MATH first page

Workshop Handouts

The following are examples of some of the handouts used during our training workshops.

We also used some graphics: the "Cone of Learning" and a couple of grids.

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DIFFERENT WAYS OF "SEEING"  

Basic Mathematic Operations

ADDITION: Most students have little difficulty visualizing a problem that involves addition, however it is necessary for them to be aware that those things they are adding (combining) are alike in some way.  It is also most important for the student to realize that the things that are being added make up parts of a whole.

SUBTRACTION: There are four ways of looking at a subtraction problem.

• 1. Take away:  If John has 10 pens and he gives 8 to Jenny how many does he have left?
• 2. Comparison: If John has 10 pens and Jenny has 8 pens, John has how many more pens than Jenny. (Comparison (er) words such as, taller, shorter, longer more than, less than are examples of "key" words.)
• 3. Missing Addend:  If John has 8 pens, how many more does he need to have as many as Jenny who has 10 pens?
• 4. The Difference Between: (A gap exists)  It is important to recognize that the difference between two quantities depends on the direction or starting point- where you are looking or starting from.What is the difference between 8 and 3?   What is the difference between 3 and 8?   Are they the same?

MULTIPLICATION: There are three ways of looking at a multiplication problem.

• 1. Repetitive addition:   If John saves 10 each week, how much money will he have saved in 4 weeks?
• 2. Array/Area:   If John has 4 rows of tomato plants with 10 plants in each row, how many plants does he have altogether?
• 3. Cartesian/Sets   If Jenny has 4 shirts and 3 skirts, how many different combinations of shirts and skirts does she have?

DIVISION: There are three ways of looking at division.

• 1. Repetitive subtraction:   If you have 40 pens, how many students will get pens if each student is to get 5 pens?
• 2. Partitioning:  If you have 40 pens and 8 students, how many pens will each student get?
• 3. Array:  If there are 40 trees in a orchard and there are 5 rows of trees, how many trees are there in each row?

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IMPACT OF TECHNOLOGY

ON THE

MATHEMATICS CURRICULUM

*

Some mathematics becomes more

important because technology

requires it.

*

Some mathematics becomes less

important because technology

replaces it

*

Some mathematics becomes possible

because technology

allows it

AMTNJ's NJ Calculator Handbook

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"Appropriate calculators should be a

available to all students at all times."

A calculator should be used when it:

• facilitates problem solving

• creates problematic situations

• supports concept development

• focuses attention on meaning

• relieves tedious computation

• facilitates a search for patterns

• promotes number sense

• encourages creativity & exploration

• removes anxiety

• provides motivation & confidence

Curriculum, and Evaluation Standards for School Mathematics  National Council of Teachers of Mathematics, 1989

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Knowing Mathematics is Doing Mathematics

Doing Mathematics

             Verbs from the Standards

                                 explore                                                            construct

                             justify or varify                                                discuss

                             represent                                                         use

                             solve                                                               describe

                             explain                                                             develop

                             investigate                                                        predict

                             discover                                                           formulate

Are your students doing mathematics?

Curriculum, and Evaluation Standards for School Mathematics  National Council of Teachers of Mathematics, 1989

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HELPING ADULTS THINK CRITICALLY

Stephen Brookfield (1987), while cautioning against the standardized, as opposed to contextual, approach to helping adults become critical thinkers, offers sseveral rules of thumb for effective facilitation of critical thinking.  These include:

• Affirming self-worth.  This can bre a difficult balancing act of respecting integrity while encouraging challenging inquiry.

• Listening attentively.  The purpose is to identify relevant past experiences and to help frame critical questions.

• Reflecting (mirroring) ideas and actions.  The purpose of reflecting is to extend self-awareness and raise assumtions to a conscious level.

• Motivating critical thinking.  This encourages critical thought and actions while assisting in the articulation of both advantages and risks: such motivation requires sensitivity and caution.

• Evaluating progress on a regular basis.  Brookfield notes that "stock-takingand reflective evaluation are a central part of critically formed action" (p.79)

• Helping create networks.  Most adults need resource networks and peer support as indicated by the success of single-parent and other support groups.

• Being critical teachers.  Critical teaching has been identified (Shor, 1993; Freire, 1971) as "assisting people to become aware of their taken-for-granted ideas about the world" (P. 80).  Risk-taking in both teachers and learners is at the heart of critical teaaching.

• Making students aware of their own learning processes. Metacognition- thinking about ones thinking processes.

Improving Thinking Skills, Pelavin Associates, Inc., Wash. D.C.; Office of Vocational and Adult Education, U.S. Dept. of Ed.

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