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What is Mathematics?

Major Unifying Themes in This Document

Syllabus

Foundational Information

Learning Theories

Mind and Body Tools

Science of Teaching & Learning

Project-Based Learning

Computational Mathematics

The Future

Recommendations

References

 

 Website Author
"Dr. Dave" Moursund

Problem-Based Learning and Project-Based Learning

ICT-Assisted Project-Based Learning and Problem-Based Learning add new dimensions to math education. And, they help move math education out of its current "stand and deliver" and "individual seat work" modes.

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While Project-Based Learning and Problem-Based Learning share much in common, they are two distinct approaches to learning. In Project-Based Learning, students have a great deal of control of the project they will work on and what they will do in the project. The project may or may not address a specific problem. In Problem-Based Learning, a specific problem is specified by the course instructor. Students work individually or in teams over a period of time to develop solutions to this problem. This instructional approach is widely used in Architecture Education, Business Education, Medical Education, and in other situations where "case study" methods provide a useful focus in teaching/learning.

Project-Based Learning

Definition: Project-Based Learning is an individual or group activity that goes on over a period of time, resulting in a product, presentation, or performance. It typically has a time line and milestones, and other aspects of formative evaluation as the project proceeds. My Website on Information and Communication Technology-Assisted Project-Based Learning is available at http://darkwing.uoregon.edu/~moursund/PBL/

Project-Based Learning shares much in common with Process Writing. The roots of Process Writing as taught in the United States are often traced back to the Bay Area Writers Project circa 1975. A six step version of Process Writing is:

  1. brainstorming
  2. organizing the brainstormed ideas
  3. developing a draft
  4. obtaining feedback
  5. revising, which may involve going bask to earlier steps
  6. publishing

Here are some general ideas about Project-Based Learning

  1. Project-based learning is learner centered. Students have a significant voice in selecting the content areas and nature of the projects that they do. There is considerable focus on students understanding what it is they are doing, why it is important, and how they will be assessed. Indeed, students may help to set some of the goals over which they will be assessed and how they will be assessed over these goals. All of these learner-centered characteristics of PBL contribute to learner motivation and active engagement. A high level of intrinsic motivation and active engagement are essential to the success of a PBL lesson.
  2. From student point of view, Project-Based Learning:
    1. Is learner centered and intrinsically motivating.
    2. Encourages collaboration and cooperative learning.
    3. Requires students to produce a product, presentation, or performance.
    4. Allows students to make incremental and continual improvement in their product, presentation, or performance.
    5. Is designed so that students are actively engaged in "doing" things rather then in "learning about" something.
    6. Is challenging; focusing on higher-order skills.
  3. From teacher point of view, Project-Based Learning:
    1. Has authentic content and purpose.
    2. Uses authentic assessment.
    3. Is teacher facilitated--but the teacher is much more a "guide on the side" rather than a "sage on the stage."
    4. Has explicit educational goals.
    5. Is rooted in constructivism (a social learning theory).
    6. Is designed so that the teacher will be a learner.
    7. Teacher plays a major role in setting the learning goals of the project.
    8. Teacher and students provide formative evaluation.
    9. Teacher, students, and others may help in the summative (final) evaluation.
    10. Rubrics created by a combination of teacher and students. These facilitate self-evaluation, peer evaluation, evaluation by the teacher, and evaluation by outside experts.
  4. From a research point of view, Project-Based Learning is supported by work in:
    1. Constructivism
    2. Situated Learning Theory
    3. Motivation Theory
    4. Inquiry & Discovery-Based Learning
    5. Cooperative Learning
    6. Individual & Collaborative Problem Solving
    7. Peer Instruction
    8. Problem-Based Learning

Activity 1. Working in small groups, share your experiences (both successes and failures) in making use of Project-Based Learning in math education. As you think about this activity, you might conclude that PBL is used much more in non-math disciplines as than in math education. Why do you think this is the case?

ICT-Assisted PBL in Math Education

As noted elsewhere in this Website, there are many possible goals for Math Education. These goals can be expressed as a quite specific scope and sequence, such as textbook series tend to do.

In addition to a scope and sequence approach, one can look at some guiding themes of principles. In the What is Mathematics? section of this Website, for example, we have listed three quite general areas of expertise that might be developed by a person studying mathematics:

  1. Mathematics as a human endeavor. Mathematics has a very long history. Mathematics has beauty. Mathematics is an important aspect of aspect of past and current cultures. Mathematics is "the queen of the sciences."
  2. Mathematics as an interdisciplinary language and tool. Mathematics can be used to help represent, communicate about, and solve problems in many different disciplines. Many jobs and other aspects of responsible adult life in our society require some mathematical knowledge and skills.
  3. Mathematics as a discipline. The formal study of and research in mathematics is at least 5,000 years old. It is a deep and wide discipline with a huge amount of accumulated knowledge.

Each of these three general areas of mathematics expertise lends itself to both ICT-Assisted Project-Based Learning and ICT-Assisted Problem-Based Learning. The next three subsections give a few examples.

Mathematics as a Human Endeavor

Formal mathematics has a 5,000 year history, going back to the time that the Sumerians developed both writing and mathematics. Some of the important milestones in math are named after specific people (for example, Euclidean geometry, Pythagorean theorem) and many others are not (for example, abacus, fractions, decimals).

Activity 2: Select a person or topic from the history of mathematics. Do a project on it designed to increase your knowledge of your selected topic, and then to effectively share your increased knowledge with your fellow students.

Activity 3: Do a project on math-oriented aids to the human mind.

Activity 4: Do a project on women in mathematics, or on people of color in mathematics.

Activity 5: Do a project on ways in which ICT is changes and/or could be changing math education.

Mathematics as an Interdisciplinary Language and Tool

Beginning at least as early as the first grade in US schools, math is taught during a specific time slot each day. In elementary school, Typically a longer time slot is devoted to teaching reading and writing. Science and social studies may be taught on alternate days, sharing a time slot.

Think of this situation in terms of transfer of learning. As an example, consider reading. The goal is to have students develop a level of reading knowledge and skills by the end of the third grade so that in future grades then can learn by reading. The expectation is that a typical student learns to read and then reads to learn while he or she is continuing to become a still better reader. Although reading is part of a language arts time slot, reading and writing gradually become part of all of the daily curriculum.

Moreover, even though television and computer games consume a great deal of young students' time, there are a large number of opportunities and situations that encourage reading through the daily life of a young student. That is, both within school and outside of school, students tend to be in environments that encourage reading as a way to gain needed information.

Contrast this with math—and especially with the type of math outside the area of computational arithmetic. Although math is every were in our world, most students make relatively little use of it outside of the time slot of formal math education in school. Both the teaching of math and the use of math are minimal in the non-math curriculum at the K-12 level. Teachers typically make little attempt to help students transfer their steadily growing math maturity into non-math areas.

Perhaps one reason for this is that there can easily be a significant difference between a student's level of math maturity and that which is needed to apply math in a non-math discipline that is being taught. As a simple example, consider third grade students who are doing a classification or sorting activity in science of social studies,. The students are able to do such an activity in working with colored blocks in their math "class." Thus, with some help in transfer of learning, they can do it in science or social studies.

However, consider the possibility of students producing a pie chart (circle graph) of the data of the science or social studies data. This is an activity that is well above the math knowledge and skills of a typical third grader, even though such students can understand the parts of a whole and differences in size of the parts in a pie chart.

ICT provides an answer. There is easy to use software that an convert data into a pie chart. Thus, the procedural aspects of developing a pie chart can be given over to a computer.

This one example suggests a general idea:

  1. Look for direct applications of math while teaching non-math topics. If there is an obvious avenue for transfer of learning—directly applying math that students have already learned—make that application and talk about this type of transfer of learning. "Transfer of learning" should become part of the vocabulary of students and a goal in all of their learning.
  2. Look for situations in which a computer system can help student understanding and the knowledge they are gaining by mathematizing in a visual or other manner some of what is being covered in the non-math course. This mathematizing process may well draw upon ideas that students can understand (such as a pie chart) but that are several years above their current math developmental level.

Activity 6: Develop at least two more examples that have the characteristics 1 and 2 given above.

Mathematics as a Discipline

Math is divided into many different subfields. Students at the K-12 level may have the opportunity to study arithmetic, algebra, geometry, probability, statistics, trigonometry, and calculus. At a university undergraduate and graduate level they may have the opportunity to study additional areas such as abstract algebra, real and complex analysis, topology, non Euclidean geometry, number theory, numerical analysis, applied mathematics, discrete mathematics, and so on.

Activity 7: Select an area of mathematics that is at or below the level at which you are currently studying. Select an audience (students and/or adults) that have not had formal study in the area you have selected. Do a project that is aimed at understanding what that area of mathematics is well enough so that you can communicate it effectively to your fellow students. The end product in this project is to be a presentation or some other activity that effectively communicates your insights to your fellow students.

Activity 8: Select an area of math that is above the level that you are currently studying. Do a project that is aimed at understanding what that area of mathematics is well enough so that you can communicate it effectively to your fellow students. The end product in this project is to be a presentation or some other activity that effectively communicates your insights to your fellow students.

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