Delarna nedan är direkt tagna från de vetenskapliga artiklarna i studien. Det som är fetmarkerat är vi själva som gjort för att få en tydlig överblick över vilka centrala delar samt kritiska aspekter som finns i tidig algebra.
Lärandeobjekt
The teacher then transitions to “meta-equations” to represent the schema and teaches step-by-step strategies that begin with identifying problem statements as combine, compare, or change schema and then building the propositional text structure. It also provides children with strategies that reduce demands on working memory and reasoning. They were told to use the strategies they learned from their classroom teachers. Linkages with pre-algebraic This occurred in two ways. First, both CAL and WP intervention explicitly focused on understanding the equal sign as a relational symbol. Second, as discussed, WP intervention taught children to represent the underlying structure of schemas in terms of “meta-equations” (Fuchs et al., 2014:12).
This “gap” leads to specific errors on a variety of problems assessing knowledge of math equivalence. For example, children often provide an operational definition of the equal sign – inferring that it means “get the answer” or “the total”. (Fyfe et al., 2018:6)
Regardless of the strategy used to solve the equations (algebraic or arithmetic), students who interpreted the equal sign relationally were more likely than students who did not to solve the equations correctly (Byrd et al., 2015:61f).
Kontrast
Third, word-problem solving reflects understanding of relationships between known and unknown quantities and may therefore be supported by pre-algebraic thinking (Fuchs et al., 2016:4).
Instead, they focus on comparing physical attrib- utes of objects. These comparisons and relationships between attributes are described with relational statements, using letters.
Only gradually are numbers introduced into the curriculum, without ever setting aside the main focus on comparisons and relationships (Brizuela & Schliemann, 2004:34).
Researchers have generally agreed on the important distinction between two understandings of equality: the ‘operational’; and the ‘relational’. Students have to develop an understanding of equivalence, as compared to equality, when using the sign (Pepin et al., 2014:40).
Separation
A well-developed conception of the equal sign applicable to elementary and middle school children is characterized by relational understanding: realizing that the equal sign symbolizes the sameness of the expressions or quantities represented by each side of an equation. There is general agreement that relational understanding of the equal sign supports greater algebraic competence, including equation-solving skills and algebraic reasoning. Because algebra is an important gateway not only into higher mathematics, but also into higher education more generally, the importance of building highquality relational understanding of the equal sign is of critical importance (Matthew et al., 2012:318).
The first stage of learning is arithmetic competence. We defined pre-algebraic knowledge as understanding of the equal sign foundational skills for the word-problem content (Fuchs et al., 2014:3).
First, students at second grade are typically unfamiliar with the multiplication sign (×), which could be confused with x (Powell & Fuchs, 2014:8).
Generalisation
This is an equal sign, and we use this in other situations as well when we calculate, and we say that it should be just as much on the left side as on the right side of the equal sign (writes ‘equal sign’ below the sign and underlines the wording twice). She also moved directly to the concept of equivalence and the scale (Pepin et al, 2014:47).
In the Nonsymbolic section, we assessed problem solving using nonsymbolic pictorial representations with accompanying story scenarios. Examiners read aloud three different scenarios from a testing script. The first scenario was about a farmer needing to put the same number of cows on either side of a fence. The second scenario featured a mother who needed to give her two children the same number of grapes. The final scenario
involved boys and girls playing a game where the same number of balls was needed on each side of a gym. Each scenario asked students to “make both sides the same” (i.e., the same number of cows on either side of the fence, the same number of grapes for each child, the same number of balls on each side of the gym) (Driver & Powell, 2015:130).
At the elementary level, solving equations with an unknown can be presented to students in several ways. Students may be asked to solve for unknowns through situations using manipulatives. For example, a teacher presents a student with four chips placed on a desk alongside a cup. The teacher asks how many chips are in the cup if there are nine chips total. For solving equations, students can also be presented with an equation (i.e., any number sentence with a relational symbol such as the equal sign [=]). The unknown in the equation may be marked with a line (e.g., 4 + = 9), a question mark (e.g., 4 + ? = 9), or a box (e.g., 4 + □ = 9). The unknown, especially in the later elementary grades, may also be notated with a letter (Powell & Fuchs, 2014:3).
Fusion
Throughout each tutoring lesson, RAs asked questions to individual students or the small group to checks for engagement and understanding. RAs set a timer at random intervals (as outlined in each lesson´s guide) and awarded a check if all students in the group were following directions and working hard when the timer beeped. At the end of each lesson, students earned individual checks for correctly answering “bonus problems”. Students did not know wich problems until the end of each lesson. This encouraged students to do their best mathematics work throughout the lesson (Powell et al., 2014:454).
In comprehending text, for example, results suggest that direct and indirect effects of foundational components of reading skill (decoding and word- recognition skill), higherorder cognitive resources (nonverbal reasoning and oral language ability), and lower-order cognitive resources (phonological processing, rapid automatized naming ability) (Fuchs et al.,2012:14).
In which the pupils said that, for much of their time in mathematics lessons they work on exercises from the textbooks. In fact, this was one of the reasons why many pupils disliked mathematics (“there is too little variation in maths”). However, evidence from this study shows that, in practice, the teachers found it hard to differentiate and provide exercises so that every pupil could access the mathematics. The pupils came to the lessons with different mathematics backgrounds (depending what they had been taught in
previous years). In the Stimulated Recall interview, one teacher described her pedagogic practices and explained her efforts “to keep the whole class together” and at the same time attend to the mathematical needs of individual pupils. It’s not always that easy to explain to students who haven’t been taught equations before and who might not be that interested in mathematics and always requires the practical side of it… you’ll get a lot of different answers (Pepin et al., 2014:43f).