Geometry Chapter 41

Geometry Chapter 4 July 28, 2008 : 

Geometry Chapter 4 July 28, 2008

Agenda : 

Agenda Brain Research Follow-up from Math & Science Center Questions / Exit Cards Big Idea: Visualization Small Group Activities Closing Discussion--Geometry

The Brain on Mathematics : 

The Brain on Mathematics from How the Brain Learns Mathematics by David Sousa

Number Sense – We’re born with it : 

We recognize the number of objects in a small collection – this is called subitizing (latin for “sudden”) Humans also possess two arithmetic capabilities: - The ability to count - The ability to use and manipulate symbols that represent numeric quantities Number Sense – We’re born with it

Beginning to Count : 

By 30 months, children have seen someone counting and understand that counting is an abstract procedure that applies to all kinds of visual and auditory objects By 3 years, children know there are separate words to describe the quantity of something. Beginning to Count

one – two – three – four. . . : 

Counting is a complex process using a one-to-one principle. It involves saying number words in the correct sequence. Cardinal principle – recognizing the last number in the counting sequence tells the total number of objects in the collection. one – two – three – four. . .

Language and Counting : 

Chinese English ten one eleven ten two twelve two ten one twenty one Recite numbers in 0.25 sec 0.33 sec Number of words 11 28 needed to count to 100 Language and Counting

Mental Number Line : 

When we compare two numbers, the speed we are able to do it depends not only on the distance between the two numbers, but also on their size. It’s easier to recognize that 74 is larger than 37 than to decide that 74 is larger than 73. It takes much longer to decide that 59 is larger than 58 than to decide that 8 is larger than 7. Mental Number Line

Implications for students : 

The speed and accuracy with which we carry out calculations decreases as the numbers get larger. (does this make sense?) Our internal number line only deals with positive integers (no negative numbers, or irrational numbers and some fractions in our ancestral environment) We have to construct the mental models that provide their understanding. Implications for students

Are Number Symbols different from Number Words? : 

Number Symbols are hardwired in the left parietal lobe (the region of the motor cortex that controls the fingers). Language number words are stored in Broca’s area, in the left frontal lobe, where our language vocabulary is processed. Are Number Symbols different from Number Words?

Slide 11: 

The human brain comprehends numerals as a quantity, not as words. Unconsciously and automatically, numerical symbols are converted almost instantly to an internal quantity. The conversion orients numbers in space, small ones to the left and large ones to the right.

Quantities to Words to Symbols : 

Quantities to Words to Symbols

Can we teach number sense? : 

YES! Just as phonemic awareness is a prerequisite to becoming a successful reader and can be learned, number sense is a prerequisite for succeeding in mathematics and can be learned. Can we teach number sense?

What can Teachers do? : 

Pair numbers with meaningful objects Use language to match numbers with objects and symbols Incorporate counting activities Provide experiences with number lines Plan meaningful estimation experiences What can Teachers do?

What can Teachers do? : 

What can Teachers do? Measure and then make measurement estimates. Use number charts Introduce materials that involve numbers or number representations Read literature involving numbers

Slide 16: 

Create magic number squares Manipulate different representations of the same quantity (0.25 = ¼ = 25%) Explore very large numbers and their representations Collect and chart data What can Teachers do?

Slide 17: 

Compare number representations in other cultures Set up spreadsheets Solve problems and consider the reasonableness of the solution What can Teachers do?

What can Teachers do? : 

Find everyday, functional uses of numbers Explore unusual numbers Model the enjoyment of numbers and number patterns What can Teachers do?

Is number sense the same as Logical/mathematical Intelligence? : 

Number sense can be considered the beginnings of mathematical intelligence Gardner’s Logical/mathematical intelligence also includes the ability to think logically, systematically and move easily from the concrete to the abstract. Is number sense the same as Logical/mathematical Intelligence?

Housekeeping : 

Housekeeping Follow-up from Math & Science Center Questions / Exit Cards

Visualization, Spatial Reasoning, and Geometric Modeling : 

Visualization, Spatial Reasoning, and Geometric Modeling “Research has found a strong correlation between spatial ability and problem-solving performance, suggesting that spatial visualization is a good predictor of successful problem solving” (Tillotson 1984; Geddes and Fortunato 1993)

Activity Rotation : 

Activity Rotation Julie Exploring Shapes with Tangrams Logos and Geometric Properties Carla Isometric Explorations Constructing Three-Dimensional Figures Ann Minimizing Perimeter Indirect Measurement

Slide 23: 

1. What makes shapes different? 2. How can I describe my location? 3. What are some different ways I can move a shape without changing it’s size? 4. When will I use geometry outside of this math class? 5. any + or we need to be aware of today

Assignment : 

Assignment Read Navigating through Measurement in Grades 6-8 Chapter 1 and Chapter 2. What are the “Big Ideas” in measurement in the middle grades? What are some possible “Essential Questions” that you could post for this unit? Final product.