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Design Project: Problematic Mathematics: PBL designed for the math classroom

Project Topic:

Problem-Based Learning in the Math Classroom:

  • This is an approach that strives to achieve mathematical understanding by solving problems or complex tasks that require exploration and discovery. Students are to reflect on the problems and their experiences individually and in groups to communicate different understandings. “These are tasks for which students have no memorized rules, nor for which they perceive there is one right solution method. Rather, the tasks are viewed as opportunities to explore mathematics and come up with reasonable methods for solution” (Hiebert et al., 1997, p. 8).

Intent of Project:

The intent of this project is to give teachers the resources needed to understand Problem-Based Learning as it relates to the math classroom and to provide authentic examples of this type of pedagogy. These examples will be in the form of lesson plans and can be integrated in a high school math classroom.

Learning outcomes and topics to be covered:

Participants must see the importance of problem-based learning in the mathematics classroom; thus in order for the goal of instruction to be achieved, participants must have positive attitudes regarding mathematical instruction. Since this is directly correlated with motivation, the individuals participating in this course must be motivated, both intrinsically and extrinsically. Learners will need to acquire knowledge regarding problem-based instruction (identify tasks that are problematic in nature) and gain the ability to create and use tasks that represent this approach (cognitive strategies). Therefore, learners will need to become very familiar with problem-based learning, since "the information is essential to these events" (Gagne,2005, p. 52)

  • What is problem-based learning in the math classroom
  • Why this pedagogy is important and useful
  • Identifying, understanding, and creating problematic tasks
  • How to effectively design assessments and provide feedback

Needs Assessment:

Instructional Problem: Students have difficulties understanding mathematics as a useful subject in their everyday lives. More importantly, math should make sense and should be a discipline of exploration and discovery; however, teachers have a difficult time designing tasks that align with such an outcome. Therefore, the problem lies in the design of tasks and the implementation of existing activities that are designed to promote understanding and transfer in the mathematics classroom.

The nature of what is to be learned: Teachers will learn about Problem-based mathematics, how to design problematic tasks, and how to implement existing curricula/tasks in a problematic manner that encourages mathematical thinking and exploration.

About the learners (Learner Profile): This course will be designed for high school math teachers looking to improve teaching methodologies and change classroom experiences. As teachers, from a variety of backgrounds, the experience level will vary from little (one-year or less) to significant (10 years or more) levels of expertise. However, these learners will share a common characteristic, they will all be intrinsically and extrinsically motivated to learn, as these participants are goal-oriented wishing to improve their teaching practices. In addition, the education level held by the participants will also differ; while they will hold a minimum of a bachelors degree in Mathematics or Education, some will hold master’s degrees and PhD’s in their specialized field. Thus, the participants will be teachers of diverse backgrounds, varying teaching experiences, and differing achievement levels.

Instructional content: All of the units will be designed using the same format. However, each unit will include different learning activities and tasks that will contribute to the learning objectives of both the individual unit and the course itself. The ideas presented in this course will include information from scholarly articles, books, surveys, and video-based media presentations of lessons designed for problem-based learning. The learners will be reading articles, analyzing existing tasks, and watching video-presentations of lesson plans that accurately portray the goals of instruction. The participants will also be discussing their ideas with peers and reflecting on the ideas presented/learned through journal entries.

Units in this course:

  1. What is problem-based learning and why is it beneficial?
  2. Assessing and providing feedback for students.
  3. Designing problematic tasks.
    1. A problematic task in the Algebra classroom.
    2. A problematic task in the Geometry classroom.

Explore the issues surrounding the instructional problem and solution: The problem is the lack of understanding and transfer in the mathematics classroom due to the ill design of learning tasks and implementation of activities. The proposed solution is Problem-Based Learning in the Mathematics classroom where teachers are introduced to a new approach of teaching, which discourages memorization and repetitive exercise and instead introduces the concept of problematic mathematics.

  • What is already known about the problem? Designing tasks that are engaging and 'problematic' in nature is a difficult and time-demanding process. Creating a classroom environment that supports such an approach is opposite of the traditional methodologies of teaching, and can as a result, conflict with the current curricula. Moreover, the need for mathematical understanding and interactive activities is only one objective that teachers have, since the pressures of standardized testing monopolize much of their time/focus. More importantly, as a result of this problem, students cannot transfer their knowledge to other disciplines or applicable problems in math since concepts are frequently misunderstood.
  • What has already been done: Teachers have attempted to create tasks that teach for understanding but they often do not align with the learning outcomes. "Understanding is being able to carry out a variety of "performances" that shows one's understanding of a topic and, at the same time, advance it." "Most school activities are not performances that demonstrate understanding: Rather, they build knowledge or routine skills" (Perkins & Blythe, 1994, p. 5).

Goals of Instruction: Pending

Performance Objectives:

As a result of this mini-course, the participants will be able to do the following things:

  1. Given a question, the participants will describe the characteristics of problem-based learning, as it relates to the mathematics classroom, in his or her own words.
  2. When colleagues use traditional teaching methodologies in the math classroom, the participants will choose to use problem-based learning strategies.
  3. Given core mathematical concepts to be taught, the participants will design tasks that are problematic in nature, in written form, to stimulate student inquiry.
  4. Given a problematic task, the participants will identify three ways to assess students on their understanding, in written form.
  5. Given a problematic task, the participants will describe different ways to provide student's with adequate feedback, in writing.
  6. Given different authentic problems/lesson plans, the participants will understand and adapt these techniques, putting them to use in their individual classrooms.

Task Analysis:


The purpose of this course is to provide participants with the information and resources needed to understand problem-based learning as it relates to the math classroom.


Below are the enabling and supportive prerequisites that correlate to three of the six objectives listed above (1, 2, & 6). These learned capabilities represent verbal information, attitudes, and cognitive strategies.

Enabling Prerequisites:

  • Recognizes and understands the term problem-based learning
  • Realizes how problem-based learning relates to the mathematics classroom
  • Discriminates between problem-based learning and other teaching methodologies.
  • Understands why tasks are problematic in nature.
  • Possesses the ability to create a learning environment that promotes problem-based learning.

Supportive Prerequisites:

  • Translates the definitions and descriptions into statements using self-generated explanations.
  • Values problem-based learning in the mathematics classroom.
  • Acquires the ability to recall the important details of problem-based learning.
  • Justifies the value of problem-based learning in the classroom.

Intellectual Skills Objectives with their correlating learning hierarchies:

  • Given core mathematical concepts to be taught, the participants will design tasks that are problematic in nature, in written form, to stimulate student inquiry.
Learning hierarchy of enabling and supportive prerequisites: File:Intellectual Skill Objective.pdf
  • Given a problematic task, the participants will identify three ways to assess students on their understanding, in written form.
Learning hierarchy of enabling and supportive prerequisites: File:Intellectual Skill Objective 2.pdf
  • Given a problematic task, the participants will describe different ways to provide student's with adequate feedback, in writing.
Learning hierarchy of enabling and supportive prerequisites: File:Intellectual Skill Objective 3.pdf

Instructional Curriculum Map:

The following link sequences the units (correlating target objectives), and the performance objectives for the course: File:Revised Sequencing objectives and units.pdf

The following link will bring you to the curriculum map for this course. It will explain the units, target objectives, and the activities to come. File:Revised ICM Problem Based Learning in Math .pdf


After reading chapters 9-12 in Gagne I have thought a lot about the different learning activities I need to use in order to gain the attention of the learner, inform the learner of the objectives, stimulate recall, present stimulus material, provide guidance, elicit performance, provide feedback, assess performance, and enhance retention. I decided to use a combination of the following activities; video tutorial, articles, lesson analyzation, lesson creation, and journal entries to encourage learning.

Arrow.png Unit 1: What is problem-based Learning and why is it beneficial?

  • Objective:
  • When given a prompt participants will be able to describe, identify, and explain important aspects of problem-based learning through a journal entry.
  • When given a question, participants will be able to characterize the benefits of problem-based inquiry through a reflected journal entry.
  • Activities:
  1. Read articles
  2. Analyze statistics within the article
  3. Reflect: How could PBL be beneficial to your classroom? How do PBL and traditional styled classrooms differ? Explain, in your own words, the important aspects of problem-based learning.

Arrow.pngUnit 2: Designing Problematic Tasks

  • Objective: Given some examples, participants will identify and describe ways to create problematic tasks while acquiring the ability to design authentic tasks in their individual classrooms.
  • Activities:
  1. Read scholarly articles
  2. Review authentic tasks developed by the instructor (Algebra and Geometry)
  3. Watch/observe at least one lesson from the following website: http://www.insidemathematics.org/index.php/home
  4. Reflect: How can tasks be designed so they are problematic in nature? What are the common components of an effective lesson that follows PBL? How can you create such tasks to implement in your existing classroom? Now try and create another authentic and problematic task for an Algebra course; once finished evaluate it using the 'rules' of problem-based learning.

Arrow.pngUnit 3: Assessment & Feedback

  • Objectives: Participants will identify and describe different methods of appropriate assessment and feedback in writing through discussions/journals.
  • Activities:
  1. Read scholarly articles
  2. Review different forms of assessment
  3. Reflection: What is the difference between summative and formative assessment? How does this relate to the ability to give feedback? What are three different ways to assess students participating in PBL? Why are they effective methods?


Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: middle school videor cases to support teaching and learning. Portsmouth, NH: Heinemann.

Gagne, R. M., Wager, W. W., Golas, K. C., & Keller, J. M. (2005). Principles of Instructional Design (5th Ed.). Belmont, CA: Wadworth/Thomson Learning. (ISBN 0-534-58284-2)

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., Murray, H., et al. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Perkins, D., & Blythe, T. (1994). Putting understanding up front. Educational Leadership, 51 (5), 4-7.