Development and use of Test Instruments to measure Algebraic Reasoning Based on Cognitive Systems in Marzano’s Taxonomy

<p style="text-align: justify;">Algebraic reasoning involves representation, generalization, formalization of patterns and order in all aspects of mathematics. Hence, the focus of algebraic reasoning is on patterns, functions, and the ability to analyze situations with the help of symbols. The purpose of this study was to develop a test instrument to measure students' algebraic reasoning abilities based on cognitive systems in Marzano's taxonomy. The cognitive system in Marzano's taxonomy consists of four levels, including retrieval, comprehension, analysis, and knowledge utilization. According to the stage of cognitive development, students are at the level of knowledge utilization. At this level, students can make decisions, solve problems, generates and test hypotheses, as well as carry out investigations that are in line with indicators of algebraic reasoning abilities. The stages in developing the test instrument were based on three phases: preliminary investigation phase, prototyping phase, and assessment phase. The study obtains a set of valid and reliable algebraic reasoning test instruments for students based on the cognitive system in Marzano's taxonomy. Through the development of an algebraic reasoning test instrument based on Marzano's taxonomy, students can build' thinking habits so that active learning exercises occurs.</p>

An empirically based practical learning progression for generalisation, an essential element of algebraic reasoning

Generalisation is a key feature of learning algebra, requiring all four proficiency strands of the Australian Curriculum: Mathematics (AC:M): Understanding, Fluency, Problem Solving and Reasoning. From a review of the literature, we propose a learning progression for algebraic generalisation consisting of five levels. Our learning progression is then elaborated and validated by reference to a large range of assessment tasks acquired from a previous project Reframing Mathematical Futures II (RMFII). In the RMFII project, Rasch modelling of the responses of over 5000 high school students (Years 7–10) to algebra tasks led to the development of a Learning Progression for Algebraic Reasoning (LPAR). Our learning progression in generalisation is more specific than the LPAR, more coherent regarding algebraic generalisation, and enabling teachers to locate students’ performances within the progression and to target their teaching. In addition, a selection of appropriate teaching resources and marking rubrics used in the RMFII project is provided for each level of the learning progression.

Generalization Process by Second Grade Students

This study is part of a broader study on algebraic reasoning in elementary education. The research objective of the present survey, namely to describe generalization among second grade (7- to 8-year-old) students, was pursued through semi-structured interviews with six children in connection with a contextualized generalization task involving the function y = x + 3. Particular attention was paid to the structures recognized and the type of generalization expressed by these students as they reasoned. In all six, we observed three phases of inductive reasoning: (a) abductive, (b) inductive and (c) generalization. The students correctly recognized the structure at least once during the interview and expressed generalization in three ways.

Engaging Algebra Early through Manipulatives: reappraising Cuisinaire-Gattegno Rods

The Cuisenaire-Gattegno (Cui) approach uses color coded rods of unit increment lengths embedded in a systematic curriculum designed to guide learners as young as age five from exploration of ratio through to formal algebraic writing. As the rods have had greater adoption as a teaching aide than the curriculum, we set out to investigate how fidelity to the seminal curriculum and pedagogy impacts learning via a meta-analysis and novel study of preparation for future learning. This meta-analysis of 23 studies (n=1968) revealed advantages of Cui over traditional arithmetic approaches (effect size = 0.55). Curriculum fidelity significantly predicted efficacy. Higher fidelity implementations were associated with large effects and lower fidelity resulted in small or null effects. To test how this curriculum prepares students for future learning, we carried out an 18-month longitudinal school-comparison study (n=114) executed to a similar fidelity level as the study with the largest treatment effect. Cui treatment accelerated learning rates measured during the school-year after treatment, and demonstrated transfer to novel tests of algebraic reasoning (effect size = 1.0). Tests of scholastic aptitude replicated aptitude by treatment interaction for both arithmetic and algebraic reasoning. While Cui provided significant learning benefits for children with higher aptitude, these benefits were significantly enhanced for children with lower aptitude. Together, these findings support the benefits of this approach, and further substantiate the importance of embedding these teaching aides within the theory-grounded curricula that gave rise to them.

Penalaran Aljabar Siswa Smp Dalam Menyelesaikan Soal Pola Bilangan

Abstrak — Penalaran aljabar merupakan proses berpikir logis untuk mencari dan mengenali pola dari suatu situasi tertentu kemudian membuat kesimpulan berupa generalisasi atas ide-ide matematika terkait situasi tersebut. Penelitian kualitatif ini bertujuan untuk mendeskripsikan penalaran aljabar siswa SMP dalam menyelesaikan soal pola bilangan. Subjek dalam penelitian ini adalah dua siswa kelas VIII dengan kriteria mampu menyelesaikan soal tes penalaran aljabar dan memiliki variasi jawaban yang berbeda. Kedua subjek memiliki karakteristik yaitu subjek pertama menyelesaikan soal dengan satu ide untuk menemukan aturan umum pola bilangan sedangkan subjek kedua menemukan gagasan baru untuk menemukan aturan umum pola bilangan.Teknik pengumpulan data pada penelitian ini yaitu menggunakan tes penalaran aljabar dan wawancara. Data yang terkumpul dianalisis berdasarkan indikator penalaran aljabar meliputi mencari pola, mengenali pola, dan generalisasi. Hasil penelitian menunjukkan siswa pada mencari pola diawali dengan mengidentifikasi unsur penyusun pola bilangan kemudian siswa menemukan hubungan tiap unsur pada pola bilangan. Pada tahap mengenali pola, siswa menyadari hubungan antar suku-suku ganjil dan suku-suku genap pada pola bilangan, kemudian melakukan percobaan-percobaan untuk menemukan rumus umum pola bilangan agar memudahkan mencari nilai tiap suku. Siswa menggunakan tabel dan melakukan pendataan tiap suku pada pola bilangan untuk menemukan rumus umum pola bilangan. Ada perbedaan cara untuk menemukan rumus umum, yaitu ada siswa yang menggunakan selisih antar suku, sedangkan siswa yang lain menggunakan selisih antara suku ganjil dengan ganjil dan suku genap dengan genap. Pada langkah akhir penalaran aljabar yaitu generalisasi siswa menarik kesimpulan dari proses yang dilakukannya di tahap sebelumnya.Kata Kunci: penalaran, penalaran aljabar, pola bilangan. Abstract — Algebraic reasoning is a process of logical thinking to search and recognize patterns of a particular situation and then make conclusions in the form of generalizations of mathematical ideas related to the situation. This qualitative research aims to describe the algebraic reasoning of junior high school students in solving numerical pattern problems. The subjects in this study were two eighth grade students with the criteria of being able to solve algebraic reasoning test questions and having different variations of answers. Both subjects have the characteristics of the first subject solving the problem with one idea to find general rules of number patterns while the second subject finds new ideas to find general rules of number patterns. Data collection techniques in this study are using algebraic reasoning tests and interviews. The collected data is analyzed based on algebraic reasoning indicators including finding patterns, recognizing patterns, and generalizing. The results showed students in looking for patterns beginning with identifying the constituent elements of number patterns then students find the relationship of each element in number patterns. In the stage of recognizing patterns, students realize the relationship between odd terms and even terms in number patterns, then conduct experiments to find general formulas for number patterns to make it easier to find the value of each term. Students use tables and collect data on each term in a number pattern to find a general formula for a number pattern. There are different ways to find a general formula, namely, there are students who use differences between tribes, while other students use the difference between odd and even terms and even terms with even numbers. In the final step of algebraic reasoning, the generalization of students draws conclusions from the processes they did in the previous stage.Keywords: reasoning, algebraic reasoning, number patterns

Types of Generalization Made by Pupils Aged 12–13 and by Their Future Mathematics Teachers

This paper seeks to establish what kind of arguments pupils (aged 12–13) use and how they make their assumptions and generalizations. Our research also explored the same phenomenon in the case of graduate mathematics teachers studying for their masters’ degrees in our faculty at that time. The main focus was on algebraic reasoning, in particular pattern exploring and expressing regularities in numbers. In this paper, we introduce the necessary concepts and notations used in the study, briefly characterize the theoretical levels of cognitive development and terms from the Theory of Didactical Situations. We set out to answer three research questions. To collect the research data, we worked with a group of 32 pupils aged 12–13 and 19 university students (all prospective mathematics teachers in the first year of their master’s). We assigned them two flexible tasks to and asked them to explain their findings/formulas. Besides that, we collected additional (supportive) data using a short questionnaire. The supporting data concerned their opinions on the tasks and the explanations. The results and limited scope of the research indicated what should be changed in preparing future mathematics teachers. These changes could positively influence the pupils’ strategies of solving not only flexible tasks but also their ability to generalize.