**1. Introduction**

In mathematics education, focus is on the interactions among the three components of an instructional unit, the teacher, material, and students. In other words, the capacity to deliver quality instruction depends not only on the individual teacher's intellectual and personal resources but also on his or her interaction with specific groups of students and materials. According to [1], all curricula exist to provide the basis for effective instruction, that is, instruction that maximizes learning. Effective instruction is a result of proper and extensive planning. Planning starts with organizing material from the mathematics content. After deciding what material will be used, the next step is sequencing that material in the way students will experience it [1]. If instruction requires all three components (the teacher, students, and materials), then the capacity to produce worthwhile learning must also be a function of the interactions among these three components. Students bring experience, prior knowledge, and habit of mind, and these influence how they apprehend, interpret, and respond to materials and teachers. Teacher's awareness of students' "capabilities, needs, and past experiences" and the ability to use this information to "create a learning situation in which students can meet their needs or solve a problem in an autonomous and independent way" is therefore important.

Assessment is the process of gathering information so as to monitor students' prior knowledge and progress and make sound instructional decisions. As such, the primary purpose of assessment is to improve student learning of mathematics. Teachers examine the standards, assess where their students are in their knowledge base through some sort of pretest, and then plan their instruction based on the data collected (diagnostic assessment). During the process of teaching and learning in the classroom, teachers do assessment for learning (formative assessment) and assessment of learning (summative assessment). They then compare assessment for learning and assessment of learning in order to determine whether or not the implemented learning activity in class should be used again (or modified) [20]. A test score as feedback that measures whether a student has attained the expected standard cannot serve as formative assessment. Teachers need students' background information in order to modify teaching and learning activities to improve their learning. Therefore, feedback that involves a focus on the detailed content of what is being learnt has a central function of formative assessment [4]. Formative assessment or assessment for learning involves a continuous way of checks and balances in the teaching and learning process. It can be done at the beginning of instruction to tap prior knowledge in order to connect concepts when motivating for upcoming new concepts. The method allows teachers to check their students' progress and shortcomings as well as the effectiveness of their own practice, thus allowing for self-assessment.

The functional role of formative assessment (assessment for teaching) is often compromised in light of growing demand for external accountability related to performance and learning outcomes. Accountability pressures put many (mathematics) teachers between striking a balance between teaching mathematics facts and calculation procedures and also developing a conceptual understanding of mathematics. Due to accountability pressures, teachers have a tendency to focus on the preparation for examinations, where they opt to provide students with the necessary skills by working out problems similar to those that have occurred in past examination papers. This approach has dismally failed because student performance in mathematics remains depressed. Mindful of that, it is deplorable that the state of affairs concerning the functional role of formative assessment (assessment for teaching) is often overlooked. During the process of teaching and learning, teachers should assess the impact of their teaching on their students with the intention to create optimal learning spaces that meet the learning needs of each student. Therefore, teachers are discouraged from thinking of assessment as pencil and paper and embrace alternative forms of assessment in the teaching and learning of mathematics [2]. They should try to check on the performance of each student by giving class daily written exercises and mark the exercise books before the next day lesson. Also they should and always carry out weekly informal tests. Carrying out formative assessments in the form of informal tests, written classwork, or homework provides continual snapshots of students' progress throughout the week, month, or school year. By using these formative assessments, teachers can target students' specific problem areas derived from qualitative feedback (rather than scores), adapt instruction, and intervene earlier rather than later. The qualitative feedback about students and their abilities are likely to improve teachers' mathematics knowledge in teaching (which is demonstrated in the class by how well a teacher uses mathematical and pedagogical knowledge to help students learn mathematics) [5]. As teachers are guided by the qualitative feedback from the formative assessment, the critical component that must be present in any intervention is an opportunity for the students to discover the joy of creating knowledge from their own experience of the subject matter. Hence the activities that the teacher creates should be studentcentered. Besides, when teachers' classroom assessments become an integral part of

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*Formative Assessment in Mathematics Education in the Twenty-First Century*

the instructional process and a central ingredient in efforts to help students learn, the benefits of assessment for both teachers and students will be boundless [5].

**2. Formative assessment in mathematics classroom thrives on teaching** 

In the process of teaching and learning, there are myriad factors that impact on student learning. However, how well a teacher uses mathematical and pedagogical knowledge to help students learn mathematics is one of the factors that influence student success. As such, teachers should strive to expose students to teaching practices that stimulate critical thinking process whose salient features are conceptualizing, applying, analyzing, synthesizing, and evaluating information [3]. It is therefore important for teachers to provide enquiry or problem-solving approaches in mathematics classes. Infusing critical thinking skills into didactic activities requires teachers to consciously integrate new knowledge with already existing knowledge schema of mathematics content [3]. Students battle to recall and apply basic concepts of mathematics in the resolution of mathematical problems, and this leads to the lack of understanding of mathematics [18]. Concept mapping can be used by teachers to stimulate critical thinking in students because it represents and organizes knowledge, helps retention and recall of concepts learnt, and provides feedback on the understanding of the concepts learnt [4]. Therefore, thinking of assessment as a task that shows a student has acquired the concept and can link with other related concepts becomes paramount [3]. Accordingly, learning is meaningful when the student comprehends the relationship of what is being learned to other knowledge. As such, there is a need for teachers to incorporate the concept mapping in the formative assessment process as this will help them diagnose students' misconceptions [4]. If students can link new information to their existing conceptual framework, they can construct new, meaningful interconnections, so that their existing conceptions are transformed, enriched, or revised, and conceptual change occurs. This is achieved by carrying out formative assessment where students are asked to summarize at the end of instruction to allow them to make connections [1]. Therefore, existing conceptions are transformed during construction of understanding [5]. Interaction, collaboration, cooperation, dialog, and discourse are key concepts facilitated by formative assessment for the effectiveness of instructional activities. As such, collaborative group learning fosters meaningful learning and

**3. Assessment for learning versus assessment of learning in mathematics**

Assessment is generally broken down into three categories: assessment before instruction (pre-assessment), assessment during instruction (formative assessment), and assessment after instruction (summative assessment). It could be argued that pre-assessment is both assessments of and for (as) learning—that is, it assesses "prior knowledge" (as a pre-assessment) and that data is then used to revise planned instruction (making it formative assessment). Assessment of learning is used to determine what students have learned, while assessment for learning is used to determine what students are learning. It should be clear that assessment for (as) learning is a process of gathering information about students learning and provide qualitative feedback to support individual student learning and improve teaching practice in the classroom. However, there is a significant overlap between assessment of and for learning. Therefore, learning for assessment

*DOI: http://dx.doi.org/10.5772/intechopen.88996*

new knowledge construction [4].

**approaches that promote critical thinking**

*Theorizing STEM Education in the 21st Century*

self-assessment.

As such, the primary purpose of assessment is to improve student learning of mathematics. Teachers examine the standards, assess where their students are in their knowledge base through some sort of pretest, and then plan their instruction based on the data collected (diagnostic assessment). During the process of teaching and learning in the classroom, teachers do assessment for learning (formative assessment) and assessment of learning (summative assessment). They then compare assessment for learning and assessment of learning in order to determine whether or not the implemented learning activity in class should be used again (or modified) [20]. A test score as feedback that measures whether a student has attained the expected standard cannot serve as formative assessment. Teachers need students' background information in order to modify teaching and learning activities to improve their learning. Therefore, feedback that involves a focus on the detailed content of what is being learnt has a central function of formative assessment [4]. Formative assessment or assessment for learning involves a continuous way of checks and balances in the teaching and learning process. It can be done at the beginning of instruction to tap prior knowledge in order to connect concepts when motivating for upcoming new concepts. The method allows teachers to check their students' progress and shortcomings as well as the effectiveness of their own practice, thus allowing for

The functional role of formative assessment (assessment for teaching) is often compromised in light of growing demand for external accountability related to performance and learning outcomes. Accountability pressures put many (mathematics) teachers between striking a balance between teaching mathematics facts and calculation procedures and also developing a conceptual understanding of mathematics. Due to accountability pressures, teachers have a tendency to focus on the preparation for examinations, where they opt to provide students with the necessary skills by working out problems similar to those that have occurred in past examination papers. This approach has dismally failed because student performance in mathematics remains depressed. Mindful of that, it is deplorable that the state of affairs concerning the functional role of formative assessment (assessment for teaching) is often overlooked. During the process of teaching and learning, teachers should assess the impact of their teaching on their students with the intention to create optimal learning spaces that meet the learning needs of each student. Therefore, teachers are discouraged from thinking of assessment as pencil and paper and embrace alternative forms of assessment in the teaching and learning of mathematics [2]. They should try to check on the performance of each student by giving class daily written exercises and mark the exercise books before the next day lesson. Also they should and always carry out weekly informal tests. Carrying out formative assessments in the form of informal tests, written classwork, or homework provides continual snapshots of students' progress throughout the week, month, or school year. By using these formative assessments, teachers can target students' specific problem areas derived from qualitative feedback (rather than scores), adapt instruction, and intervene earlier rather than later. The qualitative feedback about students and their abilities are likely to improve teachers' mathematics knowledge in teaching (which is demonstrated in the class by how well a teacher uses mathematical and pedagogical knowledge to help students learn mathematics) [5]. As teachers are guided by the qualitative feedback from the formative assessment, the critical component that must be present in any intervention is an opportunity for the students to discover the joy of creating knowledge from their own experience of the subject matter. Hence the activities that the teacher creates should be studentcentered. Besides, when teachers' classroom assessments become an integral part of

**128**

the instructional process and a central ingredient in efforts to help students learn, the benefits of assessment for both teachers and students will be boundless [5].
