10/1/15
12/12/12
Collaborative Learning at the College Preparatory School in Oakland
Thanks to Pam Patterson at La Jolla Country Day School for sending me to read this.
A2 Driving to the Pyrenees
Introduction
A2 Driving to the Pyrenees is the second investigation in the acceleration unit. If you've not done so already, I recommend that you check out the first investigation first.
General Teaching Notes
In A1 students began to visualize slope as a representation of velocity. They imagined drivers on journeys with changing velocities and created graphs of distance versus time. They also graphed velocity as a function of time. These explorations continue in A2, with one twist: They are given the desired distances and are invited to invent the velocities, calculate the time each segment takes, and then graph. Some students will initially struggle with calculating time from distance and velocity. I am opposed to giving them D = rt because it draws them formulaic while they could instead still exploit their own experience of these relationships. If needed, I prefer to talk them through a couple examples with five minutes of leading questions at the beginning of class: Suppose you want to drive to the national park 100 km away. What kind of roads are those? How aggressive a driver are you? How fast would you go? If you went that fast, how long would it take? How fast would someone else here go? How long would that take? How did you get that answer?
1) Some students may choose to convert to minutes instead of keeping the table in decimal hours. You can suggest decimal hours, but I consider it an excellent exercise for them to labor through the minute conversions, if that is their preference. They also may hiccup at the total distance when they have to add -50 km. Let them work it out. Let them know that the second graph at the bottom is a graph of velocity vs. time, and have them label it appropriately. Students will have a variety of methods of graphing velocity, from something that looks like a bar graph, to a modified bar graph that shows some acceleration and deceleration, to simply graphing single points and then connecting the dots. Each attempt is a great opportunity to engage them in debate and discussion. I do not correct students much on the velocity graph, but I do go to groups and ask lots of questions. I want them to experiment and think about how to represent the velocity graphically.
I am, however, quite assiduous in pointing out mistakes on the distance graph. It's important to set an expectation that they graph as precisely as possible while acknowledging graphs are approximations. As math teachers we are comfortable when we see rough sketches because we already know the relationships implied by the graph, but students don't necessarily know that a steeper line corresponds to a bigger slope which is representing a bigger velocity which means the car goes further in less time. They are quite capable of understanding this, but they need to work it out as opposed to simply memorizing and implementing rise over run.
2) This repeats the drive done in number 1, but of course students can invent at least some parts of it, like how long Michael takes to eat lunch, and whether or not he forgets something at the gas station and has to go back for it. They could even invent a slightly longer or shorter route with different roads and so on. Some students may initially balk at inventing anything new--encourage them to be creative.
A2 Driving to the Pyrenees is the second investigation in the acceleration unit. If you've not done so already, I recommend that you check out the first investigation first.
General Teaching Notes
In A1 students began to visualize slope as a representation of velocity. They imagined drivers on journeys with changing velocities and created graphs of distance versus time. They also graphed velocity as a function of time. These explorations continue in A2, with one twist: They are given the desired distances and are invited to invent the velocities, calculate the time each segment takes, and then graph. Some students will initially struggle with calculating time from distance and velocity. I am opposed to giving them D = rt because it draws them formulaic while they could instead still exploit their own experience of these relationships. If needed, I prefer to talk them through a couple examples with five minutes of leading questions at the beginning of class: Suppose you want to drive to the national park 100 km away. What kind of roads are those? How aggressive a driver are you? How fast would you go? If you went that fast, how long would it take? How fast would someone else here go? How long would that take? How did you get that answer?
1) Some students may choose to convert to minutes instead of keeping the table in decimal hours. You can suggest decimal hours, but I consider it an excellent exercise for them to labor through the minute conversions, if that is their preference. They also may hiccup at the total distance when they have to add -50 km. Let them work it out. Let them know that the second graph at the bottom is a graph of velocity vs. time, and have them label it appropriately. Students will have a variety of methods of graphing velocity, from something that looks like a bar graph, to a modified bar graph that shows some acceleration and deceleration, to simply graphing single points and then connecting the dots. Each attempt is a great opportunity to engage them in debate and discussion. I do not correct students much on the velocity graph, but I do go to groups and ask lots of questions. I want them to experiment and think about how to represent the velocity graphically.
I am, however, quite assiduous in pointing out mistakes on the distance graph. It's important to set an expectation that they graph as precisely as possible while acknowledging graphs are approximations. As math teachers we are comfortable when we see rough sketches because we already know the relationships implied by the graph, but students don't necessarily know that a steeper line corresponds to a bigger slope which is representing a bigger velocity which means the car goes further in less time. They are quite capable of understanding this, but they need to work it out as opposed to simply memorizing and implementing rise over run.
2) This repeats the drive done in number 1, but of course students can invent at least some parts of it, like how long Michael takes to eat lunch, and whether or not he forgets something at the gas station and has to go back for it. They could even invent a slightly longer or shorter route with different roads and so on. Some students may initially balk at inventing anything new--encourage them to be creative.
11/27/12
Falling in Love With Boat Building
Why I teach.
(A speech I gave to the Cum Laude Society at La Jolla Country Day School, later published in LJCDS360, a Country Day promotional publication, in Summer 2010. )
(A speech I gave to the Cum Laude Society at La Jolla Country Day School, later published in LJCDS360, a Country Day promotional publication, in Summer 2010. )
11/22/12
Why Collaborate? The Bullet Points
- students learn by doing
- reduce passive listening
- give students more autonomy over their learning
- empower students to express opinions, ask for help, help others
- gives me (eachother) immediate feedback on their understanding and engagement
- adapts well to students of different ability
- research and my anecdotal evidence shows it works
- it teaches soft skills: collaboration, communication, writing and creative problem solving
- CEO’s like those skills, research shows social skills are the #1 factor in future success
- forces me to teach better
- IB and Common Core are headed this way
- more interesting
11/14/12
Khan Academy, a learning institution
If you've read my post about collaborative investigation you've got an idea of what Khan Academy is all about. They've got a mission to provide instructive videos on nearly every topic, organized in an intuitively sensible way and paired with interactive practice problems. In this way, a student selects their own homework to support their work in class. These online tools allow teachers to abandon traditional lowest-common denominator lecturing and open up the classroom to deep exploration of mathematical concepts linked to real world problems. Indeed, this is the larger revolution which Khan Academy aspires to inspire.
Khan Academy is young, and they are learning as they go along. Their process is as collaborative and non-linear as that of our students. They adapt to unexpected realities, they adopt new tools when they make sense, and they model the creative problem solving which we hope to see in our students' work.
Learn how they learn here.
Khan Academy is young, and they are learning as they go along. Their process is as collaborative and non-linear as that of our students. They adapt to unexpected realities, they adopt new tools when they make sense, and they model the creative problem solving which we hope to see in our students' work.
Learn how they learn here.
11/13/12
A1 Pre-Race Warm Up
Introduction
A1 Pre-Race Warm Up is the first investigation in a unit on acceleration. In my 9th grade math class, this unit is preceded by an inaugural unit on lines and systems, but as I am introducing this material to the blog in November I'll skip lines for now. In any case, I think the acceleration unit is rather more interesting and actually stands well on its own no matter where you are in your year of work with your students. Indeed, you might use it in any high school math class.
As you may have guessed, this unit is followed by a quadratics unit. You might think that students would need to understand quadratics before modeling accelerating objects, but actually the converse is true. The acceleration unit, driven by tables and graphs, eventually draws students to develop a quadratic function. As students approximate solutions to problems, they invariably ask me what algebraic tools they can use to get more precise results. Algebraic tools are introduced when a student has developed a need for the tool.
Paintbrushes and wrenches, words and hammers, these tools of the trade are powered by the simple desire to order the world. For the toddler, before the hammer had any known application, hammering for the sake of hammering sufficed. But even the toddler's first word, "water," was arguably uttered for want of a solution to a problem, thirst. The question, "Why do we have to learn this," is evidence of a tool lacking a purpose. This acceleration unit flips the question to "Do you have any tools I can use to solve this more precisely?"
Subscribe to:
Comments (Atom)