Insights Into Algebra 1: Teaching for Learning
Exponential Functions Teaching Strategies: Affective Domain
Affective refers to those actions that result from and are influenced by emotions. Consequently, the affective domain relates to emotions, attitudes, appreciations, and values. It is highly personal to learning, demonstrated by behaviors indicating attitudes of interest, attention, concern, and responsibility.
According to the National Guidelines for Educating EMS (Emergency Medical Service) Instructors, the following words describe the affective domain: defend, appreciate, value, model, tolerate, respect.
In the mathematics classroom, the affective domain is concerned with students’ perception of mathematics, their feelings toward solving problems, and their attitudes about school and education in general. Personal development, self-management, and the ability to focus are key areas. Apart from cognitive outcomes, teachers stress attitude as the most common affective outcome.
The Taxonomy of the Affective Domain
Most psychologists describe five “levels of understanding” within the affective domain. These five levels define the path from passively observing a stimulus, such as watching a movie or reading a textbook (“receiving”), to becoming self-reliant and making choices on the basis of well formed beliefs (“characterization”).
The major work in describing the affective domain was written by David R. Krathwohl in the 1950s. In his book, Taxonomy of Educational Objectives, Handbook II: Affective Domain (1956), he described the five levels mentioned above. These five levels are restated below with definitions, based on Krathwohl’s book, as well as classroom examples.
In the mathematics classroom, and indeed in all classrooms, instructors are role models. Sometimes, we lose sight of this inherent fact, yet we must remember that our actions model the behavior that students will emulate. When focusing on content, we model the procedures and strategies that we would like students to employ when they solve problems on their own. In the same way, we must model the attitudes and behaviors that we would like students to exhibit when interacting with others and making personal decisions.
Model the behaviors and values that you would like your students to emulate, such as:
Remember that students constantly observe and scrutinize your actions, and immediately correct behaviors that do not model appropriate values. Consider affective objectives when assessing student work. Establish classroom procedures that support affective objectives; that is, through classroom rules, encourage students to be honest, punctual, fair, and so forth, and provide opportunities for them to develop as independent thinkers and self-reliant problem solvers.
Effective teachers promote inquisitiveness and perseverance, and they do not make statements such as “This is an easy problem.” Successful teachers establish good relationships with students by acting more friendly than formal, and they share personal anecdotes about their own problem-solving that reveal their strengths and weaknesses. Effective teachers hold students accountable for performance and base assessment on strategies and communication of conjectures, not simply on finding the correct answer.
Read what Mike Melville has to say about developing student confidence:
Transcript from Mike Melville
The grouping starts at the beginning of the unit. I pass out cards, and in this class there are eight groups, so I use the numbers Ace to 8. Each student gets a card, and they are supposed to sit by the designation of the card – either spades, clubs, diamonds, or hearts. And then I’ll use that fact that they are in that particular seat to actually choose periodically who is going to be making presentations. If I ask for volunteers all the time, I would hear from a select few voices … To make sure that everybody gets a sense that they have an equal share in the class, I pick. One day it will be hearts, and the next day it will be clubs, and the next day it will be spades, and then it will be the diamond person in the group, and then in between I’ll be having volunteers do it. But before they change groups again, every one of those different suits – diamonds, hearts, clubs and spades – will be asked twice to make a presentation to the class. It’s to make sure that they all have a voice in the class and they feel part of it.
As seen in the video for Workshop 6, Part II, Mike Melville creates an atmosphere in which students feel safe to share their feelings, an environment in which students are able to develop emotionally. All students in his class are required to share their thoughts with group partners and to present their ideas to the entire class. Although students may feel intimidated by these activities at the beginning of the year, by the end of the year, they develop confidence in their abilities to discuss mathematics, to present their ideas to others, to disagree when appropriate, and to ask questions when they do not understand.
The National Council of Teachers of Mathematics (NCTM) asserts in Principles and Standards for School Mathematics (PSSM):
Learning mathematics is stimulating, rewarding, and at times difficult. Mathematics students, particularly in the middle grades and high school, can do their part by engaging seriously with the material and striving to make mathematical connections that will support their learning. If students are committed to communicating their understandings clearly to their teachers, then teachers are better able to plan instruction and respond to students’ difficulties. Productive communication requires that students record and revise their thinking and learn to ask good questions as part of learning mathematics. (PSSM, p. 374)
While this is likely true, it may at times be difficult to convince students to make connections, ask questions, and communicate their understandings. A classroom in which students are free to share their thoughts and express their ideas – like the classroom that Mike Melville has established – will go a long way in ensuring that all students learn.
Read what Jane Schielack has to say about the classroom environment:
Transcript from Jane Schielack
It was clear that the environment in [Melville’s] classroom was very conducive to students feeling safe about sharing their questions with each other. In particular, the discussion about subtracting when you are dividing … at the end [of the video] led to the mathematical validation of the law that they had created about exponents and division. And the opportunity for them to discuss that question among themselves, and then Mike paying attention and recognizing that the discussion was going on and could be useful to the rest of the class, provided something from the students to the whole group.
Read what Mike Melville has to say about encouraging participation from all students:
Transcript from Mike Melville
I have some students who are very, very interpersonal and so they keep the groups going. And I have some other students who just want to do it themselves, and I have to push them out of their shells to share their ideas with other people. And I have the students who are very graphic. And so all of that works well together in the kind of interactive setting we have, because then each of the different ways of learning gets a chance to show itself and be shared with the class.
What policies, rules, and regulations have you enacted to help your students proceed through the five levels of the affective domain? In other words, what do you do to help your students progress from being passive recipients of knowledge (“receiving”) to being self-confident thinkers with admirable values (“characterization”)?
The Affective Domain in the Mathematics Classroom
Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction … Students confidently engage in complex mathematical tasks chosen carefully by teachers. They draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine, and explore conjectures on the basis of evidence and use a variety of reasoning and proof techniques to confirm or disprove those conjectures. Students are flexible and resourceful problem solvers. Alone or in groups, and with access to technology, they work productively and reflectively, with the skilled guidance of their teachers. Orally and in writing, students communicate their ideas and results effectively. They value mathematics and engage actively in learning it. (PSSM, p. 3).
The above passage advocates a mathematical education that contains rich mathematics, complex tasks, and the use of technology. Yet NCTM does not promote that those three components are enough to successfully teach all students. The phrases highlighted above in bold (emphasis added by this author) show that NCTM pays attention to the affective domain – that is, the council recognizes that how students perceive mathematics is at least as important as the topics they study.
As the affective domain is concerned with student attitudes and beliefs, one goal for teachers should be to make students believe that mathematics is useful, interesting, and tangible. In addition, teachers should promote self confidence by helping all students experience success in the classroom.
Consider the adjectives used to describe mathematics students in the passage above: confident, flexible, resourceful, productive, reflective, active. Of course, there are also the implied adjectives: persistent, determined, open minded, resolute, cooperative. List at least three things that you do to create confident students, persistent problem-solvers, and active learners.
The Role of the Curriculum
The curriculum is the single most important factor in whether students will find mathematics both exciting and necessary. As PSSM states:
A school mathematics curriculum is a strong determinant of what students have an opportunity to learn and what they do learn. In a coherent curriculum, mathematical ideas are linked to and build on one another so that students’ understanding and knowledge deepens and their ability to apply mathematics expands. (PSSM, p. 14.)
In addition to providing the content of what they learn, a solid curriculum also provides motivation for learning. As stated in a recent issue of the journal Educational Leadership:
A teacher’s personality, voice, and style of instruction are not key factors in producing boredom. Instead, boredom is primarily an effect of curriculum. Curriculum design based on four natural human interests – the drive toward mastery, the drive to understand, the drive toward self-expression, and the need to relate – will not only reduce student boredom, but will yield boredom’s opposite: abiding interest in the content that students need to learn. (“Boredom and Its Opposite.” Educational Leadership, September 2003: pp. 24-29.)
Students express their desire to make sense of the world by raising questions, pointing out errors, insisting on explanations, and sharing their opinions. Conversely, when these behaviors are not present, it’s likely that students are uninterested. The following student-interest rubric can be used to evaluate your classroom, to assess your curriculum, and to gauge your effectiveness in the affective domain.
Data suggests that students generally believe that mathematics is important yet difficult, that it is based on a set of rules, and that it is skill-oriented. Researchers note that while teachers do not share these same beliefs, a poorly designed curriculum may contribute to students’ negative attitudes toward the discipline. As a consequence, students’ narrow views on mathematics may weaken their ability to solve non-routine problems, especially if they believe problems should always be completed quickly.
To supplant these beliefs, teachers must invigorate the curriculum with activities that promote student engagement and that require thought and deliberation at an appropriate cognitive level. For students to grasp that mathematics is necessary and attainable, they must participate in mathematical simulations that foster conceptual understanding, realize that the material they are learning is necessary, and experience real world examples that make the mathematics tangible.
Using Mathematical Simulations
Simulations, such as the Skeeters activity Orlando Pajon used in the video for Workshop 6 Part II, help to foster positive feelings toward mathematics. In general, students enjoy simulations because:
- They are tactile and fun.
- They typically represent a situation in the real world.
- They are tangible and usually more comprehensible than abstract ideas.
- They promote student success because most students will require a similar amount of time to explore. This differs from solving simple, routine problems, which some students do quickly while others struggle.
With each shake of the box containing Skeeters, students were able to see and understand the results. They were able to notice that each time, approximately half of the Skeeters were showing a mark, indicating that the population would grow by one half at each stage. Because of the tactile nature of the simulation, students were able to form the mathematical connections between the actual numbers and the exponential model that described the situation. This is in concert with NCTM’s belief that students must recognize the mathematical connections between the ideas they learn. A curriculum which fosters these connections is imperative.
According to PSSM:
Students will be served well by school mathematics programs that enhance their natural desire to understand what they are asked to learn. From a young age, children are interested in mathematical ideas. Through their experiences in everyday life, they gradually develop a rather complex set of informal ideas about numbers, patterns, shapes, quantities, data, and size, and many of these ideas are correct and robust. (PSSM, p. 20.)
Read what Orlando Pajon has to say about a teacher’s role during simulations:
Transcript from Orlando Pajon
One of the most important components of this curriculum is that the teacher serves as a facilitator of the learning, which means that the kids explore different concepts and the teacher helps them get to the goal … A teacher doesn’t have the answers to everything, but [is] somebody who can give you a hand, somebody who can show you ways to explore, in order to get something accomplished.
Hear what Orlando Pajon has to say about student explorations in the affective domain:
|Listen to audio clip of teacher
Transcript from Orlando Pajon
First of all they are required to work in teams, which means that before they give an answer to a topic, they have to share their individual views about that question. At the end, they have to come up into an agreement, based on what everybody is thinking. That forces them to share, that forces them to talk about the topics or to talk about mathematics…
After these kids graduate from high school and they get a job, or they get into the university, they always are going to have to work as a team. They always are going to have to come into an agreement. Sometimes they will not agree 100 percent on what they want, but something needs to be done, and in order to accomplish it, we have to do it as a group.
So whenever we are having these types of discussions, I always emphasize the fact that they have to give me their own opinion. What do they think? After that, I try to ask somebody else for agreements or disagreements. I always try to emphasize the fact that it’s okay to be wrong, that it’s okay to have different opinions. I’m trying to create an atmosphere in which every kid will feel free to express his idea without fearing that somebody will say, “That’s wrong” or “That’s not right.”
Explain Why the Material Is Necessary
No mathematics classroom is free of the question “When are we ever going to use this?” Students ask this question all the time, and unless we are able to provide acceptable answers, students may believe that mathematics has no use in their lives.
At the same time, answering “Why do we need this?” can be difficult. When teaching the rules of exponents, for instance, it’s not always easy to explain why students must know the rules regarding a product of powers. And the justifications – “Because it’ll be on the test” or “Because you’ll need it for other math classes” – generally don’t satisfy inquisitive teenagers.
Questions about the necessity of a topic, however, can often be diffused at the outset. Prior to a unit or lesson, an explanation of why the material students are about to learn is important should set the stage. You may also encourage and challenge students to investigate the topic for themselves and to bring potential applications and uses to the class. If the material is mostly needed for advanced study or to improve their understanding of the structure and language of mathematics, offer this explanation before students question the need for the topic. This way, you may avert classroom-management and motivation issues.
Present Real-World Examples
The students in Mike Melville’s class experienced an activity that had fewer real-world applications than Orlando Pajon’s population simulation. Students in Melville’s class considered the size of Alice when she ate cake (and her size doubled) or drank beverage (and her size reduced by half). Yet the fact that it was tangible – that students could understand how Alice’s height changed, and that they could imagine her growing and shrinking as she consumed – made the mathematics of the activity concrete. Most importantly, the situation allowed students to see how exponents influence the value of a number.
Hear what Mike Melville has to say about helping students understand abstract concepts:
Transcript from Mike Melville
I think for most students mathematics is abstract and the way it’s been approached by them in their past has been from an abstract basis. In any kind of mathematics that really works with kids, I think they have to have a context to understand, well what does that mean in a real kind of situation. We all know cakes, we all know how to drink beverages, so it’s a context that they can understand. They all know the story of Alice in Wonderland, so they’ve seen pictures of her growing bigger and they’ve seen pictures of her shrinking smaller. It gives them a context within which to understand what would normally be an abstract kind of thing. And the net result of that is it no longer is abstract. And where we’re headed with this is that they’ll have a situational way of explaining every one of the exponential issues that we look at … We’re developing rules and definitions and conventions, and we’re seeing why those work, given the model we have. But we’re also seeing why those work, given a pattern of calculations based on the context that we’ve set up. When they’re through with that, they’ll always come back to explaining it situationally, but they know how to compute it, they know how to calculate it. And I think that frame of reference is extremely important for them.
Research suggests that many of the instructional strategies that promote mathematical achievement also promote growth in the affective domain. Teachers should incorporate group work and assign tasks more compatible with the development of higher-order thinking skills. In their work, students should experience the wonder of discovery in mathematics, and teachers need to present more problem-solving and fewer skill-based assignments in the classroom. To help alleviate problem-solving anxiety and to expand student attitudes about the length of time required to complete a task, teachers should assign problems that require and foster research skills and that may have more than one possible solution.
In short, to foster positive student attitudes regarding mathematics, the activities and assignments in which they engage ought to challenge them; require them to struggle, persist, and succeed; and show them the beauty of mathematics that math educators already see.
Explain how the activities, tasks, and problems in your curriculum relate to the affective domain.
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.