Curriculum Platform: Introduction

The place: A typical mathematics classroom in grades 7-14.

The time: Sometime in the future.

You open the door to the classroom and your initial reaction is shock. It seems that chaos reigns as you notice students out of their seats as they talk with one another. Furthermore, the desks are not neatly arranged in rows, but seem to be randomly clustered into small groups as the students use the desk tops for what appears to be some sort of technical equipment set-up. Where is the teacher? You are not sure. Concerned with the outward appearance of this classroom, you enter the classroom to investigate. You stop at the first group of students and try to figure out what is going on. Quickly you realize that the topic of their conversations is highly technical. They are hypothesizing the result of an experiment where two tuning forks of different frequency are struck and the combined frequency is measured. Would the frequency be the sum of the two tuning forks? The difference? Or some strange combination that will be difficult to determine? They follow through with the experiment using a data collection device and a graphing calculator and the students begin to analyze the results. They are discussing ideas and concepts foreign to you as they search for the beat pattern, calculate the period and frequency, determine the equation of the sinusoid which fits over the beat pattern, and ultimately discover the relationship between the frequency of the individual tuning forks and the combined frequency. You ask one of the students, "why are you doing this?" Annoyed by the interruption, she responds that they are trying to solve the problem at hand where, in the situation given, an old World War II mine needs to be deactivated. This can only be accomplished by directing the proper sound frequency at the timing mechanism in the bomb in order to disable it. You find it refreshing to see students motivated to solve a problem and notice that students are simultaneously using technology, paper and pencil calculations, and teamwork to find the solution to a problem that has obviously engaged the interest of the students.

You move on to the next group of students. They are equally engaged in what appears to be a different problem. You first notice that the students are huddled around a graphing calculator screen. Suddenly, a shout of celebration is heard as students triumphantly give each other "high-fives". You get the attention of one of the students and ask what they are doing. He explains that their team was given data related to the height of waves in the ocean during a particularly dangerous storm. Their job was to create a scatter plot of the data and determine a sinusoidal regression model for this data so that they could calculate the period, frequency, and height of the waves at the moment in time a small touring vessel was capsized. Given the manufacturers specifications of this vessel, the students would be able to determine if it was built to withstand such a storm or not. This would aid in the investigation and eventual lawsuit brought by passengers of the vessel who believed that the vessel's skipper used poor judgment in venturing out into the storm.

Impressed by the level of engagement of the students, the high-level discussions that were taking place between students, and the interesting combination of technology and paper-and-pencil calculations that were taking place, you still have one more concern: where is the teacher?

You move on to the next group and find that they are working with tuning forks also. Peering in to see what this group is discussing, you find that one of the members of the group seems to be asking questions. "What would you know about the frequency if the period is 0.123 seconds?" "Do you see a pattern when you consider the different combinations of two tuning forks and their resulting beat pattern?" "Can you explain the difference between period and frequency?" You realize that this is the instructor asking probing questions to the students to help them construct their own understanding of the mathematical concepts being explored.

Standing back to observe, as a whole, the classroom setting that initially brought disgust (nearly), you now feel joy (nearly) as you realize the high engagement of students as they study difficult math (and physics) concepts. But the students seem to be enjoying it. Could this really be a math classroom? What curriculum development and implementation steps were employed by the instructor in this classroom to get to this point?

The purpose of this paper is to describe my curriculum platform and to what extent it would include the linear, holistic, laissez-faire, and critical theorist traditions. I describe this from the perspective of a mathematics educator who is interested in the mathematics classroom in grades 7-14.

Linear Tradition

I appreciate the guidelines set out by Ralph Tyler for effective curriculum design. It is a systematic and efficient way to guide us towards an effective curriculum that can be implemented with students. Specifically, the order he has described makes sense and that we, in the mathematics education community, need to spend some time in discussion of the principles that Tyler has discussed. Currently, the mathematics curriculum in grades 7-14 prepares students to be mathematics majors instead of preparing students to effectively use mathematics in their personal and professional lives. We need to work on Tyler's step 1: What educational purposes should the school seek to attain, specifically in the mathematics discipline? Quoting directly from Tyler, "What can your subject (mathematics) contribute to the education of young people (or returning students at the community college, in my case) who are not going to be specialists in your field; what can your subject contribute to the layman, the garden variety of citizen?" We need to create a set of goals that are inferred from studies of the learner and from studies of life outside of school.

In my ideal curriculum, these goals would include:

• Communicating mathematically
• Becoming more efficient problem solvers
• Thinking mathematically
• Applying mathematics to practical situations to solve problems and make predictions

Subject specialists should be involved, but must obtain input from users of mathematics. Students should experience, actively (as described in the opening vignette), the study of mathematics in order to model real world phenomena, improve critical thinking abilities, and solve meaningful problems. Currently, the predominant mathematics classroom, grades 7-community college, is a classroom where students passively receive instructions to perform meaningless skills and solve silly "story problems". Having visited dozens of classrooms, I rarely see what I consider an effective, exciting classroom experience for students.

The evaluation of learning in the type of classroom that I am advocating should be diverse. Traditional paper and pencil tests can be used to evaluate students' ability to apply the skills and demonstrate an understanding of the concepts studied. But, also, students should demonstrate their abilities in alternative ways such as projects, portfolios, interviews, and learning reflections.

I am a linearist to the extent that I believe that curriculum should be developed by

• creating practical objectives based on the needs of the learner;
• creating active, varied experiences to help students obtain these objectives;
• organizing these learning experiences to create an effective unit, course, or program of study;
• evaluating the learning in a variety of ways.

Here, I use the term practical objectives to describe what I believe students should learn. I am using the term to mean the specific things that students should learn, but in the context of a problem situation or application. For example, I believe that students should learn to solve equations in an algebra class. The objective might say, "students will be able to solve linear equations symbolically, graphically, and numerically." These equations, however, will not be given in isolation, but will be connected to an application that is, hopefully, meaningful to the students.

When I mention the needs of the learner, I am referring to the gap theory. There is a gap between what my algebra students know about algebra and its applications and what I think they ought to know as educated citizens. I have determined that gap and fill it throughout the course by using the curriculum established in that course.

Again, using the example of solving equations, I think that students should be able to solve an equation in a meaningful situation. Here is a specific example. Most algebra students could solve the equation .35x + 20 = -.15x + 35. This equation does not have any meaning and addresses the isolated skill of solving for the variable, x. What I mean by "practical objectives" is this:

The number of gallons of soda consumed by American's can be modeled by the function

y = 0.35x + 20, where x is the number of years since 1970. The number of gallons of milk consumed by American's can be modeled by the function y = -0.15x + 35. When was the number of gallons of soda consumed equal to the number of gallons of milk consumed by American's? In addition, other questions can be addressed such as

• What is the current trend of soda consumption/milk consumption in the US?
• What implications does this have on our society?
• When will then number of gallons of soda consumed be double the number of gallons of milk consumed?
• Do you think that these trends will continue?
• What is the slope of each line? What would the slope mean in this situation? What would be its units of measure?

Holistic Tradition

Students come into the mathematics classroom with a variety of backgrounds and experiences. They are not just empty vessels waiting to be filled by the "expert" teacher as in the traditional mode of education. As stated in Dewey (p. 18), "the traditional scheme is, in essence, one of imposition from above and from outside." In my experiences, this does not work. Students learn when they are actively engaged in meaningful activities that have connections to a motivating context. The opening vignette describes what I mean by this. This belief is what puts me into the holistic tradition. I believe that the experience should be more organic (not all students are doing exactly the same thing at the same time), integrated (connect the mathematics to a real-life context), but still structured (students will arrive to a common understanding of the concept). This is where I move into the linear tradition. As in the example, the goal was for all students in the class to come to understand the concepts of frequency, period, sinusoidal modeling, amplitude, etc. This was done by preparing an experience where all students would arrive at this understanding as a result of the experience.

The planning of these types of experiences is challenging since, as Dewey's states (p. 58), "the planning must be flexible enough to permit free play for individuality of experience and yet firm enough to give direction towards continuous development of power." In this type of classroom, the teacher is not the dictator, telling students what they need to know, but rather a facilitator of a meaningful learning experience. Again, I agree with Dewey (p. 59) when he states that "the teacher loses the position of external boss or dictator but takes on that of leader of group activities." My ideal curriculum for all teachers at my school would include these activities and allow teachers to take on the role of facilitator or leader of group activities.

In this role, teachers need to be careful that the experience is truly educational and not mis-educative. I believe that the experience in the classroom must carefully be planned so as to avoid (Dewey p. 25-26)

• arresting or distorting the growth of further experience;
• causing the students to fall into a groove or rut;
• creating a careless attitude in students where the quality of subsequent activities will be affected;
• disconnected activities which can cause a student to become "scatter brained".

This can be a challenge for the teacher in the holistic classroom. Contrast this holistic planning with a more traditional approach. Upon completion of a more traditional classroom experience where the teacher acted as the "external boss", telling and showing students how to perform isolated skills, a teacher can feel as though they did their job. The teacher told the students what they are supposed to know and the students diligently copied the notes from the board. In this situation, an instructor may not consider the four mis-educative opportunities listed above.

In a more progressive classroom, a teachers work begins outside of the classroom when he/she prepares the activity in which the students will be engaged. These activities must not be just fun and games, but an intelligent activity which is "within in the range of the capacity of students and that is such that it arouses in the learner an active quest for information and for production of new ideas." (p. 79) And, after creating a lesson such as this, the instructor needs to facilitate the classroom experience where students are constructing their own conceptual understanding rather than being told how to do something.

This is very difficult and makes teaching in this way very challenging, which is why many teachers do not teach in this way. Dewey says "this tax on the educator is another reason why progressive education is more difficult to carry on than was ever the traditional system." (p. 40)

The result of these educational experiences must still be assessed. Are students learning what the teacher wants them to learn? How do you know? This is where I tend to believe in the linear tradition. Even though the experiences I have mentioned should be integrated, organic, meaningful, and structured, the final product, I believe, is predetermined. For example, when I evaluate a written project, a presentation, reflection or portfolio completed by my students, there is a prototypical product in mind. In fact, the rubric used to evaluate these assignments describes this prototype. Students are judged according to this prototype and a value is placed on their work. In fact, samples of student work are available on my web site for students to model their work after.

Laissez-Faire Tradition

Even in my ideal curriculum, I would have difficulty making room for the laissez-faire approach. I don't believe that students will eventually come around and desire to learn the concepts that they need to be well-educated, independent learners. Another main problem that I have with this approach is the timing. I realize that this is supposed to be my ideal curriculum, but, students have a finite amount of time to spend on school related issues. They need to be directed, based on a needs assessment, to learn the desired curriculum. However, if you think back to my opening vignette, students do have some degree of flexibility in their problem experience. They can choose to solve the problem at their own pace and in their own way rather than have an instructor work out the solution at the board.

According to the laissez-faire tradition, "children are good and if you leave them alone, they will grow". I do not believe that this is true. I believe that inherently, people are not good and need to have a subject specialist (an educator) help them to learn a particular topic (in my case mathematics).

Critical Theory

Our world is a great example of teamwork. Some people can perform surgery, others put out fires, still others manage businesses that people need, and, most importantly, others teach (I had to throw that inÉ). Each of us has a role (I believe it is a God given gift) to assume which allows us to help "construct our society into a more democratic, just, and caring place to live" (Goodman, p. 126). Schools provide the environment in which many students discover this gift and provide the arena for nurturing students intellectual talents.

This environment is established as teachers assume the role of curriculum specialists. I agree with the notion that we have a "teacher-centered curriculum" in that teachers play a major role in curriculum development. In my ideal curriculum development platform, teachers would have the ability, freedom, and power to adjust, define, write, and adapt curriculum to fit the needs of their particular audience. That is, teachers need to view students and curriculum dynamically by learning to make classroom-based developments within the curriculum implementation process. Curriculum developers (textbook authors) have a certain learning experience in mind, but teachers need to be willing and able to adapt that experience to fit the needs of their particular student population. To this extent, I would want my curriculum to be teacher-centered.

My curriculum would need to be delivered to students in an interactive, collaborative learning environment. The opening vignette attempts to describe a classroom situation where this is done. Students are talking to each other, depend on one another, and feel as if they were in control. This is important to me because I know that this is how I learn and problem solve best. Even in everyday decisions, I need to talk things out. Thank God I have a wife who doesn't mind listening to me talk out loud as I make decisions involving my coursework or my classroom. In my experience, most students appreciate this opportunity. On many occasions, when the scheduled class time has expired and students are allowed to leave, many will stay and continue to discuss the problem situation and the methods of solution. In this collaborative setting, students feel like they are a part of something worthwhile and this sense of community, I believe, motivates them to persevere in solving the problem, an important aspect of my "hidden curriculum".

Another aspect of my hidden curriculum is the students' attitude toward mathematics. As described in the opening vignette, students were excited about learning, felt that the final product had meaning, and in most cases were more active in the small group setting than they would have been in a whole class discussion/activity. If a tally was kept to count the number of times an individual's voice was heard, all students would have scored. Some students may have been domineering within the group, but the classroom culture can be created so that every student has a voice and should be recognized. This interaction between students and teacher is beneficial to student learning and creates a positive attitude toward learning.

Conclusion

As one of the authors of the Maricopa Mathematics Modules, I have had the wonderful opportunity of actually developing my ideal curriculum. Because of this continuing experience, I feel that it is appropriate to conclude this paper with the philosophy that we have used to create our curriculum (taken from the Preface). You will see the underlying philosophy that I live by and have used in describing my ideal curriculum platform.

We believe that we should teach what is most important. Therefore, we set our sights on six student outcomes that guide our work as authors and teachers.

Connect

Learners employ a variety of methods (visual, symbolic, numeric, verbal, et al) to represent and explore mathematical ideas. They construct and apply models that connect mathematics to the world. Learners evaluate the soundness of their methodologies and results.

Reason

Learners demonstrate clear reasoning in analyzing information. They develop and apply logical thinking skills to formulate and support conclusions.

Express

Learners read, write, listen to, and speak mathematics both individually and in teams.

Appreciate

Learners value the power of mathematics. They are confident, flexible, and persistent lifelong learners and users of mathematics.

Tap into Technology

Learners discover the benefits and limitations of current technologies. They utilize technologies as resources for learning and problem solving.

Establish a Firm Foundation

Foundation skills provide a basis for continual learning. Learners acquire and develop a core of content-specific knowledge and abilities, as well as strategies for learning.

In order to achieve the (C.R.E.A.T.E) outcomes above, we believe that students need to gain a strong conceptual foundation of mathematics: that learning mathematics means to build strong connections among various topics of mathematics. We recognize that students do not build conceptual knowledge quickly, nor do they build a conceptual foundation by becoming proficient at doing template exercises and problems. Rather, students build conceptual knowledge by reflecting on their own knowledge and concepts.

To be successful in academic pursuits, as well as in life after college, students need to learn more mathematics than algebra. In the Maricopa Project Modules, mathematical topics such as probability, sampling, geometry, and functions are interwoven with the development of algebra skills. In each module, students learn mathematics and mathematical language in a context of use. The mathematics in each module is grounded in the application of that mathematics. With the Maricopa Project Modules, students do not wonder how this mathematics would ever be used.

We believe that all students can achieve the C.R.E.A.T.E. outcomes.

I decided to force myself to place myself on the continuum that we discussed in class. Notice I made the "x" large enough to encompass a large area on the continuum and also have reserved the right to slide the "x" as I grow and change as a professional educator.