Insights Into Algebra 1: Teaching for Learning
Systems of Equations and Inequalities Teaching Strategies: Building Understanding
People who have developed expertise in particular areas are naturally able to think effectively about problems in those areas. Understanding such expertise is important for teachers because it provides insights into the nature of how people think and solve problems. Research about how experts learn new things in their fields has identified several important factors that help all learners comprehend new concepts and transfer their learning to other situations. Allowing students to discover concepts, instead of telling them the concept, is important. Giving them motivation to learn is another factor, as is allowing them to learn new concepts in a meaningful context. Encouraging students to think metacognitively about their own learning is crucial; in plain terms, that means we want students to think about how they think. It’s also important to provide activities that encourage students to transfer what they learn in school to their everyday lives.
When her students are beginning to build their understanding of a new concept, Jenny Novak prefers to give them time to observe and explore.
|Listen to audio clip of teacher
Transcript from Jenny Novak
I think that my students today did get the big picture of what was happening. Today’s lesson is a beginning of the unit. I know that we are going to be doing a lot more reinforcing of the major key concepts, and we’re going to be looking at different strategies and different ways to solve systems of equations and analyze graphs throughout this unit. But I think for their first day and their first intro to the lesson, I think they did a pretty good job.
Key Principles of Expert Learners
What differentiates a novice from an “expert?” Research shows that it’s not memory or intelligence, nor is it the use of general strategies. Rather, experts have acquired extensive knowledge that affects what they notice and how they organize, represent, and interpret information. This, in turn, affects their abilities to remember, reason, and solve problems.
Consider these key characteristics of expert learners and the potential implications for the classroom:
- Expert learners notice features and meaningful patterns of information that novices do not.
- Expert learners have acquired a great deal of content knowledge that is organized in ways which reflect a deep understanding of the subject.
- Expert learners’ knowledge cannot be reduced to sets of isolated facts or propositions. Their knowledge is fixed because of its applicability to them and their circumstances.
- Expert learners are able to retrieve important aspects of their knowledge with little effort.
- Experts demonstrate flexibility in their approach to new situations.
It is important to be realistic about the amount of time it takes to learn complex subject matter. On average, world-class chess masters require from 50,000 to 100,000 hours of practice to reach that level of expertise. Much of this time involves the development of pattern-recognition skills that support the fluent identification of meaningful patterns of information, plus the knowledge of their implications for future outcomes. Although many people believe that “talent” plays a role in determining who becomes an expert in a particular area, even seemingly talented individuals require a great deal of practice in order to develop their expertise.
Students often face new tasks that seem to lack apparent meaning or logic. It can be difficult for them to learn with understanding at the start; they may need to take time to explore underlying concepts and to generate connections to what they already know. Attempting to cover too many topics too quickly may actually hinder learning. Students will either try to memorize the facts as isolated pieces of information, or they’ll misunderstand the organizing principles because they don’t understand the specific details well enough. Researchers have found that students who “grapple with specific information relevant to a topic” before listening to a lecture understand the lecture much better than students who listened to it without any prior experience with the topic, according to How People Learn: Brain, Mind, Experience, and School: Expanded Edition (National Academy Press, 2001). By introducing the topic of systems of equations with an engaging activity – the Right hand/Left hand experiment – teacher Jenny Novak provided her students with an opportunity to grapple with new ideas in a meaningful way.
How can you incorporate the key principles of experts’ knowledge into your teaching so that you can help your students learn better?
One important aspect of building understanding is to provide students with opportunities to discover the key concepts and underlying principles of a mathematical topic. Learning by discovery takes place most readily in problem-solving situations where students can draw on their own experience and make connections between their prior knowledge and the new material. Jenny Novak explains how she used a conceptual approach in the workshop video.
Transcript from Jenny Novak
When I was going through the different ways that two lines on the same graph could be plotted and what the relationship could be between those lines, I thought it was important to start out with a conceptual approach where we just had a graph without any type of scale. [That way,] students didn’t get bogged down with what’s the starting point of these lines, when is it going to be touching each other. They could [just] figure out, in general, what was happening so that when we transitioned into the more specific example, they’d have a more rich understanding of the basic concept so that they could apply it and develop more solid conclusions.
Noted education scholar Jerome Bruner contends that students, starting in elementary school, should learn the structure of a body of knowledge rather than simply memorizing facts. He also asserts that students should be taught and encouraged to discover information on their own. In other words, teachers should try to help their students “learn how to learn” and develop confidence in their ability to learn.
Bruner does not claim that students have to discover every bit of information by themselves. What they need to discover is the interrelatedness between ideas and concepts by building those new ideas onto what they already know. According to Bruner, “Emphasis on discovery in learning has precisely the effect on the learner of leading him to organize what he is encountering in a manner not only designed to discover regularity and relatedness, but also to avoid the kind of information drift that fails to keep account of the uses to which information might have to be put.”
Think about a lesson that you have taught in which you used an approach that allowed students to discover a new concept or concepts. Describe the components of the lesson. Explain why the lesson was or wasn’t successful.
Students are open to learning new things when they’re engaged in the lesson. Effective lessons are designed to make students want to think more creatively and take an active role in their own learning. One way to motivate students is to “hook” them with questions and situations that take them by surprise or otherwise stimulate their natural curiosity. Students will be motivated to investigate the situation and solve the problems to satisfy their curiosity. Jenny’s students were motivated because they enjoyed performing the experiment and comparing results with their classmates. They were also motivated because they could see how the graphs had meaning in terms of the experiment.
Transcript from teacher Jenny Novak
I’ve always enjoyed doing this activity with the students because first of all, everybody has their own unique data that is special for them and meaningful for each person.
Another aspect of motivating students is finding appropriate tasks. Remember that challenges must be at the proper level of difficulty; tasks that are too easy become boring, and tasks that are too difficult can cause frustration. Here, of course, it helps to know your own students’ individual styles. Those who are more “learning oriented” enjoy new challenges and will stick with them longer; those who are more “performance oriented” may get discouraged quickly because they’re worried about making mistakes. Jenny Novak describes the challenge of keeping a large, diverse class motivated.
Transcript from teacher Jenny Novak
The challenge I sometimes face is keeping the students who already know a particular topic motivated while making sure that the students that are seeing this for the first time can spend enough time on it.
Social opportunities also affect students’ motivation. When they see that what they do can have an effect on others, students’ eagerness to learn increases. For example, young children are highly motivated to write stories and draw pictures that they can share with others. People of all ages are more motivated when they can see the usefulness of what they’re learning and when they can use it to do something that has an impact on others – especially their local community.
Consider an activity or lesson you teach that motivates your students. What are some of the components of the activity or lesson, and how do these elements work to increase motivation?
Many students learn better when a subject is taught in a context. However, if a concept is taught in only one context, students may have difficulty applying it in other situations. Research shows that when teachers present material in multiple contexts and include examples that demonstrate a variety of ways the material can be used, students are more likely to pull out the relevant features and the key concepts. The goal is to help students understand general principles that they then can transfer flexibly to other situations.
One way to do this is to ask students to solve a problem in a specific context and then provide them with an additional, similar case to investigate. Another way to improve flexibility is to let students learn in a specific context and then engage them in several “what if” problem-solving scenarios. Moving from the concrete to the abstract is essential. Students must be able to understand and communicate mathematical concepts in general terms. In the workshop video, Jenny made sure that her students had several different contexts in which to consider systems of equations.
Transcript from Jenny Novak
I think they came up with a lot of interesting scenarios, some things that I would not have thought of right away. And I was pretty impressed with some of the groups – particularly, one of the groups had talked about different sports scores and comparing men’s data and women’s data. And we do look at that a little later in our unit with some of the different sports, so we’ll have them gather some data on their own.
In learning mathematics, students need to study problems in more than one context. Additionally, they need to discover the abstract mathematical concepts on which the concrete examples are based. Write about whether Jenny accomplished this in her lesson and explain your thinking.
In plain terms, metacognition means thinking about how we think. Students understand concepts better and are able to transfer their learning successfully to other situations if they pay attention to their own learning strategies and resources, and assess their own abilities. In fact, metacognitive approaches to instruction have been shown to help students transfer prior learning to new situations without needing to be prompted. The following excerpt from the North Central Regional Educational Laboratory’s Strategic Teaching and Reading Project Guidebook provides a few ideas on how to help students increase their metacognitive abilities:
Metacognition consists of three basic elements:
- Developing a plan of action
- Maintaining/monitoring the plan
- Evaluating the plan.
When you are developing the plan of action, ask yourself:
- What in my prior knowledge will help me with this particular task?
- In what direction do I want my thinking to take me?
- What should I do first?
- How much time do I have to complete the task?
When you are maintaining/monitoring the plan of action, ask yourself:
- How am I doing?
- Am I on the right track?
- How should I proceed?
- What information is important to remember?
- Should I move in a different direction?
- Should I adjust the pace depending on the difficulty?
- What do I need to do if I do not understand?
When you are evaluating the plan of action, ask yourself:
- How well did I do?
- Did my particular course of thinking produce more or less than I had expected?
- What could I have done differently?
- How might I apply this line of thinking to other problems?
- Do I need to go back through the task to fill in any “blanks” in my understanding?
Describe what you can do to help your students understand their math learning experiences in terms of metacognition.
Transfer Between School and Everyday Life
Helping students learn mathematics and apply their learning to other contexts and situations is important. Ideally, students should be able to recognize that what they learn in school is applicable and helpful to them in everyday life. It’s hard for them to make that connection when what they do in school doesn’t resemble their reality.
There are three distinct ways in which most school environments differ from other environments:
- In school, working individually is more important than it is in most other settings.
In most workplaces, people work collaboratively and share their ideas and expertise. For example, many discoveries that have come out of genetics laboratories have involved in-depth collaboration among scientists. Similarly, decision making in hospital emergency rooms is distributed among many members of the medical team. In the workshop video, Jenny first allows her students time to think and work out a problem individually, but she then gives them an opportunity to collaborate with a partner or a team to decide on a solution path.
Transcript from Jenny Novak
Another reason, particularly for this class, that I put students in cooperative groups is because we do have so many students in here that have difficulty with English. So if I put them in the groups, they find a way to communicate with another student and they show me, usually through the written work, that they understand it. But then they can show their classmates. They do find a way to tell them. And it’s always nice to see that the students are working together in trying to break these barriers.
- In everyday settings, people usually have tools to help them solve problems, while in school the focus is on “mental work.”
New technologies, however, now make it possible for students in schools to use tools very much like those used by professionals in workplaces. Proficiency with relevant tools may help students see how they might use what they’re learning in other settings. Even though Jenny’s students have just begun developing the concept of systems of equations, they’re already seamlessly integrating the use of the graphing calculator as a tool. As they progress through the unit, they’ll probably spend less time graphing by hand and use the calculator to analyze and compare more graphs.
- Academic subjects require students to reason abstractly, while contextualized reasoning is used more often in everyday settings.
Studies have shown that students’ ability to reason improves when teachers embody abstract logical arguments in a variety of concrete contexts. For example, think about how Jenny tried to closely match her students’ school experiences with what they will experience in everyday life.
Transcript from Jenny Novak
I think that systems of equations is a very important topic in this course, and it’s also a topic that involves a lot of real-world applications. This is the type of scenario that they are going to actually be seeing when they are in the real world. So a lot of the applications I came up with were things like hiring companies or hiring DJs, so that they could say, “Well, I might have to do this one day and I need to be an informed consumer.”
A major goal of education is to prepare students to adapt flexibly to new problems and settings. Students’ ability to use their knowledge in new situations provides an important index of learning that can help teachers evaluate and improve their instruction. Many approaches to instruction look similar when the only measure of learning is how well the students recall the information that was presented. But instructional differences become apparent when one looks at how well students can apply their learning to new problems and settings. This is why creating an intellectual bridge between school and everyday environments is the ultimate purpose of classroom learning. Remember, however, that it is important to avoid instruction that is overly dependent on one particular context. Helping learners choose, adapt, and invent strategies for solving problems is one way to facilitate transfer while also encouraging flexibility.
Describe ways in which you can improve the connection between what students learn in class and the situations they encounter in life outside of school.
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.