• by Jack Dieckmann, M.A. • IDRA Newsletter • May 2003
With a dramatically growing number of English language learners in the United States, there is a rising demand for skilled teachers who are trained to serve current and former limited-English-proficient (LEP) secondary students. Often, teaching methodology presented to English as a second language (ESL) teachers is so general that they are forced to make it work with all of the subjects they teach. Teachers need to be able to see how ESL instruction fits into content area subjects. In particular, they must be able to see how language develops alongside mathematics.
ESL students are rarely challenged with meaningful and comprehensive instruction in math and science courses. Generic ESL strategies need to be seen in the context of a real content lesson with a mix of English dominant and English language learning students. Furthermore, students must communicate to make meaning at every stage of the inquiry process.
This article is the first in a three-part series on the role of language in teaching mathematics to English language learners. It describes a sixth-grade geometry lesson that engages all students in meaningful mathematics. The second article will examine how teachers observing the geometry lesson were guided through a follow-up conversation. Finally, the third article will address some of the challenges and solutions for effective communication and capitalizing on students’ language in math.
The Intercultural Development Research Association (IDRA) hosted an event recently for teachers participating in its ExCELS project. ExCELS (Educators x Communities = English Language learners’ Success) is an innovative professional development program that creates learning communities of schools, families and communities for English language learners’ academic success. Two secondary schools in San Antonio are the partner schools in this U.S. Department of Education Title VII program (see box below).
At the special event, trainers modeled a sixth grade lesson on the sum of angles. Teachers observed: “Students couldn’t wait to test their hypothesis. There was a buzz in the room.”
What follows is an abbreviated transcript of the lesson including what the teacher said and student comments that illustrate what was happening in the class. In the dialogues presented, notice that the teacher talks and the student responds and how these illustrate the math being learned, the language used, and the metacognitive or thinking-about-thinking parts.
The 45-minute geometry lesson was conducted for 19 sixth-grade students, four of whom are intermediate-level English language learners. The objectives of the lesson were to: investigate and predict sums of angles (mathematics); define shapes and identify Spanish cognates (language); discuss, explain and compare patterns (language); explain to student peers and the teacher how they arrived at predictions and conclusions (language); and identify mathematical thinking patterns and speak to the thinking processes experienced in mathematical experimentation (metacognitive). Following is a conversation between the students and teacher.
Orientation to Experimentation
Teacher: “Mathematics involves the noticing, predicting, and testing of patterns in numbers and shapes. You will be thinking, creating and talking about patterns. Where do you notice patterns?”
Student: “What do you mean?”
Teacher: “First what is a pattern?”
Student: “When something repeats over and over.”
Student: “Like colors in clothes: red, red, black, then red, red, black.”
Teacher: “Yes, patterns have the property that foretell in a very predictable way. You can predict what will happen without seeing the new shapes.”
Setting a Context for the Experiment
Teacher: “Today we are going to do an experiment with triangles and other shapes to see what happens when we join the angles. (Teacher shows scalene, isosceles, equilateral and right triangles.) Tape one of your triangles on the board. Compare and classify your triangle in contrast to the other triangles. What do these triangles have in common and what is different? Discuss this in your groups.”
Group A: “We didn’t know what to call them.”
Group B: “That one has a right angle so it’s a right triangle.”
Group C: “One is an equal-angle triangle.”
Group D: “We didn’t know what to call them either.”
Teacher: “We can classify triangles by the sides and by their angles. For instance this one is called an equilateral. Why?”
Student: “Oh, lados…lateral” (lados means “sides” in Spanish).
Teacher: “These triangles are isosceles and scalene.” (Properties are clarified and discussed.)
Posing the Problem
Teacher: “I’m going to give each group a different triangle. Tear off the corners of your triangles.” (Write on the board: “Corners are called vertices.”) “See what shape you come up with when you try to join all three vertices. Make sure everyone in your group agrees and see how many shapes you can generate.”
Conducting the Experiment
The teacher interacts with one group.
Group A: “We think we have three shapes. This one (points to angles placed in 1, 2, 3 order [see box above]). This is another (rearranges the angles into 2, 3, 1 order). And here is one more. Is that right? Did we get it?”
Teacher: “Well, how are these shapes the same and how are they different?”
Student: (Pause) “They all make a straight line.”
Teacher: “OK, how are they different?”
Student: “The numbers are in a different order. Is that right?”
Teacher: “So are they the same or different in shape?” (Points to the straight line formed.)
Student: The same.
Student: Different. Is that right?
Teacher: (Speaking to the whole class) “Now take your shape and tape it next to your original triangle on the board. Go to the board and see how your group’s shape is the same and different from the others.”
Analyzing and Interpreting Findings
Student: “They all make a line.” (Student draws line with marker.)
Student: “It’s not really a line. It’s more like a half-circle.” (Student draws semi-circle with marker.)
Student: “Sir, who’s right? Is it a line or a half-circle?”
Teacher: “All the answers are very interesting, but we want to focus in on the ‘tips’ where the angles touched. What shape do you see when the angles come together?”
Students: “A straight line.”
Teacher: “Even though the triangles were different, what happened when we put the angles together?”
Student: “They all formed a line.”
Teacher: “Did it happen in every case, with every triangle?”
Some students respond yes, others are quiet. It seems some do not understand the question. When polled, the majority voted for “All triangles will form a straight line if the angles are joined.”
Teacher: “Remember what I told you about mathematicians thinking about and looking for patterns? We’ve just finished an experiment with triangles. In one sentence, what can we say about the joining of the angles of any triangle?”
Students: “They form a straight line.”
Students recount in small groups what they have done and what they can conclude. In whole group discussion, they review their comparison and labeling of triangles and how they cut and rejoined them in new shapes that were a line or a semicircle They are asked about conclusions and generalizations given the variety of triangles. With prompting, students are able to state that regardless of the triangle’s shape and size, the shape resulting from joining all corners will always look the same.
Teacher: (Showing a larger triangle to the class) “Look at this much larger triangle. If I cut these corners up and join them, do you predict that three corners will form a straight line?”
Students: (Confidently) “Yes, we predict that they will form a straight line.”
Teacher: “Why do you predict that?”
Student: “Because it worked with all the other triangles.”
Student: “Your triangle is just like one of ours… it’s just bigger.”
Teacher: “OK. Now we are thinking like mathematicians. We’ve done some experimenting. We’ve looked for some patterns, and now we are going to see if the pattern holds for this triangle. This is the exciting part of mathematics.” (Tears the triangle and joins the three angles.)
Students: “We were right!”
As the teacher recreates the activity with a larger right triangle and asks the students why this was being done, they are able to say that mathematics is about pattern-finding and predicting patterns.
Application to Quadrilaterals: Setting a Context for the New Experiment
Teacher: “You are good mathematical thinkers because you noticed important patterns, made predictions and tested the predictions.”
Student: “What is this, science class?”
Teacher: “The thinking is very similar between a good scientist and a good mathematical thinker. Now are we ready for another math experiment?”
Posing the Problem
Teacher: “Our last experiment was about triangles. This one is with quadrilaterals. (Five different quadrilaterals are posted on the board.) Why did I call them quadrilaterals? Talk in your small groups.”
Group A: “Quadrilateral means four sides, like a rectangle.”
Group B: (English language learner) “Cuatro lados.”
Teacher: “Cuatro lados. Four sides.”
Teacher: “What is the same and what is different about these quadrilaterals. Do they have special names?”
Group A: “Square and rectangle.”
Group B: “Trapezoid.”
Teacher: “These two are trapezoids. One is a special kind of trapezoid. Can you guess what it is?” (Pointing to the right angle of the right trapezoid. No student could guess.)
Teacher: “This is called a right trapezoid because it has a right angle.” (Teacher uses the same process with a rhombus.)
Student: “Sir, let’s get to the experiment. I think I know what it’s going to be.”
One student was already tearing a rectangular sheet of paper to try out the experiment before completion of the discussion of predictions.
Teacher: “Before I give you the quadrilaterals and you tear them like you did the triangles, what do you predict the figure will look like? Talk in your groups.”
Group A: “It’s going to be a straight line, like with the triangles.”
Group B: “It’s going to be a circle.”
Group C: “It’s going to be part of a circle.”
Teacher: “Good guesses. Now when you do the experiment and join the corners, will the same shape apply to all the quadrilaterals?”
Students: (Pause) “Yes, because we got the same thing when we tried the different triangles.”
Conducting the Experiment
Teacher: “OK. Now I’m going to give each group a figure, and as with the triangles, you will tear it so that you can join the angles and see what shape is formed.”
Analyzing and Interpreting Findings
Teacher: “Tape your shapes on the board. Who predicted correctly?”
Student: “We said a circle, and it was the right answer.”
Student: “We formed two lines like this” (two angles joined in two pairs of a rectangle).
Teacher: “That’s interesting, but all the corners have to be touching.”
Summarizing What Was Learned
Teacher: “OK. What did we learn today?”
Student: “We cut up triangles and quadrilaterals.”
Student: “To find patterns.”
Teacher: “What was important about how we were thinking today? What were we trying to do with those patterns?”
Teacher: “What did we say mathematics was about?”
Student: (Pointing to the board) “Noticing and predicting patterns and numbers and shapes.”
Teacher: “So today we did some experiments to look at some patterns and shapes. In your groups tell me what we noticed about the joining of the corners of triangles and quadrilaterals.”
Student: (After brief buzz session, synthesis of group responses.) “When you cut the corners of a triangle, they form a straight line. When you do the same thing with quadrilaterals, they form a circle.”
Teacher: “Write in your journal the most important thing you learned today.”
The lesson engrossed the attention of all the students. Conversations were dynamic and energetic. No student was tuned out of the lesson, and the English language learners participated fully. Rhythms varied in the rate of response, and sometimes probing questions required further prompting of student responses but this did not diminish the individual and collective interest in the tasks, the predictions and the discussions of the findings. The lesson succeeded in all three areas: math content, language development and metacognition as demonstrated by student answers.
As demonstrated here, guideposts for effective learning of mathematics concepts include the following:
- The natural curiosity of students is a powerful hook for an experimentation or discovery driven lesson.
- Establishing relevance communicates a fundamental aspect of mathematics in contrast to the student perception of math as capricious, arbitrary and, ultimately, unknowable.
- Critical mathematical understanding requires that students determine what stays the same (invariant) and what changes (variant).
- Mini-reviews connect the activity (experiment) to the mathematical conjecture and verification process of doing and knowing mathematics.
- Repeating the process of mathematical exploration facilitates grasping increasingly complex mathematical shapes and properties.
The teacher’s planned assessment was an observation of student products and noticing of their conversation before, during and after the experiments. The critical elements of this measurement were the verbalization and comparison of patterns on angles in groups. This was not accomplished because of an interesting intervention by the teachers observing the lesson to be reviewed in the next article in this series. ESL strategies used in this lesson include:
- Integration of content, language and metacognition is a core feature of effective instruction for academic language development for English language learners.
- The use of visuals, universal shapes that are clearly understood irrespective of language ability.
- Introduction of vocabulary as needed and connection to cognates in the first language.
- Small group sharing presents further opportunities to communicate in English.
- Careful construction of mathematical tasks and questions allows for meaningful natural student talk that accelerates the development of mathematical concepts.
This was the thumbnail sketch of a specific lesson modeled for a group of middle school teachers who have English language learners combined with other students. It gives glimpses of the interaction within the framework of the lesson steps and then summarizes the strategies and the reasons for their use.
The lesson demonstration showed the following:
- ESL students along with the other students grasped important geometric concepts.
- Students communicated to make meaning at every stage of the inquiry process.
- ESL strategies were integrated in the lesson seamlessly so that all students benefited.
The next article in this series will focus on classroom observation as a learning tool.
Educators x Communities = English Language learners’ Success
Excels is an innovative IDRA professional development program that creates learning communities of schools, families and communities for English language learners’ academic success. Funded by the US Department of Education, the project is focusing on improving teachers’ capacity to address curriculum, instruction, assessment and parent involvement issues that impact the achievement of limited-English-proficient (LEP) students.
The project is comprised of five components that contribute to student success, as supported by the literature:
For more information contact IDRA at 210-444-1710 or firstname.lastname@example.org.
What is the importance of reading and writing in the mathematics curriculum?
Research and Best Practice
Reading, writing, and mathematics are, or should be, inseparable. Hands-on mathematics can stimulate curiosity, engage student interest and build important prior knowledge before students read or write about the topic. The more students know about a topic, the better they comprehend and learn from text on the topic. Prior knowledge is the strongest predictor of student ability to make inferences from text.
Hands-on mathematics, though, must be combined with minds-on activities. Reading and writing activities can help students analyze, interpret and communicate mathematical ideas. These are skills needed to evaluate sources of information and the validity of the information itself, a key competency for mathematically literate citizens.
Many of the process skills needed for mathematics are similar to reading skills and, when taught together, would reinforce each other. Examples of common skills are predicting, inferring, communicating, comparing and contrasting, and recognizing cause and effect relationships. Teachers who recognize the interrelatedness of mathematics and literacy processes can design instruction that reflects these similarities. Becoming a Nation of Readers suggests that the most logical place for instruction in most reading and thinking strategies is in the content areas rather than in separate lessons about reading.
The importance of writing in the mathematics classroom cannot be overemphasized. In the process of writing, students clarify their own understanding of mathematics and hone their communication skills. They must organize their ideas and thoughts more logically and structure their conclusions in a more coherent way. Competency in writing can only be accomplished through active practice; solving mathematics problems is a natural vehicle for increasing students’ writing competence.
Motivating and engaging students to speak, ask questions, learn new vocabulary and write their thoughts comes easily when they are curious, exploring and engaged in their own mathematics inquiry. Teachers can take advantage of students’ innate wonder and inquisitiveness to develop language skills while learning mathematics concepts. Integrating literacy activities into mathematics classes helps clarify concepts and can make mathematics more meaningful and interesting. Teachers can use a wide variety of literature, including trade books, texts and fiction. Selecting a fiction book with a mathematical theme both provides information and captivates student interest. Fiction works successfully with young learners by embedding cognitive learning in imaginative stories.
Asking students to write mathematics journals about their problem-solving experiences or to articulate and defend their views about mathematics-related issues provides opportunities to clarify their thinking and develop communications skills. Other ways to integrate writing in mathematics are recording and describing situations that involve mathematics, and writing persuasive letters on social issues like the use of sampling by the Census Bureau. National Council of Teachers of Mathematics provides annual lists of outstanding new literature and multimedia materials.
For English language learners, instruction in mathematics can be enhanced by the use of hands-on materials. Interacting with materials and phenomena enables English language learners to ask and answer questions of the materials themselves and use the materials as visual aids in conversation with the teacher and peers. Visual and auditory clues should be plentiful – charts with pictures of materials and key procedures, for example. Teachers should select vocabulary carefully, repeat key words often, and refer to charts with the written words. Working in pairs or small groups makes native language support by peers or instructional aides more feasible.
Mathematics teachers can help all students increase their comprehension of mathematics texts by activating their prior knowledge through brainstorming, discussing the topic, asking questions and providing analogies. Specific attention to vocabulary is often necessary to enable comprehension of mathematics texts. Teachers should introduce new vocabulary and use a graphic organizer, concept or semantic map or collaborative peer study techniques to develop understanding of new words.
Anderson, R.C., and E.H. Hiebert, I.A.G. Scott. Becoming a Nation of Readers: The Report of the Commission on Reading (Washington, D.C.: The National Institute of Education, 1984).
Barton, M.L., and C. Heidema. Teaching Reading in Mathematics, second edition (Aurora, Colo.: Mid-continent Research for Education and Learning, 2002).
Billmeyer, R., and ML Barton. Teaching Reading in the Content Areas (Aurora, Colo.: Mid-continent Research for Education and Learning, 1988).
National Commission of Excellence in Education. A Nation at Risk: The Imperative for Educational Reform (Washington, DC: US Government Printing Office, 1983).
Reprinted with permission from EDThoughts – What We Know About Mathematics Teaching and Learning, edited by J. Sutton and A. Krueger (Aurora, Colo.: Mid-continent Research for Education and Learning, 2002) pp. 50-51.
Jack Dieckmann, M.A., is a senior education associate in the IDRA Division of Professional Development. Comments and questions may be directed to him via e-mail at email@example.com.
[©2003, IDRA. This article originally appeared in the May 2003 IDRA Newsletter by the Intercultural Development Research Association. Permission to reproduce this article is granted provided the article is reprinted in its entirety and proper credit is given to IDRA and the author.]