What is the difference between a Problem and a Project? While it is difficult to draw a definitive line that separates one from the other, the attributes of each and their differences as I see them are:
- Require less student time to complete (usually less than an hour)
- Focus on a single task, with fewer than 10 questions relating to it
- Can involve open-ended questions, but more often does not
- Are often one of a series of problems relating to a topic
- Look similar to many exam questions
- Can be used to introduce new concepts (Exeter Math)
- Can be used as practice on previously introduced concepts (most math texts)
- Require more student time to complete (hours to weeks)
- Focus on a theme, but with many tasks and questions to complete
- Provide an opportunity to acquire and demonstrate mastery
- Ask students to demonstrate a greater depth of understanding
- Ask students to reach and defend a conclusion, to connect ideas or procedures
- Can introduce new ideas or situations in a more scaffolded manner
Why Use Problems?
The lesson plans I find most interesting, both to read and to teach from, have both “public” and “hidden” learning objectives. The public objectives focus student attention and help interest students in the problem: they need to be short, to the point, and tightly related to the problem or project at hand.
The “hidden” objectives are the focus of teacher attention. They reflect the skills and concepts that the teacher hopes to see students grappling with, discussing with peers, and mastering over time while working on successive problems. If students are informed about a teacher’s list of objectives in assigning a task, students are likely to use only that list in their work. By not publicizing the teacher’s objective list, students are more likely to try a wider variety of approaches to solving a problem. I think the problem solving process starts with determining which concepts and skills seem relevant to the problem, therefore keeping the teacher’s objective list hidden helps students become better problem solvers.
The list below covers topics typically taught over a large percentage of the school year, so not all objectives are appropriate at any given point in the year. However, by the end of the year hopefully most of the following objectives will have been mastered by most students in a class:
Continue reading “Hidden” Learning Objectives for a Linear Equations Problem or Project
The title of this posting is the title of a chapter in “Making Learning Whole”, by David Perkins (2009), which I mentioned in my previous posting. I recommend it highly.
What is the “hidden game” in High School mathematics? What mindsets, approaches, techniques, etc. do those comfortable with the work asked of them rely upon, yet perhaps neglect to Continue reading Uncover the Hidden Game
One of the hardest questions for many math teachers to answer in a way that is relevant to students is: “why do I need to know this?” “For the next course you take”, the easiest answer in many cases, does not answer the question that was usually being asked. My answers to this question obviously depend on the topic being studied at moment, and I don’t have “good” answers for all topics… but here is my list of key life skills I learned directly or indirectly from math class, with some examples of situations where I find them indispensable.
Continue reading Life Skills Learned In Math Class
Towards the end of the unit(s) on Linear Equations and their graphs, students can feel a bit overwhelmed. The following is an attempt to summarize and link the key concepts you need to be comfortable with.
What is the least amount of information you need to know in order to be able to identify a line exactly? Two pieces of information: a point that the line passes through, as well as either a second point on the line or the line’s slope.
Continue reading Analyzing Linear Equations: a summary