed as isolated, individualistic, or competitive. Mathematics problems are ideally suited to cooperative group discussions because they have solutions that can be objectively demonstrated. Students can persuade one another by the logic of their arguments. Mathematics problems can often be solved by several different approaches, and students in groups can discuss the merits of different proposed solutions (Robertson, Davidson, & Dees, 1994). For this and several other reasons mentioned throughout this chapter, cooperative learning takes a central place in mathematics instruction.
Teaching and Learning Meaningful Math
The lesson template for ExC-ELL is the same as that described in previous chapters. It begins with stating the learning objectives, back­ ground and vocabulary building, and ends with evidence of learning of vocabulary, concepts, and mathematical processes. We will provide an example here.
Reading Comprehension in Math
Planning the lesson. Teachers can begin planning lessons by using a set of questions such as the following to work in the text and with the objectives and standard s they have selected:
Semantic Awareness.
Which Tier 1, 2, and 3 words will I teach? Which techniques will I use to teach each one?
How can I make sure students interact with each word at least 5 times prior to the lesson? And use the words 7 or more times during the lesson?
Connecting students' prior knowledge with new knowledge.
What are the important mathematical concepts of this lesson? How will I help students link these goals with previous work and with our math standards?
Do the problems or exploration allow for multiple strategies, perspectives, and solutions?
Is the text appropriate, or do I parse it or find another?
What do the students know about this topic? How can I find out?
What type of motivation and props can I use to connect to what we are about to read and learn? How can I make this lesson interesting, accessible, and challenging for students at all levels of mathematical understanding?
What visuals do I use to represent these problems?
Metacognitive Awareness.
Which reading strategies are most appropriate for comprehending the text?
How do I present them in my read-aloud?
Which is the best activity for the students to practice these skills?
How will I debrief what they have learned? What open-ended questions might extend students' thinking?
What questions should I model when introducing the lesson?
Active and engaged reading.
What are the best sections for partner reading? What strategy should I use for partner reading? How can I make sure they use the new vocabulary?
Is there a graphic or cognitive organizer they can use?
What follow-up cooperative learning strategy should I use to consolidate knowledge and develop more language skills?
Ample discussions to anchor domain knowledge.
What are my questions for before reading? During reading, where do we stop and debrief?
What are my debriefing questions for the final discussion after reading?
What are appropriate sponge activities for teams that finish early or for the next day?
Written product for assessment.
How can I link assessment with this instruction?
What is the best genre for writing up the content they have learned?
What evidence of vocabulary, grammar, critical thinking, and other skills do I want them to exhibit?
What rubrics should I use? How will I test vocabulary?
Do other types of assessments fit here?
After background building, the teacher reads two or so para­ graphs to model reading comprehension strategies. First, the teacher makes one of the statements below and then proceeds to model how to perform the statement. For instance, a teacher would say: * I'm going to visualize and think aloud about what I just read. * I'm going to read chunks I can handle and then summarize. * I'm going to change the title and subheadings into questions. * I'm going to make predictions before I start to solve the problem . * What are some possible strategies I might try to solve these problems? * What has helped teams in the past? * What could that word mean? Let me reread. * I'm going to stop and reread confusing parts of this sentence. * I'm going to put a Post-it note after this sentence so I can ask for clarification. * What kind of test question would the teacher ask from this paragraph? * How does this relate to the paragraph above? * How does this relate to the problem below?
Vocabulary Development for Transmittin'1 Ideas
Math words do not occur in isolation. They are learned in the context of using math concepts during the preteaching of vocabulary before the teacher 's read-aloud, explanations on board with students, partner reading, and concept consolidation activities. Thompson and Rubenstein (2000) identify key vocabulary issues that become major hurdles for ELLs and other students. They provide the following examples for teachers because these are easily overlooked and are rarely taught explicitly.
Mathematics and everyday English share some words, but they have distinct meanings:
Number: prime, power, factor
Algebra: origin, function, domain, radical, imaginary Geometry: volume, leg, right
Statistics/probability: mode, event, combination Discrete mathematics: tree
Some mathematics words are shared with English and have com­ parable meanings, but the mathematical meaning is more precise:
Number: divide, equivalent, even, difference Algebra: continuous, limit, amplitude, slope Geometry: similar, reflection
Statistics/probability: average
Discrete mathematics: array, edge, and, or
Some words have more than one mathematical meaning: Number: inverse, round
Algebra: square, range, base, inverse, degree
Geometry: square, round, dimensions, median, base, degree, vertex
Statistics/probability: median, range
Discrete mathematics: sequence or arithmetic sequence, reasoning or circular reasoning.
Some mathematical phrases must be learned and understood in their entirety:
Number: at most, at least Algebra: one-to-one
Geometry: if-then, if-and-only-if Statistics/probability: stem-and-leaf
Discrete mathematics: if-then, if-and-only-if
In particular, it is important to highlight to ELLs that some mathematical phonemes are homophones with other everyday words in English. When you are explaining or presenting a lesson, look out for words such as: sum some pair pear whole hole sine sign cosine cosign pi pie dual duel plane plain leaf leave complement compliment graph graft
Cognates. Mathematics, like other subject areas, are high-cognate languages. English math words are cognates with other languages like Spanish, French, and Italian because they are derived from Latin and Greek roots. That is, they are similar. For instance: English Spanish sum suma complement complementar perpendicular perpendicular factor factor equation ecuaci6n mathematics matematicas algebra algebra graph graf gram gram
False cognates or terms to watch out for:
Spanish English restar is not rest, it means remainder or “subtract” resto means remainder or "I subtract" Redondo round (circle) Redondear round off
Although most mathematics appears to be a universal language, in English-speaking classrooms a variety of language patterns can occur. Margot Gottlieb (2006) uses the following example to illustrate the many ways a teacher can talk about addition and subtraction: Addition Subtraction and take away plus minus more less more than less than altogether diminished by increased by are left sum remain in all fewer total from combine not as much as difference
This means an ELL who has learned only "add" and "subtract" for addition and subtraction may be totally confused 40 out of 41 instances when the teacher is explaining or giving instructions.
If ELLs have studied math in their countries, they will be able to keep up with math much faster than ELLs who have not. This is where differentiation is important. Sometimes, ELLs with rich mathematical back­grounds appear disinterested because they already know the concept. All they need is the language to express what they know. In this instance, differentiated instruction calls for rich vocabulary and oral development.
For ELLs who do not have a mathematical background, additional one-on-one instruction with you or a bilingual instructional resource person will be necessary in order to help the student catch up with both language and content. Simply pairing with another student will not be enough. Pulling the student out of the class with an instructional aide or someone who is not trained in mathematics will not help either. The student needs to be immersed in regular math instruction with quality support through immediate intensive interventions beyond the class­ room. Math fluency is like reading fluency. Automaticity of basic skills needs to be achieved before the student can truly comprehend higher math concepts. If the student doesn't know his multiplication tables or understand the intersect between multiplication and division, learning the words will not help him understand. Someone needs to teach him the basics. Just as decoding skills are basic for reading comprehension, basic operations and concepts are the "decoding skills" of math.
Partner Reading
Any textbook or text can be used for partner reading. After the teacher models reading and thinking about reading math, partners are assigned a reading portion, which includes rereading what the teacher read aloud. Each partner reads one sentence at the beginning of the class period. "If a student does not know how to read mathematics out loud, it is difficult to register the mathematics" (Usiskin, 1996, p. 236).
After reading a small section ask the whole class: Turn to your partner and discuss, "What did you notice about the data?"
Next, ask students to continue reading in small chunks. Then stop and: 1. Work together to solve the problems. 2. Decide on a strategy for reading and a strategy for problem solving. 3. Be sure you both agree to the solution before you go to the next problem. 4. Be sure you can both explain how to get your solutions. 5. Be prepared to present problem, process, and solution to the class.
Partner Interview. One way to ensure all students have mastered the concepts, and vocabulary and can present, is to have partners prepare by using the partner interview strategy. Pairs interview each other on the information to be learned, then switch roles. Pairs can then dis­ cuss points of weakness and disagreements or agreements, make notes, and role play the interviews again for additional practice.
Teacher and Students
Debrief/Reflect After Partner Reading!
Reflection includes mathematical and social aspects of group work. Students need to talk about and hear the strategies, problems, and successes of others. After partner reading and activities, open­ ended questions such as these can be asked: * What strategies did you try? Why? * What was your solution? Is there another solution that might work? * What problems did you have? Were you able to solve them? How? * What are some ways to work that you would recommend to other partners? * What strategy was particularly helpful to you or your partner?
The debriefing should take no more than 7 minutes. That includes any recommendations you as a teacher might have for your class. Debriefing should, however, be conducted often throughout the period, or after each instructional activity.
Students Work in Teams of Four
There are many cooperative learning strategies for teamwork. All problem solving, explorations, investigations, and creative projects can be done in teams of four. These experiences work best when each student is assigned a specific aspect of a problem to solve or investigate. A note of caution: when roles such as timekeeper, artist, etc., are assigned, the work is typically left to the "problem solver." When a task is evenly distributed among the 4 members, including the ELL, the cooperative process and learning go much more smoothly.
Working in teams gives ELLs more opportunities to "talk mathematics" and teachers opportunities to listen in and "record learning." Teachers can also conduct instructional conversations with teams as they work in teams. They can reinforce correct usage of vocabulary and help students rephrase mathematical ideas, or simply ask questions that elicit the use of key words and concepts.
Teachers like to post or hand out lists of strategies such as the ones below as reminders.
Problem-Solving! Strategies * Act out or use objects * Use or make a table * Make an organized list * Make a picture or diagram * Use or look for a pattern * Work backwards * Use logical reasoning * Make it simpler * Brainstorm * Guess and check
Higher-Order Thinking and Formulating Questions
There are several cooperative learning strategies that can be used with the whole class: Numbered Heads Together, Tea Party, Corners, Line Ups, and Roundtable are some that adapt to many learning tasks and content.
Numbered Heads Together. Each student in a team is numbered from one to four. Then, the teacher posses a question, problem, or issue. Students "put their heads together" and talk this over with the team and make sure all are prepared to respond, especially the ELL. The teacher then calls upon students by number to represent the team. Some teachers give points for correct answers; others just set the context for a challenging game.
Line Ups. Students can be given problems on cards to solve. Then they line up in sequence of results. Or students can line up based on specific categories, such as birthday dates. After this, they line up in bar graph style. For instance all the students born in January stand in a line, all born in February form a second line, and so forth. To find the median birth date and the percentage or fraction of the students born each month, they can huddle in their lines or go back and transfer the information to paper.
Jigsaws. Anything that can be divided into sections can be jigsawed. For instance, a different portion of a text can be given to each team to read, summarize, and prepare to teach to the other teams. Problems at the end of a chapter can also be split up among teams this way. There are different types of jigsaws. There is a simple in-team jigsaw where something is divided into 4 parts and each team member becomes an expert on that part and teaches the others. In another, each team studies a portion and presents or teaches the information to the other teams in a whole-class presentation or by rotating from team to team. A more elaborate jigsaw typically has 4 steps: 1. Task division: A task, passage or text, or a problem is divided into equal parts. 2. Home groups: Each team member is given a topic on which to become an expert. 3. Expert groups: Students who have the same topics meet in expert groups to discuss the topics, master them, and plan how to teach them. 4. Home groups: Students return to their original groups and teach what they learned to their team members. 5. All students take the same test, but individually.
Corners. This is a quick activity that gets student out of their seats and gives them an opportunity to interact with other students since the small groups are randomly selected. All students in the class number off from 1to 8 or the number that will enable students to stand in different parts of the room in triads or groups of four. The teacher poses a question or problem, and they discuss it. Volunteers share their results with the other groups. As they get ready to go back to their seats, each triad invents a good-bye or 2-line rhyme using terminology they have been discussing.
Roundtable. Each team has only one piece of paper or worksheet and one pencil. One student in the team writes down an answer, says it aloud, and then passes the paper to the person on his right. The process continues until the teacher says stop (usually from 3 to 5 minutes for math). No student is allowed to pass. However, a team can help a student who asks for help. Roundtable can be used with problems such as: * Find combinations of 2 numbers that add up to 42. * Give examples of the rule (am)n = amn. * How many terms in geometry can you list? * Draw and label as many geometric shapes as you can.
Whatever the strategy used, students will benefit from the inter­ action. Students who teach other students learn more. ELLs who need help understand better when they have peers to mull over with or to help articulate their thinking.
Final Debriefing and Anchoring of Knowledge
For a final debriefing 5 to 7 minutes before the class ends, teachers usually ask:
What did you learn today? What made it easier?
What made it difficult?
How can you improve your learning?
How can your team improve for next time?
Student Writing to Anchor Comprehension
Some team writing can be simple compositions prepared as oral recitations to be presented as raps or choral readings. Sometimes students even find ways of making these rhyme, or they borrow lines from other subjects. For instance:
How can we use fractions? Let me count the ways …
What are some variable attributes of … Let me count the ways …
What are the differences among the rectangle, square, rhombus, and parallelogram? Let me count the ways …
Other team writing. Individual writing can be simple summaries on exit passes or learning logs of what they learned that day. Other times, specific questions can be written on board: * How many polysemous words did you find today besides similar? * Compare and contrast a rectangle and a rhombus. * What is the difference between the square of a number and the square root of a number?
As part of a homework writing assignment, ELLs can add visuals or drawings to the definitions or descriptions. These visuals help students see the relationships that they must learn to verbalize. There may be certain processes that are very difficult for ELLs to write about, but they might be able to illustrate them.
When it comes to assessment, performance-based assessments are best. Rubrics will give you a lot of information for individual and team products. These rubrics can be used for you and your students to assess: * Cooperative learning activities * Exit passes and learning log entries * Summaries and reports * Presentations of math processes
Rubrics are the basis for establishing inter-rater agreement among teachers for performance tasks. Standards-based rubrics can and should be adapted for ELLs. When traditional tests are not logical or practical ways of assessing ELLs, the rubrics or observation protocols become the best way to assess ELLs. For example, in math, an observation protocol might focus on: 1. Does the student use the first language to make meaning? 2. Does the student require more than one explanation? 3. Does the student use pictures to help meaning? 4. Are objects and manipulatives helpful to the student? 5. How often does the student ask peers for help? 6. What type of help does the student ask from peers? From the teacher? 7. Does the student read the problems aloud? 8. Do examples help the student? How? 9. Does the student prefer written or oral directions? 10. Does the student check work done?
Portfolios with an ELL's work help gauge individual progress on a monthly basis. They also serve as evidence for AYP, and as a tool for discussing progress or lack thereof with other teachers and/ or the student's parents.
Summary
* Math is a language that needs to be pretaught before students can comprehend math texts. * Math problems are ideally suited to Cooperative Learning. * Math is filled with cognates shared between English and Spanish and bilingual teachers can capitalize on these by explicitly pointing them out. * It is more beneficial for students to learn selected basic concepts and to learn them well than it is to cover many math topics superficially.