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?
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
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
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.
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:
- Communicate effectively, and with respect
- Be resourceful
- Be accountable
- Know how to learn
- Think critically
- 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.
There are a few additional skills that I would advocate adding to, or being more explicit about in the above list:
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?
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
A continuum of activity types are used in classrooms around the world. They range in duration from long (weeks or months) to short (seconds or minutes), from “projects” to “problems”. There are differing styles of activities, ranging from context-rich to almost context-free. There are also differing roles for activities in a curriculum: they can serve as warm-ups, practice, assessment, and/or a primary means of instruction (as at Phillips Exeter Academy and elsewhere).
The activities that seem most effective to me tend to require at least 15 minutes, if not more, to complete unless the context for the activity has been previously introduced. Longer (or recycled from the recent past) activities require fewer mental transitions for students, and hopefully lead to greater focus on the core concepts and skills. Obviously short duration activities cannot include as many of the attributes below as longer ones might. The “laundry list” of attributes I consider when creating or modifying an activity now includes:
1) A warm-up section which:
– engages the Continue reading Eight Attributes of Effective Activities, Problems, or Projects
Everyone likes textbooks
Teachers like a textbook because teaching from a text can require less work than other approaches. Many texts provide extra resources such as Chapter Tests, worksheets of extra problems, and project support materials that save time for a teacher. Furthermore, a teacher’s edition of the text can also remind teachers of alternative approaches to a topic, give guidance on sequence and timing, and make it easier to coordinate with other teachers who are teaching the same course.
Parents like a textbook because it shows them what their child has learned and will learn. It also explains the approach being used in greater detail than their child probably can, which is vital when the approach differs from the one the parent learned when they were in school. If a child has questions which the parent cannot readily answer, the parent can use the textbook to help the child figure out the answer.
Students like a textbook because Continue reading To Use A Textbook, Or Not?
I have started a separate blog devoted to helping students learn to find mistakes in worked problems (their own, or someone else’s). If this is of interest, check it out:
7/17/11 Update: There can be great value in work that contains mistakes. Learning to catch your own mistakes is a critical life skill, as is learning to review other people’s work while seeking to understand it fully (the best way to do this is by looking for mistakes).
Along these lines, I came across an interesting blog posting by Kelly O’Shea. She came up with the idea of insisting that each group who is presenting try to sneak a mistake in their work past their peers. Brilliant!
After reading a number of blog postings about Standards Based Grading (SBG), I tried a hybrid version of it during the Fall semester of 2010 in an Algebra I class and three Algebra II classes. What follows is a description of how I approached things, what worked, and what didn’t.
Approximately 40% of each student’s semester grade was based on SBG quiz scores, 30% on traditional chapter test scores, and 30% on the semester exam.
I was not obligated to test to specific standards, so I picked quiz “topics” which were timely and allowed for challenging questions. My goal was to have most questions be challenging enough to make perfect scores unlikely on the first try.
Each student’s best two scores for each topic counted towards their overall Quiz grade. All lower scores for the topic were dropped, regardless of the sequence in which the scores were obtained.
Each answer received a maximum of five points:
– One point for attempting the problem
– One point for using a valid approach to the problem
– Three points for working the problem to a solution
– One point taken away (up to three) for each algebra or arithmetic error
My grade book consisted of a three-ring binder with one page for each student. Each page had a Continue reading Standards Based Grading Trial
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
I am researching student response clickers, in hopes of using them in Algebra I and II classes this fall.
To date I have come across six companies that offer them:
I have found internet references to the following vendors Continue reading Student Response Clickers
I just came across this 1 hour and 20 minute video of Eric Mazur talking about using the Force Concept Inventory to teach physics, advocating “peer instruction”, and presenting data from his courses before and after he changed his instructional techniques.
This is great stuff. My teaching and learning experiences have convinced me wholeheartedly that these ideas apply equally well to mathematics instruction and should be implemented at all levels. Having said that, I also recognize that it can be very difficult to implement some of these approaches in early Algebra, particularly when using an existing textbook that takes a traditional approach…
Life is full of alternatives. Would like fries or coleslaw with your meal? Should you put on your right or your left shoe first? Should you attempt to solve a math problem using algebraic procedures, or your intuitive sense of the situation?
Life is also full of false choices: there are many occasions when you do not have to make a choice unless you wish to. You could have fries with a side order of coleslaw. If you wear loafers, you could slide your feet into both shoes at the same time. And many math problems can be solved quite successfully using a combination of intuitive reasoning and algebraic procedures.
Continue reading Procedural vs Intuitive Approaches