             Single Digit Arithmetic (Siegler & Alibali, 2005)    - 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    