- using strategies to add or subtract single digits

Example: 3 + 4 = 7

**Strategies used to solve single digit arithmetic:**

- retrieve answers from memory

- counting on fingers from number one

- counting from the larger of the two addends (numbers)

Example: 3 + 4 = 7 ( 4, 5, 6, 7 )

- use previous knowledge to determine other related problems

Example: 4 + 5 = 9

- child would think: 4 + 4 = 8; so: 4 + 5 must equal 9

- decomposition of a difficult problem into two easier ones

Example: 8 + 7 = ?

- child thinks: "7 + 7 = 14, so if I add 1, I get 15." So they
know that 7 + 8 = 15

**How strategies develop**

- as children gain experience, the strategies they use change

- there is an increase in use of retrieval

- after years of adding and subtracting, most children can retrieve answers to most of the basic
arithmetic

- during the same time period,
children begin to solve problems faster and more accurately because as they gain experience, they develop the ability to choose
the quickest strategy for each arithmetic problem

**Choosing a strategy**

- when faced with a problem, children must choose a retrieved answer or a back-up strategy,
which are strategies used when retrieval is unsuccessful

- young children tend to use retrieval for easier mathematic problems and use counting for harder
ones

**Complex Arithmetic **(Siegler &
Alibali, 2005)

- It is very important
that children understand concepts that underlie math procedures. If students memorize procedures without understanding they
will have great problems with more complex math.

**Fractions**

- many children have diffucilities
in fraction arithmetic

- they tend to add the two
numerators and the two denominator

Ex: 1/2 + 1/3 = 2/5

- this problem happens because
children do not think of the amount represent by each fraction.

- children also have problems
when dealing with decimal fractions as well

- fourth and fifth graders
would say that the larger number is the one with more digits to thr right of the decimal point.

Ex:
they would say 3.258 is larger than 3.26

- this is because children
know that when dealing with whole numbers the number with more digits is bigger so they apply that rule to decimal numbers.

**Research Study on Fractions**

- The researcher in the study by
Rittle-Johnson and Koedinger (2005) assessed prior knowledge of grade 6 students on adding and subtracting
fractions. They implemented an intervention to scaffold three kinds of knowledge, then tested achievement after the intervention.

*The
Intervention:*

Use computer
software to scaffold 3 areas of knowledge for problem solving:

Contextual
– real world, story contexts

Conceptual
– learned content, relative size of numbers, add facts

Procedural
– subcomponents of a correct procedure, step-by-step actions

*Scaffolds:*

Contextual
– candy bar story

Conceptual
– fraction bars to model value of fraction

Procedural
– provided the common denominator for each problem

*The
Findings*

Students’
test scores for adding fractions improved after intervention

Students
used conventional procedure more accurately after the intervention

Students
did not need the scaffolds for the same lengths of time

Researchers
learned more from error analysis than from test scores

**Algebra**

- a branch of mathematics
in which mathematical relations are explored by using letters or symbols to represent numbers

- children often
have many difficulties learning algebra because they cannot simply generalize their knowledge of arithmetic to algebra

- it is also difficult
because in algebra children must learn to read symbolic expressions

- because algebra
involves problem solving, children have to learn the rules