Algebraic thinking is an essential mathematical element involving paying attention to important pattern aspects, generalizing mathematical ideas and the patterns with individual’s experiences with number and computation, and using symbols to explore them. Additionally, algebraic thinking forms the basis for mathematical reasoning and allows students to use algebra as a language to generalize these patterns and ideas. Acknowledging teachers’ presence and adopting this generalization to students’ education enables learners to enhance mathematical reasoning. Moreover, algebraic thinking allows learners to understand mathematics beyond the procedural application of formulas and the result of specific calculations. This enables them to operate on any unknown quantity as if the quantity were known.
The first core element is algebraic thinking, as generalizing arithmetic involves arithmetic and generalizations in increasingly systematic, conventional symbol systems. It allows students to develop reasoning about number operations and properties and understand the underlying mathematical arithmetic structures by identifying arithmetic patterns.
There are three ways in which learners develop algebraic reasoning. Firstly, they can explore properties and relationships and consider arithmetic rules as general principles to assist them in determining arithmetic results patterns rather than focusing on the results of the specific calculations. This enhances students’ ability to contemplate the underlying generalization by analyzing arithmetic cases rather than memorizing the properties. Secondly, they can explore equality as a relationship between quantities to understand the meaning of the equal sign as a symbol denoting the relationship of equality rather than a signal for doing something. These enable them to develop algebraic reasoning in determining the relationship in algebraic expressions. Thirdly, they can use symbols, including letters as variables, to make mathematical generalizations to understand multiple variables in discerning manipulation of various variables.
The second core element is algebraic thinking, as functional thinking, and involves identifying change and exploring and recognizing the relationship between different quantities through pattern analysis.
There are various ways in which students can develop functional thinking. Firstly, they can develop it using generalizing patterns by using function machines to select input numbers to generate output, connect visual representations with symbolic representations, and develop generalizations from repeating sets and growing patterns. Patterns allow students to engage with mathematical quantities, and it helps them elicit multiple solution strategies. Additionally, it enables them to reflect on mathematical structure and engage them in offering their assumption and prove their thoughts. Secondly, they can use inverse operations to select and use the appropriate strategies in solving mathematical problems with unknown quantities.