positive algebra from arithmetic to algebra jaap den hertog freudenthal instituut universiteit...
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Positive AlgebraFrom arithmetic to
algebra
Jaap den Hertog
Freudenthal Instituut
Universiteit Utrecht
“I used to be good at arithemetic, but now I don’t understand anything anymore.”
Counting in primary school grows into advanced and more sophisticated counting
You cannot maintain what you never learned
When do you use your calculator?
Continuous learning trajectories
To introduce negative numbers and to use them
Knowledge about fractions as a preparation to working with algebraic expressions
Rules, patterns, structures
27 – 38 = ….?
5 × -3 = -15 -1 × -3 = 3
4 × -3 = -12 -2 × -3 = 6
3 × -3 = -9 always 3 more
2 × -3 = -6
1 × -3 = -3
0 × -3 = 0
A pattern
What is the power of algebra?Reasoning and generalizing: is it always?
Are you sure? Is it certain?
Not only knowledge of (f.e. number system) but also knowledge about
Development of thinking models
A continous learning trajectoryDeveloping a fraction languageReasoned dividePerform operations within the contextTo relate ‘Part of’ to multiplicationTowards the development of routine
proceduresFractions on the number lineAnd what is next …?
Two thirds of 4500
2/3 times 4500
× 45002
3
A learning process and struggles
π/4; 1/4π; π ÷ 4; they are all the same, but differentAdd up the same number with the nominator and
the denonminatorYou divide a number and the result is larger. Why?Add up the nominators and the denominators. Is the
new fraction bigger or smaller than the sum of the fractions?
Is there a smallest fraction greater than zero?How is the number system extended?
15---
35---
25---
45---
A square of 1 bij1. Write the area of each piece as a fraction and add up.
When is formal arithmetic with letter fractions introduced?
For which students is it important?
In which grade do we start?
What are the preparations for the students?
Which formula is equivalent with…
2 3
1y
x x
3 1
2 53 5
2 15( 3)
2( 1)
x x
x
xx
x
Are there more examples?Is there a formula?
1 14 1 4 1
3 3
Simplify fractions
1
2
a
a
Reasoning with formulas
Adjust / prepare formulas yourselfDiscus the effect of changes in variables and /
or numbers
Recommended maximum heart rate
For years, the following formula was used:
Maximum heart rate = 220 – age
Who has a higher maximum heart rate, someone in your class or one of the teachers?
Recommended heart rate
Recently the formula has been changed
Maximum heart rate = 208 - (0.7 x age)
What are the consequences of using this formula: is your heart rate higher or lower than the recommended rate?
Summary
Continuous learning trajectories from Primary school and Secondary school
Introducing negative numbers in primary school, but the formal operations in secondary school
Fractions are not “ready” after the primary schoolFractions in secondary school Do not avoid fractions in secondary education, but
also include lettersLearning processes in developing and adapting
formulas