Like a good essay, a good lesson has a beginning a middle and an end. The even better essay or lesson, makes the point then revisit it and extends it in multiple formats and though different learning styles.

• Play, Warmup, & Intro
• Activity or Lesson w/Debriefings: Do Then Discuss
• Extended Summary & Extension Component

      This engagement process begins with play, the intro, and warmup. Making a warmup part of the class routine is a plus since it begins the class even before the bell has rung and because the completion of a one task actually stimulates the start of work on another task. Making it a task that enhances the lesson or presents a small challenge is also a goal.

      In an age in which it is inappopriate to welcome students with "Happy Hanukkah" or Happy St. Patrick's Day, Halloween, Valentine's Day or anything else that "might offend," an age in which experts say it is important to immediately write a behavioral objective on the board, this author suggests the opposite.

      Make appropriate greetings, make your students welcome, and challenge their minds and hands even before class begins. Yes, at some point discuss and write the objectives on the board. TO ENGAGE THE MIND AND HANDS IS MORE IMPORTANT than to inform the mind and hands initially the name of their task.

• Assign the lesson objective, in task form, before an algorithm is introduced by the instructor.
      This introduces the possibilities of discovery and independent thought.
• Extend the previous lesson by completing all work mentally. Manipulatives are designed to become obsolete. Mental work facilitates that.
• Pose a computational problem or problems that might slow the day's lesson.
      This saves time in the long run and provides a variety of lengths of time for students to work on one task.
• Set a task that reviews a topic needed to complete the day's work. Even the best tv series now begin with a quick summary of the last episode.
• Review "old" or "high-states test" material so that the routine is followed daily. Never waste a moment. Kill two birds with one stone.
• Have students create and answer a problem to be used on the next review or test.
      Creating is the BEST way to have students claim ownership of material.
• Exchange and grade last night's homework.
• Pose a problem not all students will be able to complete. Perhaps use material that all will master in the future.
• Provide a set of completed problems including the work and the answer. Have student supply the verbs and / or objectives.
• Provide a set of completed problems but include one or more with the WRONG WORK and the WRONG ANSWER. Have student FIND THE ERROR AND CORRECT THE PROBLEM. Make some of the errors, easy some hard.
      This works particularly well with work completed even earlier than the previous lesson -- old, but important, stuff.
• Each week assign a student in each row to show any student in their row the work they missed yesterday when they were absent.
      The teaching student speaks math (an abstract language) and the absent student hears math spoken by someone other than the teacher. It also provides practice prior to in-front-of-class presentations.
• Quiz the students quickly and routinely through a warmup.

      Most teachers have the presentation of a lesson mastered. This author suggests the frequent inclusion of periods of reflection, summary, and debriefing IN THE BODY OF THE LESSON as well as its conclusion.

      As stated earlier in "Manipulatives As A Language:     Introduce in the concrete (manipulative). Reinforce and review in the abstract (pictures, written and spoken words and symbols). Remember, a good story is worth retelling. Tell it in as many languages or formats as possible.

      Think VISUALIZE, VERBALIZE, DEBRIEF even during the body of the lesson.

      Introduce with Term Tiles. Reinforce and review with spoken words and symbols. Then, continue.

Extended Summary & Extension Component

Think in terms of the higher order skills of Bloom's Taxonomy.

EVALUATE - judge

SYNTHESIZE - produce

ANALYZE - identify errors or differentiate facts from assumptions

APPLY - use in a new setting

COMPREHEND - understand

KNOW - remember, acquire

      See A Journey with Self-Assessment as a Compass," © 1995, 1998, 2001, A. Azzolino, at www.mathnstuff.com/papers/ai/invent.htm, for a rather complete and sorted list of math verbs. An assortment of higher-order verbs and corresponding tasks is suggested below.

EVALUATE - judge

argue

Argue for and against writing the coefficient 1 in the term 1x.

compare

Compare the expression 2x + 4 to the expression 2x - 4.

decide

Decide which is greater x + 3 or x + 4 or "you can't tell." Decide which is greater 3x or 4x or "you can't tell." In each case, explain your reasoning.

decide

Decide which is greater x + 3 or x + 4 or "you can't tell." Decide which is greater 3x or 4x or "you can't tell." In each case, explain your reasoning to another student.

discuss

Discuss the expressions 2x and x². Use vocabulary like coefficient, base, exponent, degree. Perhaps evaluate the expressions for some values of x.

interpret

Interpret the meaning of the symbolic phrase 3x - 2.

SYNTHESIZE - produce

create

Create a poster to explain how to simplify an expression.

create

Create a Power Point presentation on how to factor a binomial.

design

Design a rectangular area of 24 square units that has the smallest perimeter.

extend

Extend what you know about combining polynomials like x + 3 and x - 4 to polynomials like x² + x +3 and 3x² + x - 4 and then complete this computation.

illustrate

Illustrate with tiles 2 examples of what is meant by "the square of the sum of …" and explain each to a student.

translate

Translate "a little smaller than 5" into algebraic symbols.

translate

Translate "5 - x" into words without using the words subtract, decrease, minus.

write

Write a linear equation with no solution.

      Some on the assignments above are short enough to be completed in class. Many require more thought or organization and are not suitable for in class work or assessment.

      Out of class writing is a valuable learning tool. Having students write, make up, problems, quizzes, their own tests and cheat sheets are assignments in which the process is as valuable, if not more valuable, than the product.

      Term Tiles & Tokens Spread Sheets encourage and support such work.

      Right now, even before the logistics of Term Tiles & Tokens use is discussed, see the resources provided for CREATION and INTERPRETATION assignments.

• Make Experts Using Creation and Interpretation Activities.