## Projects vs Problems in Math Class

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:

### Problems

• 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)

### Projects

• 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

## Linear Equation Activity Ideas

Once a set of learning objectives have been settled on for an activity, problem, or project, what should the problem’s context be? Since linear equations model situations where there is a constant rate of change, common contexts for linear equation projects often include the following:

• Steepness, height, angle
Examples: road grade, hillside, roof, skateboard park element, tide height over the two weeks before (or after) a full moon, sun angle at noon over a six month period
• Estimating time to complete a task (setup plus completion)
Examples: mowing a lawn, painting a wall, writing a research paper
• Purchase and delivery costs of bulk materials
Examples: mulch, gravel, lumber
• Purchasing a service that charges by consumption
Examples: cell phone, electricity, water, movie rental, etc.
• Total earnings over time from differing wage and bonus plan structures
Examples: hiring bonuses, longevity bonuses
• Energy use over time
Examples: calories burned, electricity, heating oil, gasoline
• Game points accumulated over time
Examples: by a professional athlete, a team, a video game player
• Pollutant levels over time Continue reading Linear Equation Activity Ideas

## “Hidden” Learning Objectives for a Linear Equations Problem or Project

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:

## Peer Instruction Network

I recently came across a start-up organization called the Peer Instruction Network. It sounds like it is seeking to expand on Eric Mazur‘s teaching approach, something which would be very interesting to me on the Mathematics side of things.  Check out their web site, and sign up to be included in their network if it sounds interesting.

## Ten Skills Every Student Should Learn

A recent eSchool News article by Meris Stansbury lists ten skills cited by its readers as being most important for today’s students to acquire:

2. Type
3. Write
4. Communicate effectively, and with respect
5. Question
6. Be resourceful
7. Be accountable
8. Know how to learn
9. Think critically
10. Be happy

The list is interesting to ponder. I would not argue that any skills on the list should be dropped, however I suspect we could have endless debates about what order to list them in or how to best group them. I am happy to note that all of the skills are beneficial in studying just about any subject or discipline.

## Integrating Mathematics With Other Subjects

What if most activities in school asked students to “reach and defend a conclusion”?

•  in Math, about quantitative or geometric relationships, about measurements of worldly phenomena, etc.
• in Music, about the effect of a melody line, about a particular mix of instruments, etc.
• in English, about effective use of language or metaphor, about storytelling techniques, etc.
• in Visual Arts, about the effective use of color or negative space, about how a work can be interpreted, etc.
• in History, about a set of events, about relationships between societies, etc.
• in Physical Education, about the effects of various activities on the human body, about the effectiveness of various strategies in a sport, etc.
• in Science about whether two measurements are related in some way, why they might be related, the consistency with which they seem related, about cause and effect, etc.

What might our schools look like under such an approach?

## Game-like Engagement

A New York Times Magazine article titled “Games Theory” (September 19, 2010) mentioned some interesting points:

– “going to school can and should be more like playing a game, which is to say it could be made more participatory, more immersive and also, well, fun.”

– One way to “make school more relevant and engaging” to those who find it boring and are therefore at risk of dropping out is “to stop looking so critically at the way children use media and to start exploring how that energy might best be harnessed to help drive them academically”

– Games provide “‘failure-based learning,’ in which failure is brief, surmountable, often exciting and therefore not scary.” Students will “Fail until they win.”

– “Failure in an academic environment is depressing. Failure in a video game is pleasant. It’s completely aspirational.”

– “When it comes to capturing and keeping Continue reading Game-like Engagement