The article highlights the major reasons for using the growing T pattern in teaching algebraic patterns. The growing T pattern enables the early graders to identify repetitive patterns easily, and the visual patterns give rise to an assortment of expressions. The article uses scaffolding questions to present the common issues that hinder algebraic problem-solving in early graders. The article highlights the basic structure of framing patterns to ensure students adopt an algebraic understanding.
The basic steps stipulated in the article include identifying patterns, describing the patterns, and expressing the patterns as algebraic symbols. The development of algebraic equations is guided by scaffolding questions and pattern exploration, which leads to a rich understanding of how algebraic equations are expressed. Teachers exploit these techniques to elicit algebraic comprehension and introduce symbolic algebra learning in middle school. The article stipulates how teachers can help develop students’ conceptual understanding of algebraic equations through questions.
Teachers first ask students to identify a pattern in a drawing presented in the classroom but should be keen on students’ misconceptions. Teachers should be keen since students are prone to imposing patterns on drawings where patterns are non-existent. Imposing non-existent patterns negatively affects problem-solving skills development as it can lead to mathematical errors. Pattern work helps develop rich algebraic thinking by enabling multiple expressions of similar patterns. This skill ensures students have the necessary knowledge to deal with symbolic manipulation and equivalent expressions.
The growing T pattern assists students to identify a long list of possible algebraic combinations, thus increasing their flexibility in moving from one configuration to the next. The growing T pattern can be used by teachers to help students decode the next pattern in a drawing. The student’s expression of the T pattern provides a different view of the pattern; thus, students understand the development of the algebraic equation. The patterns stimulate students to find less complicated techniques for performing algebraic calculations (Lee & Freiman, 2006). Students focus on the number of stars in each diagram in simple algebraic expressions. However, complex expressions require students to discover techniques for quick execution instead of the tiring work of adding stars to each diagram.
Students develop a conceptual understanding of algebraic expressions by eliciting different techniques to solve complex expressions. The article suggests changing the focus to the number of stars in each shape leads to the development of various techniques designed to solve similar equations. Teachers should discuss different approaches since some techniques may lead to a mathematical errors. Drawing T shapes and counting stars can lead to inappropriate mathematical reasoning, a significant source of errors in pattern drawing. Drawing T shapes offer an appropriate technique for dealing with this error since students can draw and count the stars, and teachers can point out the errors.
The next step is to encourage students to incorporate pattern perceptions focusing on the number of stars in the T shapes to answer complex algebraic questions. The switch from drawing T shapes to counting the number of stars in a specific T shape elicits different demands in students, stimulating problem-solving skills. The students conceptualize rules that help them remember the number of stars in a given T shape. The next step entails highlighting how many stars make up a specific figure. This step requires students to write symbolic writing to express themselves and communicate the techniques used. The verbal and written expressions may not conform to defined mathematical conventions, but it helps students present pattern perceptions and explain the meaning of each.
The next step entails choosing the right option from several try-and-error algebraic expression techniques. Students often omit this step when formulating their expressions but questioning the eligibility of each formula is a vital step in deciding the right option. This step allows teachers to introduce equivalent expressions since students will have to determine which expression is correct. In this step, students express flexibility in incorporating different pattern perceptions that can easily be expressed as symbols (Lee & Freiman, 2006). This step ensures students identify correct algebraic expressions seamlessly and, therefore, acts as the foundation of symbolic manipulation, which uses natural numbers.
Symbolic manipulation lays the foundation for solving relations and functions. Symbolic manipulation engages students to conceptualize how natural numbers can be interchanged to show that each expression is equivalent to a defined structure. Symbolic manipulation identifies a specific number capable of solving equations by finding the unknowns in algebraic equations. A natural number substitutes the unknown to solve the equation. The most vital step is deciding the natural number that can replace the unknown to balance an equation.
The principles and standards for School Mathematics outline essential components of a high-quality school mathematics program. The standards identified in the article include the relationship between pattern exploration and algebraic thinking. The standards explore how students develop meanings of operations by using algebraic reasoning as they investigate patterns and relations among a set of numbers. Algebraic expressions are best learned as a concept of mathematical thinking for formalizing patterns; thus, students are encouraged to use algebraic reasoning to study the evolution of numbers. The standards encourage students to apply problem-solving skills in various complex concepts and adapt these strategies to other mathematical contexts (Seeley, 2018). The standards ensure that students prove their mathematical understanding by noting algebraic patterns in diagrams.
In my lesson, the students will learn to recognize and understand the fundamental problem related to algebraic equations. The students will learn how to apply the gained knowledge in real-life situations. My lesson will conform to the mathematical standards to enable students to demonstrate an understanding of the lesson activity by representing the algebraic equation in multiple ways and completing the task fluently. In addition, my lesson plan will adhere to using patterns and structure to help students understand concepts and assess their reasoning for algebraic solutions. The classroom materials include algebra tiles, equation solving balance mat, equation solving ordering card, a blackboard, and chalk.
I will assess the student’s progress at each stage to ensure struggling students are identified for further consultation. I will start by leading the students to use the algebra tile to model solutions by describing the equation properties. When students are comfortable with the modeling, I will transition to writing the solution while naming the steps to justify their reasoning. Students will holp up their tiles to show how they completed each step. After viewing the student’s progress, I will give the students an equation-solving ordering card to emphasize that solving an equation entails completing each step.
I will write shorter sentences that do not compromise the academic content, use words that ELLs have used in the classroom, and state word problems in active voice. These strategies will accommodate the ELLs as they better comprehend the context and meaning when simple tenses are used (Conrado, Terri, & Margarita, 2011). In addition, I will repeat words to clarify the sentences and help the ELLs understand the concept without introducing new words. I will use illustrations and visuals since they are beneficial to the ELLs as they have fewer words to understand. An illustrated concept eliminates the language barrier since the ELLs students can understand the concept without the spoken word.
Conrado L. Gómez, Terri L. Kurz, & Margarita Jimenez-Silva. (2011). Your inner English teacher. Mathematics Teaching in the Middle School, 17(4), 238–243. doi.org/10.5951/mathteacmiddscho.17.4.0238
Lee, L., & Freiman, V. (2006). Developing algebraic thinking through pattern exploration. Mathematics Teaching in the Middle School, 11(9), 428–433.
Seeley, C. (2018). A journey in algebraic thinking – National Council of Teachers of Mathematics. Nctm.org.