Types of lessons

A good shop teacher has to allot sufficient time for students to work independently, building interesting projects.

However, the students at some point need to be introduced to the basics of operating the drill press and table saw. They need to know when that tool is called for, and they need to receive enough basic instruction to apply and explore the tool's capabilities without frustration.

The students also at some point have to return to each major tool -- once they are familiar with it and have found it repeatedly useful -- to understand it as an object in itself. At this point they need to generalize their experience using a tool in this or that project to a full understanding of the tool's operation, capabilities and limitations.

For certain important tools -- ones that are repeatedly called upon -- the students may find it necessary to actually practice using the tool, not in the service of building an object but instead so that the operation of that tool becomes automatic, ingrained and speedy.

Similarly, my students will at various times

  • Work individually and in groups on projects, producing work that is interesting important and useful. (ex. Minimum Wage or Ready Reckoner.)
  • Engage in 'short works:' one- or two-class long lessons that give a context-based introduction to a mathematical method or tool. For example, surveying the school courtyard to learn the Law of Cosines or using parametric equations to model running routes in football.
  • Return to a mathematical process to
    For example, after students have met quadratic equations in several different guises (in the course of both extended investigations and short works) it's time to step back, point out the centrality and utility of quadratic functions, and understand the powerful shortcuts available to solve them.
  • Engage in focused drill on certain vital procedures

 

 


As mentioned before, I expect to make major changes to the time periods described here as I gain experience

Extended Investigations

The course is structured around a series of extended investigations.

These projects, which have individual, small group and class/community components, are carried out over about two weeks each. Three out of five class days and a corresponding fraction of the students' homework time are devoted to working on these investigations.

An example is Minimum Wage. The class goal is to prepare a comprehensive analysis of the minimum wage, along with a policy recommendation, that they will share with their congressional representative. Students break into groups, each charged with a different fork of the analysis, that they present and defend to the class as a whole. Students are individually charged with recapitulating the main findings of each other group for their personal portfolio.

Each group submits a benchmark (formative) assessment at the end of the two-week period. They are given later time to improve their work and for the class to integrate the groups' findings into one whole document. The class sends their final recommendations to their representative; each student adds her group's section and her summary of the other groups' findings to their final portfolio.

These projects are intended to require standards- and goals-aligned skills in context. Students will want (and need) to learn necessary mathematical processes in the course of completing each extended investigation.

 


 

Short Investigations

  • Students work independently on short (est time of 1-2 classes + 1-2 homeworks) investigations.
  • There are a variety of projects available. As the semester progresses (and students' toolkits grow) I will add more to the pool.
  • Students complete these at their own pace.
    • As they complete them they turn them in, and receive goals-oriented feedback.
    • They're free (and encouraged) to improve and resubmit them.
    • These works are added to their final portfolio. The only summative (final) assessment is derived from the works that appear in the final portfolio.
    • There are minimum number and quality standards that students must meet. For this portion of their portfolio grade, however, fewer at high quality and more at average quality will contribute similarly.
  • It's expected that different students will attack different investigations, and in different order.
    • Students are free to consult with others who have completed or are also investigating this topic, within the bounds of community guidelines on cooperation vs. plagiarism.
  • While extended investigations are organized around a topic and are highly open-ended, short investigations are organized around deploying one or a small number of skills in the context of a topic.
    • As such, they will be often used to introduce students to a given skill.
    • Later in the semester, they will be more focused on topics/tools that connect to other subjects or material, or are interesting but lie outside the core material.
  • Examples:

 

Skills Practice

We want to answer real-life questions,

but at the same time, you're smart enough to begin to master some more powerful tools that are so general -- they're broadly applicable to so many things you'll encounter going forward -- that you should start using them and appreciating their power as soon as possible.

This is a difficult balancing act.

It will also, frequently, require drill.
When you're learning football you have to stretch,
you have to run laps,
and you have to run through the little staggered array of tires.
No one enjoys this (or at least, no one competes to be quarterback of the "running through the little staggered array of tires" team) -- but I think it would be an interesting experience to convince your high school football coach that it is unnecessary

 


 

Hard Problems

Given in the Moore method: I pose a small number of problems, students are left to work on them (and agree not to consult outside resources).

These are short problems that demand high-level pure mathematical reasoning like those from Challenging Mathematical Problems With Elementary Solutions or Math Olympiad Problems (more). A couple examples:

  • Does New Year's Day occur more often on Saturday or on Sunday?
  • For positive reals x, y, z where x+y+z=3, show that √x + √y + √z ≥ xy + yz + zx. Discuss in context of the triangle inequality.
  • A, B, C, D, E, F are points on the graph of y = ax3 + bx2 + cx + d such that ABC and DEF are both straight lines parallel to the x-axis (with the points in that order from left to right). Show that the length of the projection of BE onto the x-axis equals the sum of the lengths of the projections of AB and CF onto the x-axis.

We will occasionally reserve a short period of class time to discuss the problems, and for students to present solutions, headway, failed solutions, ideas, extensions and strategies.-9-=

Students can write up solutions to these problems for their portfolio. Some recognition is made for students who solve the problems independently.

 


 

Class Activities and Homework

  • Project Days
  • Skills-focused practice
  • Short (individual) investigations
  • Hard problems
  • Tests
  1. What is a typical class like?
    1.  
    2. Class time should be focused: in math class, students should do math
    3. Students should find a predictable daily/weekly structure
    4. Students should also have an interesting mixture of activities and freedom to explore
  2. What work is expected of students
    1. Readings; Texts
    2. Homework
    3. Tests
    4. Projects


Where can I find banks of SAT and TAKS problems?

Classwork and Homework

  • Problem types:
    • Mini projects that demand contextual reasoning and problem solving
    • Focused drill to build proficiency using specific mathematical processes.
    • TAKS test-like problems
    • SAT test-like problems
    • Short problems that demand pure mathematical reasoning like those from Challenging Mathematical Problems With Elementary Solutions or USSR Olympiad Problems. (e.g. Does New Year's Day occur more often on Saturday or on Sunday?)
    • Fermi problems
    • Metacognition exercises: explain-to-family-member, explain-to-dog, describe-your-approach.
  • Assign 1 hour/day of homework (honors/AP), 45 min/day (regular).
  • A pool of standing difficult problems are assigned on the Moore method: time is set aside for students to share progress/conjectures/solutions

 


 

Class environment

 

Responds flexibly to students during a lesson, adjusting instruction as needed depending on student progress.

While teaching the Modern Physics lab, I tried to tailor the course to the needs, abilities and interests of each student.

The first time I taught the Football Running Routes lesson, we realized that students were thrown off by some minor details in the very first part of the exercise (plotting the route of their player). They spent a lot of time debating how to choose axes and scales on their graphs, and in many cases drew illustrative rather than schematic lines (i.e. not straight lines). When we retaught the lesson to a later class, we stopped the class after a few minutes and had students demonstrate and justify the choices they had made. Teams that felt they had chosen badly were given new sheets. This gave people the freedom to investigate and possibly fail, but without sabotaging the larger goals of the lesson.


 

Responds flexibly to students during a lesson, adjusting instruction as needed depending on student progress.
While teaching the Modern Physics lab, I tried to tailor the course to the needs, abilities and interests of each student.
The first time I taught the Football Running Routes lesson, we realized that students were thrown off by some minor details in the very first part of the exercise (plotting the route of their player). They spent a lot of time debating how to choose axes and scales on their graphs, and in many cases drew illustrative rather than schematic lines (i.e. not straight lines). When we retaught the lesson to a later class, we stopped the class after a few minutes and had students demonstrate and justify the choices they had made. Teams that felt they had chosen badly were given new sheets. This gave people the freedom to investigate and possibly fail, but without sabotaging the larger goals of the lesson.


 

 

 

 


Engaging students in a variety of interesting, challenging, and worthwhile activities. The first step in learning something is to want to master it

Helps students link new content with their prior knowledge and gain insights into their misconceptions.
The lesson on parametric equations begins with a free-form description of a running route and asks students to graph that route. Students can either graph the route directly (by finding given points and drawing as a path connecting them), or can construct an algebraic representation of the route and then plot a graph of that function. In either case, the prior knowledge

Develops clearly-stated objectives that are age-appropriate and able to be assessed.

Guides students in using appropriate technology to gather, organize and display data.

 

Selects or designs a variety of worthwhile assessment instruments, some of which involve student self-assessment.

 

Creates learning activities that emphasize collaboration and teamwork.
"positive interdependence": for example in the Football Running Routes lesson, students had to analyze whether their individual routes crossed, and if they collided. This meant that each student relied on their teammates correctly determining their player's path and location in time.

 


 

Calling on Students

My basic approach would draw on a popular teaching tip: have each student create a popsicle stick, clothespin or playing card and hold a random election to see who will answer.

I have a good idea for a richer technology solution that I'd like to implement over the summer

Create program with

  • face array
  • seating order
  • names

Take attendance (here / late / absent)

I can hit a button and have it randomly select from all the students who are present.

  • Each student has between 1 and N tokens in the pool. I think 10 is a good number.
    • The number of tokens is never less than one: if I perform an election there is always some chance that any given student might be called upon.
  • Each time they volunteer an answer I mark off 1+1d3 (2 to 4) tokens.
  • If they're called on to answer I mark off 1d3 (1 to 3) tokens.
    • Volunteering is more valuable than getting called on, but not drasticly so.
    • It takes about two excellent/volunteered answers, three excellent/called-on answers, or four average answers to get to the minimum.
  • It's very easy to mark off answer tokens: type in student's unique name with completion, hit tab, #, enter. (face shows up to confirm.)
    • alt: show seating chart, click, #, enter.
  • The election process displays an interesting animation that
    • shows a small pool of students' names/faces
    • randomly and excitingly selects from that pool.

At the end of class, the program prints a report showing

  • Attendance state
  • Number, type and quality of responses
  • A participation heuristic based on those.

 

More ideas for calling on students:

  • "I like to use the equity card method (names on cards). I let the students decorate their own equity cards because it helps them buy into the technique. Before I draw a card, I have the students do a think-pair-share. They turn to their neighbor and discuss the answer for 30 seconds. Then I call the person on the top card to share. If a student doesn't know the answer or needs more time, I put the card aside and come back to it later. And I NEVER accept "I don't know" as an answer.
    ...
    Sounds like a good technique. I especially like the fact that you come back to the student who didn't previously know the answer."
  • "... Calling on nonvolunteers usually results in a dead conversation and an embarrassed student. The good thing is that the student may learn to pay attention and be ready to be called on, but I find I get so many blank stares and shrugs when I call on someone without their hand up. I'm not saying it's bad, I'm just saying, what next?
    1. Make sure you [have] taught the material before you question on it.
    2. When a student struggles, have them "dial a friend" for help, or ask another student. Most importantly, go back to that struggling student and make sure he can respond correctly before proceeding.
    3. If the question is high order Blooms, make them think you will call on nonvolunteers after asking the question. Then after a sufficient wait time, go ahead and ask for volunteers.
    4. If you get more than one shrug per question, then you need to go back and re-teach the material -- the kids' didn't "get it."
    5. As for embarrassment, it is better for them to be embarrassed now then behind later. Once this method of questioning becomes part of the regimen the number of shrugs will vanish."
  • "I would use playing cards, and let the kids pick the cards. They love it! Just don't present the cards until the students have had time to think of an answer, otherwise their mind will be on picking the card and not the question."