What an algorithm IS

A set of steps that can be applied to a wide range of problemsA mechanical method for solving problems represented symbolically

What an algorithm is NOT

A means to help children understand mathematical concepts

A pack of Yummy Gum has 15 sticks. There are 20 packs in one box. There are 4 packs in another box. How many sticks of gum are there in all?

1520

+ 4

39

What an algorithm is NOT (often)

The most efficient way to perform a calculation

30,000– 1

36x 25

Worthwhile Mathematical Tasks

»Engage students’ intellect;

»Develop students’ understanding and skills;

»Stimulate students to make connections;

»Promote the development of students’ dispositions to do mathematics.

Why do we teach algorithms?

That’s what’s in the book.

What else is there?

What if my calculator batteries die?

Informal Problem Solving Strategies

Direct Modeling--Children model and act out the problem

Counting--Children store one quantity and then count on or count back

Derived Facts--Children use known relationships to figure out problems

Invented StrategiesExamples for Basic Addition Facts

8 + 9 = ?

Doubles

Make tens

Compensating

8+8 = 16 8+9 must be 17

8+2 = 10 10+7=17

Take 1 away from 8 and give it to 9. 7+10 = 17

Addition Strategies (Carpenter, et al., 1998)

38 + 26 = ?

Sequential

Combining Units Separately

Compensating

30 + 20 = 50 and 50 + 8 = 58

58 + 6 = 64

30 + 20 = 50 and 8 + 6 = 14

50 + 14 = 64

Take 2 from 26 and give it to 38. 40 + 24 = 64

Application in Problem Solving

Algorithms Invented Strategies

Students do not know which operation to use in problem solving. They can only apply the steps when the problem is set up for them.

Invented strategies arise from meaningful contexts; children can recognize other similar contexts to which their strategies apply.

Confidence

Algorithms

Confidence is undermined when students are unable to do these “easy steps.” Students are more dependent on the teacher

Students have a sense of ownership of their procedures. They are much less dependent on the teacher.

Invented Strategies

Place Value Understanding

Algorithms Invented strategies

Students do not need to understand place value to perform the steps correctly.

In reasoning through situations, students build their concept of place value. They refer to the actual value of numbers in their calculations.

Accuracy

Algorithms Invented Strategies

Wrong answers resulting from faulty application of the steps are often completely unreasonable.

Wrong answers are usually close to the correct answer.

Flexible Thinking

Algorithms Invented Strategies

Students rely extensively on the steps--even when doing mental computation. They have difficulty thinking flexibly about number.

Students are better able to understand and apply other strategies--even those they did not invent.

Number Sense

Algorithms Invented Strategies

Students will often justify a result by simply repeating the steps.

Students verify correctness by looking at the reasonableness of results within the given context.