Basic Facts Understanding and Automaticity

John SanGiovanni

http://jsangiovanni.hcpss.wikispaces.net

Let’s Take a Test…

Traditionally, how have we taught (or learned) basic facts?

They don’t know their facts.

Drill on arithmetic facts does not necessarily lead to recall…….

Drill must be preceded by sound instruction.- Brownell and Chazal,

19351935

Has our approach to teaching basic facts met the needs of

all of our students?

Has our approach met the needs of ANY of our students?• Understanding• Number sense

Has it aligned with our ideas of good teaching?

Why hasn’t memorization worked?

Turn to a “shoulder buddy” and identify all of the skills

needed for this everyday task.

In math…

2 + 4

½ + ¾ 2y + 4y

0.2 + 0.462 + 34

Have we confused memorization with

AUTOMATICITY?

Processing of new information makes heavy use of working memory.

As skills are repeated, the brain recognizes the information and can process it more quickly and with less effort.

Automaticity reduces the load of working memory by as much as 90% (Schneider, 2003)

Computation in the “real world” is done mentally 84.6% of the time.

- Northcote and McIntosh 1999

With this in mind…

Solve the next in problem mentally.

49 + 27

Does memorization contribute to

understanding?

Does knowing your facts mean you

understand?

If our approach that uses memorization doesn’t help kids develop number sense

and/or computational fluency?

When do they develop it?

If we memorize 6 + 8…

Will it help with 56 + 38?

If I understand…

Multiplying by 5 is the same asMultiplying by ½ of 10 4 x 5 is the same as ½ of 4 x 10

I know 68 x 5

If I understand…

Multiplying by 9 is the same asMultiplying by 1 group less than 10

8 x 9 is the same as(8 x 10) – (8 x 1)

I know 37 x 9

New Tricks for an Old Dog orOld Tricks for a New Dog

• Foundational Understanding• Instruction in Context• Intentional Practice• Independent Practice• Assessment

•What +/-/x/÷ means..•Commutative property•Associative property *

Foundational Understanding

Picture Removed

FoundationalUnderstanding

Picture Removed

Picture Removed

Pictures Removed

Patterns andThe Beauty of Mathematics

Develop Understanding+1/+2 (counting on)

38 + 1

74 + 2

Develop Understanding+1/+2 (counting on)+0

198 + 0

0 + 56

Develop Understanding+1/+2 (counting on)+0+10

10 + 23

45 + 10

Develop Understanding+1/+2 (counting on)+0+10Doubles

33 + 33

45 + 45

Develop Understanding+1/+2 (counting on)+0+10DoublesMake ten

4 + 66

53 + 7

Develop Understanding+1/+2 (counting on)+0+10DoublesMake ten

+8/+9 (Using Ten)

38 + 7

59 + 4

Develop Understanding+1/+2 (counting on)+0+10DoublesMake ten

+8/+9 (Using Ten)Using Doubles

33 + 34

45 + 46

Develop Understanding+1/+2 (counting on)+0+10DoublesMake ten

+8/+9 (Using Ten)Using DoublesUsing Knowns

Develop Understandingx 2 (doubles)

12 x 2

210 x 2

Develop Understandingx 2 (doubles)x 10

12 x 10

Develop Understandingx 2 (doubles)x 10x 5 (1/2 of x10)

33 x 5

(33 x 10) ÷ 2

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1

14 x 1

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

x 3 (x2 + x1)

6 x 3 (12 + 6)

14 x 3 (28 + 14)

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

x 3 (x2 + x1)x 4 (x2)(x2)

7 x 4(7 x 2) x 2

23 x 4(23 x 2) x 2

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

x 3 (x2 + x1)x 4 (x2)(x2)x 6 (x3)(x2)

7 x 6(7 x 3) x 2

42 x 6(42 x 3) x 2

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

x 3 (x2 + x1)x 4 (x2)(x2)x 6 (x3)(x2)x 8 (x2)(x2)(x2)

8 x 5(5 x 2) x 2 x 2

31 x 8(31 x 2) x 2 x 2

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

x 3 (x2 + x1)x 4 (x2)(x2)x 6 (x3)(x2)x 8 (x2)(x2)(x2)x 9 (x10) – (x9)

6 x 9(6 x 10) – (6 x 1)

18 x 9(180) - 18

Develop Understandingx 2 (doubles)x 10 (double x5)x 5x 1x 0

x 3 (x2 + x1)x 4 (x2)(x2)x 6 (x3)(x2)x 8 (x2)(x2)(x2)x 9 (x10) – (x9) x 7 ---- just one

Good Mathematics Teaching

ContextProblem Solving

PracticeAssessmentPicture Removed

Draw 10 meatballs on one of your paper plates.

Draw 9 meatballs on your other paper plate.

Picture Removed

Intentional Practice

Picture Removed

Intentional Practice

1.Fold a piece of paper in half and then in half again.

2.Label your columns a number, double it, double it again, double it again.

3.Write some numbers in the first column and complete the other columns.

Intentional Practice

Engaging Practice: Games

Picture Removed

Engaging Practice: Games

Engaging Practice: Games

Match ‘Em Up

Assessment

What is our purpose?

Picture Removed

Traditional Tests

• Negative impact• They don’t teach

anything• Can reinforce

inefficient strategies• Avoid overuse Picture Removed

Picture Removed

Data from Observation

Picture Removed

Student Progress

Positive Attitudes

• Recognize progress• Provide support• Reasonable

expectationsPicture Removed

All students are able to master the basic facts-including children with learning disabilities. Children simply need to construct efficient MENTAL tools that will help them. - Van de Walle, 2005

All students are able to master the basic facts-including children with learning disabilities. Children simply need to construct efficient MENTAL tools that will help them. - Van de Walle, 2005

But we have to:• teach them in context with problems. • practice them in an intentional and engaging way.• assess them in a fair way that promotes growth.• believe that they can.