Insights Into Algebra 1: Teaching for Learning
Systems of Equations and Inequalities Teaching Strategies: Strategies for Teaching English Language Learners
According to the U.S. Department of Education, if the population of children in this country were evenly distributed geographically, the typical American classroom would look like this:
- 10 students would be from racial and ethnic minorities
- 10 would be poor
- 6 of the above 10 would be from families where a language other than English is spoken
- 2 to 4 of the above would be English language learners
- 50 percent of this last subgroup would be immigrant students.
Source: U.S. Department of Education, NAEP 1996 Mathematics Report Card for the Nation and the States.
The demands of teaching in diverse classroom environments are both challenging and rewarding. Fortunately, researchers have identified strategies and pedagogical practices teachers can employ to enhance student learning in these heterogeneous settings. They include:
- Having high expectations of all students
- Focusing instruction on developing meaning or understanding
- Building bridges between the various student cultures and the culture of the classroom
- Using contextual problems to help alleviate language barriers
- Using comprehensible communication to facilitate the learning of English and the learning of mathematics in tandem.
Teacher Patricia Valdez and her students in the video are all native Spanish speakers. However, this isn’t true of all of her classes, just as it may not be the case for yours. Here she offers some advice on overcoming language barriers.
I did have a student last year who spoke Swahili and I don’t speak Swahili. And so one of the things that really helped me is that I use everything in context. We were talking about fractions and I brought in some brownies and we cut up the brownies in different ways. So any time you can bring in real-life situations, something to connect their life with the abstract world of mathematics and bring them together, I think that really is going to help.
Recent studies suggest that teachers who hold high expectations of their students actually contribute to their pupils’ success in school. For example, researchers found that working-class Latino students made substantial academic progress in classrooms where teachers assigned rigorous, intellectually challenging work and assumed they were capable of handling it. It is important that teachers communicate their high expectations to students and provide a variety of activities so students with different strengths can experience success in mathematics.
|Listen to audio clip of teacher
I had a student in my class who failed my class. And he came back the next year and I saw him walking in and I greeted him at the door and I said, “Oh, you are back, why don’t you try another teacher?” Because he had failed with me. And he said, “But I like you, I’m just not good at math.” And I had these new ways, new strategies, and on the first day I gave them some puzzles where they had to – together, in a group – come up with a building made out of cubes, and they had to read these clues and they had to work together. And I saw that he was really a good problem solver, and I thought, “You know, he’s intelligent.” And I had thought of him as a totally different kid, but then I realized it’s just the way that I’m teaching and he’s a good problem solver, he could explain things to other kids, he could tell me why this was true, why that wasn’t true. And I thought, you know, “You shouldn’t have an F in algebra.” He should be able to understand the algebra. I just need to teach it in a different way. So I started learning how to teach algebra in a whole different way.
Developing Meaning and Understanding
Several studies have found that minority students have greater success in classrooms where they’re expected to participate actively, work collaboratively with their peers, and develop their own understanding of concepts. They have less success with teachers who expect students to memorize rules and practice skills without a context.
When students work with data and devise their own ways of solving challenging problems, they see that mathematics is relevant to their lives, and they begin to recognize and develop their intellectual power. Advocates of mathematics education reform agree that these are important goals for all students. In the video for Part II of this workshop, Patricia’s students tackled a real-world problem with two variables and several constraints and discovered how to maximize profit. Throughout the year, she’s used a variety of strategies to help her students develop their understanding of new concepts.
Transcript from Patricia Valdez
Any time you use a new word or concept – like area of a triangle – draw pictures, have [the students] draw pictures, maybe use geo-boards, use manipulatives, use stuff so they can actually understand it. Any time they are doing something, they are going to learn more. Anytime they explain it to someone else they are going to learn it more, so having them present is really important. And then, having posters all around the room that they themselves put together, kids always refer back to them. In fact, they want me to keep those posters up there because they actually do refer back to them.
Engaging students with hands-on activities helps enhance understanding, particularly for English language learners. It’s also significant that Patricia has her students work on these activities in small groups. This gives them a chance to develop their mathematical understanding and their language skills at the same time. Patricia circulates around the room, encouraging students to help teach and explain concepts to one another.
|Listen to audio clip of teacher
In order to get to all the groups – especially when you have nine groups in a normal sized class – you really don’t have that much time with each group, and so if one person really understands it, in one group, then I can ask that student to explain it to the one who doesn’t understand it. And it serves two purposes. First of all, it really cements into the mind of the kid that really understands it, because they have to think about it and they have to explain it again. Also, if I’ve explained it to a student two or three times, and he still doesn’t get it, he might get it if another student explains it to him. And then that also gives me a chance to go to another group and make sure that all of the kids are getting it or most of the kids are getting it.
Transcript from Patricia Valdez
For the first few years that I’ve been teaching mathematics, I taught it the way that I was taught. I used the textbook, and on one day I would teach them one concept, the next day I would teach them the next concept, and so on. I noticed after a few years that my students were intelligent, were able to solve problems, but they got confused when they substituted numbers into an equation or they solved an equation for y, or they solved two equations to look for the intersection. They said, “Is this when we do this, or is this when we do that?” And I didn’t understand why they couldn’t separate the concepts. And it was, I realized, because they are just memorizing things. They really don’t understand what they are doing. And I became a teacher to help students really understand math.
Building Bridges Between Cultures
Classrooms in which teachers have high expectations and give students opportunities to make meaning of relevant mathematics are rich learning environments. However, this style of instruction requires sophisticated use of language. First, the students have to read a problem and be sure they understand what it asks them to do. They then have to carefully consider the information presented in the text and identify what parts of it are relevant to solving the problem. Throughout their investigation, they must be comfortable exchanging ideas with their peers and the teacher. Students who aren’t prepared to participate will be left struggling. Therefore, “bridge building” is a critical component of activity-based instruction in multicultural classrooms. The goal is to enable students to bridge the gaps between their home and classroom cultures.
An effective way to do this is to choose activities and contexts that have relevance to students’ own cultures. Another is to listen carefully to what students say so you can build on their common sense and their informal approaches to solving problems.
In order for students to understand the culture of your classroom, you may find that they need explicit instructions about the types of interactions and behaviors you expect. Some of these behaviors include appropriate talk, peer collaboration, mathematical reasoning and justification, and participation and effort. Focusing on the process involved in students’ mathematical activities lets them know that how they work on a problem is as important as the solution itself. Patricia Valdez monitors her class to ensure that students are engaged in the mathematical activities, and she provides more support for students who aren’t involved at expected levels.
Transcript from Patricia Valdez
I’ve been working on how do I get these kids to focus more and to do their homework on a daily basis, because if they don’t do their homework, no matter what I’m doing, it’s not going to help them. And I’ve been really struggling with this for many years now. But for the last two years, I’ve called every parent. It’s a small class, so I was able to do that. So I’ve been focusing on this class, calling every single parent whenever a student doesn’t do homework. And I’m exhausted, to tell you the truth, but it’s really made a difference. They are doing their homework. The other thing is that I used to get upset about homework when they didn’t do it, and they would see that and that didn’t help either. So now I tell them why I want them to do their homework. I tell them that I love them, I really do tell them, “I love you guys.” And I think they understand I want them to do better. I’d say that 90 percent of the time, 90 percent of the kids do their homework.
Patricia made it a point to contact parents to keep them informed, and to solicit support from them. Think about how you can structure time and activities to help build more bridges in your classroom.
Using Contextual Problems
In general, learning a mathematical topic in context is particularly helpful to English language learners. Consider using real-world examples whenever possible, and physical models to make sure all students understand the context. When students apply new concepts to familiar situations, they build a stronger understanding of the mathematics and improve their language skills. Patricia Valdez elaborates on this idea:
Transcript from Patricia Valdez
We can look at algebra and how it can help us solve a real-world problem. And I always go back to: What does the variable represent? So students that have taken my classes and have gone on to other classes come back and tell me, “You know, I really learned algebra a lot better than my other friends, because I can explain to them what x and y mean in a particular problem.” And they can come up with their own stories of what the variables represent. So having a context really helps them understand.
How often do you use contextual problems in your teaching? Describe some ways in which you could incorporate more of this into your regular classroom lessons.
Using Comprehensible Communication
One way to describe what English learners are doing as they learn mathematics is that they are mapping the meaning of words and expressions between two languages. This mapping model (though admittedly oversimplified) focuses on how students learn to use new mathematics vocabulary and everyday vocabulary. For example, the word “set” has a mathematical meaning, such as in “a set of objects,” and an everyday meaning, such as in “set the table.” English language learners need to learn to match the expressions in their native language with the different ways in which an idea can be expressed in English. Because the associations among words, meanings, and concepts are different in each language, students are making multiple connections and mapping meanings from everyday language in their native language to mathematical language in English. Words and phrases with multiple meanings in either language can lead to misunderstandings in mathematics conversations. Sorting out these differences is something English language learners have to do continuously in an English-speaking classroom.
The following tips for effective communication with English language learners were adapted from Supporting Children’s English Language Development (Focus on the Learner) by Scott D. Enright (Prentice Hall, 1992):
- Use clear, normal speech in communicating with English language learners.
- Moderate your speed if you are a fast talker. It may be necessary to repeat yourself or rephrase what you said.
- Help to shape what the student wants to say.
- Use non-verbal cues (such as gestures, pictures, and concrete objects) in your teaching to assist comprehension.
- Make sure that English language learners are seated where they can see and hear well. Give them maximum access to the instructional and linguistic input that you are providing. Involve them, in some way, in all classroom activities.
- Fill your classroom environment with print and with interesting things to talk, read, and write about. Creating a language-rich environment will allow your English language learners to learn even when you aren’t directly teaching them.
- Keep in mind that the English to which English language learners are exposed in your classroom is of crucial importance to their language development.
- Encourage English language learners’ efforts to participate by celebrating their contributions and searching out opportunities for them to take part directly in learning activities. But allow for the “silent period” that some students go through.
- Correct the content of what they say if necessary.
- Provide opportunities for English language learners to use the language and concepts you are teaching them in meaningful situations. Include a variety of ways of participating in your instruction – in cooperative groups, for example. Encourage all students to work with and help English language learners.
- Treat English language learners as full members of the classroom community.
- Help them to feel comfortable, and integrate them as quickly as possible. Refer to them often and make it clear to them (and to the class) that you expect them to work and learn just like everyone else. Then, ask for more and more participation and work as these students become better able to accomplish it.
Another strategy Patricia Valdez uses is having students read or restate problems out loud in class.
Transcript from Patricia Valdez
Today is Monday, so it’s been a few days since Friday, and I don’t know if they really remember what this problem was about. So sometimes I ask them to restate the problem – “What’s the problem about?” But in this case, there were a lot of little details that were really important, and so I wanted them to read. And they need to practice reading in English and pronouncing the words and all of that vocabulary is going to help them anyway.
Why might comprehensive communication be a useful teaching strategy for all students? What new ideas are you willing to try with your students to improve communication and understanding?
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.