These goals include and extend the TEKS (the state-mandated goals).

 

Course Goals

  • Work with your classmates to solve several interesting and important problems of significant scale.
  • Master a small core of processes -- ones that you will use so many times (in this class, in later classes, in real life, and on the state exams) that you must practice them until they are automatic.
    • Rearrange simple symbolic equations effortlessly, quickly and correctly.
    • Algebraically solve for the parameters of linear and quadratic functions
    • Make clear and reasonable plots of equations in standard form
    • Find and evaluate a reasonable linear fit to a set of data, by computer or by eye.
  • Acheive your goals on the state exam (TAKS) and college entrance exams (SAT).
  • Demonstrate that you can draw on your full range of mathematical tools (tools from this and past years) to solve non-trivial problems you’ve never met before.
  • Understand how these ideas are applied in real-life situations, and how this course fits in with previous & future math topics
  • Master the core Algebra II skills:
    • Build on your core of basic understandings:
      1. Foundation concepts:
        • basic understandings of number, operation, and quantitative reasoning;
        • patterns, relationships, and algebraic thinking;
        • geometry;
        • measurement; and
        • probability and statistics
      2. Algebraic thinking and symbolic reasoning.
      3. Functions, equations, and their relationship.
      4. Relationship between algebra and geometry.
        • Equations and functions as tools to represent curves and figures
        • Geometric figures as tools to illustrate algebraic relationships
        • Perceive the connections
        • Use the tools of one to solve problems in the other
      5. Tools for algebraic thinking.
        Students become nimble problem solvers, capable of using
        • a variety of representations (concrete, numerical, algorithmic, graphical), and
        • a variety of tools and technology (including, but not limited to, graphing calculators and computers)
        to model mathematical situations and solve meaningful problems.
      6. Underlying mathematical processes:
        • Problem solving
        • Computation
        • Language and communcation
        • Connections within mathematics
        • Connections outside mathematics
        • Reasoning
        • Multiple representations
        • Applications and modelling
        • Justification and proof
    • Solve messy problems:
      • collect data and record results,
      • organize the data,
      • make scatterplots,
      • fit the curves to the appropriate parent function
      • interpret the results,
      • proceed to model, predict, and make decisions and critical judgments.
    • Become familiar with the full family of elementary functions:
      • Linear, Inverse, Quadratic, Square Root, Rational, Exponential and Logarithmic functions, as well as absolute value, piecewise linear and simple periodic functions to the extent they arise naturally in the course of work.
      • Specifically,
        • Represent them graphically, numerically (table) and algebraically, and translate among representations;
        • Skillfully manipulate them in symbolic form
        • Describe symmetries, asymptotic behavior and other important global features of a given function.
        • Use them appropriately to model, predict and solve Interesting, Important and Useful problem situations.
        • Describe their domain and range, and interpret the "reasonableness" of solutions accordingly.
        • Conceptually connect elementary functions to their inverse, and understand this connection in graphical, algebraic and practical terms.




 

Learner-Centered Instruction

My teaching is oriented around four general principles of Constructivist Learning:

  • Learning occurs in context -- I design activities that are situated in "complex, realistic, and relevant environments." I identify areas that students are experts in and capitalize on that expertise, and I forge connections with everyday tasks, other classes, and future topics.
  • Emphasis on discovery -- I encourage and accept student autonomy and initiative. Students in my classroom use raw data and primary sources, and critically evaluate the worth of the information they use. I ask more than I tell, and am happiest when students start posing complex questions (and demanding insightful answers) of each other.
  • Learning is social -- students and teachers form a Classroom Community that "encourages ownership in learning."
    I encourage students to engage in dialogue, with me and with one another. Student responses may drive lessons, shift instructional strategies, and alter content.
  • Learning is reflective -- instructors should "nurture self-awareness of the knowledge construction process" I engage students in experiences that might engender contradictions to their initial hypotheses and then encourage discussion, and provide time for students to construct relationships and create metaphors.
(Synthesized from [Driscoll, Psychology of Learning for Instruction (2000) (quoted in ERIC Digest)], [Driscoll, "How People Learn (and What Technology Might Have To Do with It)" (2002)], and [Brooks, J. & Brooks, M. In Search of Understanding: The Case for Constructivist Classrooms (excerpt in ERIC Digest) (1999)])

Curriculum

Numeracy

http://www.maa.org/ql/mathanddemocracy.html http://pdonline.ascd.org/pd_online/memory/el199910_steen.html http://www.stolaf.edu/people/steen/Papers/numeracy.html http://www.stolaf.edu/people/steen/Publications/papers.html

 

 


(note: I know this schedule is horribly naive and simplistic -- I will refine it during the school year and in consultation with master teachers.)

School Year Planning

Roughly speaking a semester is (discounting holidays and special events)

  • 90 days
  • 18 weeks

The course is divided into

  •  6 days of  2 weeks x 6: extended investigations
  •  5 days of  1 week  x 1: midterm reflection
  • 14 days of 4 weeks x 1: major project
  •  5 days of  1 week  x 1: final reflection
    • which leaves roughly 30 days throughout the semester for:
  • 'Focused topic', 'individual investigation' and 'hard problem' lessons
  • Test-taking strategies
  • Tests
  • ... other activities

 

I'd like my classroom to be an exciting and free-flowing place: I will allocate significant time for independent work by students. (Benchmark evaluations will encourage and help them track their performance). I'd like the class experience to be varied but not chaotic. While respecting the vicissitudes of an unpredictable school schedule, given a MTWΘF schedule I would devote MWF to extended investigations and TΘ to independent work, focused topics and hard problems. (Obviously this balance may change as I put more of my curriculum in place).

 


 

Rough outline

  • Ready Reckoner lesson: students build benchmark measure of their universe.
  • Rich Data Sets lab: students use statistical data and tools from GapMinder.org, Data360, ManyEyes, Baseball-Reference, the US Government and the like to solve messy problems (collect data and record results, organize the data, make scatterplots, fit the curves to the appropriate parent function, interpret the results, and proceed to model, predict, and make decisions and critical judgments.) In doing so they will meet several members of the elementary-functions family, and use them to model, predict and discuss causality and correlation. The lesson will focus on why we fit functions to data, how to critically evaluate a fit, and what this process can tell us about our world.
  • Begin individual investigations that involve modeling a real-world question with linear and quadratic functions and solving for values. Once students' individual investigations have highlighted the general importance of solving for values,
    • Connect the general problem of 'solve equation' to the simpler problem of 'find roots.'
    • Begin to investigate various methods for root-finding.
  • Connect simple symbolic transformations (x -> -x, f(x) -> f-1(x), etc.) to transformations in graph (reflection, inverse, etc).
  • ...
  • (I need to draw on existing resources and prepare more materials to flesh out this curriculum)
  • ...
  • Minimum Wage extended investigation.

 

 

 


 

Standards for Work

 

Writing

A major focus of the Modern Physics lab course I helped design was to have students produce clear, insightful scientific papers describing their investigations. Writing accounted for 25% of their grade, and through evaluation and instruction we saw dramatic improvements over the course of the semester in students' ability to present and organize material.

 


 

A few students will go on to careers in Mathematics, Engineering, and Science, and those students should

  • Receive a solid preparation for their future mathematics training.
  • Be consistently challenged to gain a deep understanding of the foundations of the ideas they’re learning

Many students will not.

  • Mathematics as exposition and argument
  • Use mathematics to organize and sharpen thinking
  • Proficiently estimate unknown quantities; carry out order-of-magnitude and rough-guess calculations; identify quantities that require precise quantification; use resources at hand to do so; derive confidence of a working quantity from such factors as the inherent uncertainty of an input quantity and the perceived reputation of the source for an input quantity.
  • Create and Interpret graphics that are interesting and informative
  • Use computers and the internet to acquire, analyze, and manage large and interesting datasets in support of their position
  •  

Students should appreciate the beauty and elegance of mathematical ideas, and be exposed to the central mysteries of mathematics

Students should know that

  • Mathematics is not complete.
  • Mathematics is a conversation (and sometimes an argument): not a doctrine.
  • The topics of that conversation center around interesting, important, and accessible questions

Starts with making a subject

  • Real
  • Important
  • Interesting

Something is relevant if student can see their current or future self drawing on process to accomplish a natural goal.

  • Context of a professional or personal task
  • Use modern, professional, task-appropriate tools

Lead students in constructing their understanding of content

  • Discovery
  • Responsibility to teach selves, each other

Mathematics as social enterprise

  • Communicate results of exploration