People have remarkable abilities including flexibility of mind and creativity. Despite this fact, some are often afraid of number manipulation and mathematical problems in ordinary situations, which is caused by drawbacks of educational system and teaching approaches in particular. Teaching for learning and teaching for a test are two options to follow during the educational process. This paper focuses on the comparative aspect of these two approaches. The teaching for learning inspires a child to explore a concept and understand what he/she is learning. The teaching for a test measures the level of knowledge, not the curiosity and understanding. This approach is related to traditional ways of teaching, which do not foresee the innovative and creative aspects of mathematical learning.

The classroom culture and atmosphere define pupils’ interaction and achievements during the process of teaching and learning. Structured classroom deprives the students of a reward for cognitive processes of reflecting. Many factors influence the whole process of teaching. The classroom environment along with the speed of test taking and conception of one right answer negatively influence students’ aspiration for knowledge. Only if the students explore some ideas, ask questions, do not feel embarrassment for wrong answers, and do not have limitations for reflection, they will be eager to learn and consider multiple perspectives. The recognition of the sense instead of memorizing fosters the work environment and helps student’s memory store information for a long time. The computation fluency as well as the ability to calculate and count are the key concepts of any teaching process. The first one is the ability to produce correct answers in a short period. A student who possesses such skills can easily perform several number calculations.

The teaching for a test presupposes traditional classroom as well as standardized testing. The latter became widely spread in the United States. This method encourages the innovative ways of thinking and enhances pupils’ performance. The aim of the teaching for a test is to compare the results of pupils and decide whether they are ready for a college or professional scholarships. The results of students of the same age or depth of knowledge are compared and contrasted. The significant amount of time is devoted to preparation for the tests and practice. However, the effects of teaching for a test are two-sided. On the one hand, the students improve their performances. On the other hand, this teaching approach presupposes significant pressure placed on teachers. They have more responsibilities as they collect, organize, and analyze data, group the students, and supervise their assignments. Teachers have to praise the best children in the presence of those who have not done well in the test. These students face the decrease of confidence and incertitude. This approach narrows the curriculum since such subjects as art, music, and foreign languages are not included in the test. The children do not develop their creativity and imagination. They lack the skills important for future life including sociability and perseverance. What is more, teaching for a test does not foresee the rhetorical and composing dimensions. The pupils focus more on multiple choice or short answer’s procedures rather than development of comprehensible writing skills. In general, this approach does not cover and measure all skills to make the overall conclusion. The teaching for a test does not control the level of creativity, curiosity, and inspiration that are the most important aspects of students’ development. Lacking teacher’s support and good piece of advice, the students are concentrated on endless sets of tests to check their theoretical knowledge and not practical skills. In general, teaching for a test lacks individualization as pupils perform the same tasks during limited time.

In contrast, teaching for learning is characterized by sense-making classroom. There are certain differences between the traditional and sense-making classrooms. To start with, in the second one, mathematics is a way of thinking, not the collection of rules to remember without their understanding. Furthermore, in this type of classroom, students discover and explore explicable and apprehensible things. Moreover, students in sense-making classroom are always active and ready to analyze details. It is a student who provides right answers, not the teacher. This teaching method presupposes that a pupil is the owner of the truth. In sense-making classroom, students show innovative thinking and comprehension of significant and valuable information. The recognition of achievement and personal enrichment are the rewards for their brainwork. The diversity of teaching and learning techniques as well as greater involvement of pupils and teachers in the task contribute to the success of the mathematics lesson. The problem solving approach motivates students to search for the sense and accept only meaningful information. According to the information that was carried out by case brief writer, there is a difference between these two notions.

In conclusion, teaching for learning offers the students an exciting and meaningful world of algebra and geometry. This method develops critical thinking and problem solving skills. The students understand the hidden sense of procedures and tasks they perform. The teachers’ planning holds the interest and curiosity of the students. Consequently, they will be able to apply these skills in ordinary situations without fear or uncertainty. Teaching for learning provides the practical approach of learning boring formulas and performing mathematical operations. The rewarding experience along with emotional stability and psychological security will turn an ordinary student into confident, highly motivated, and responsible individual. He/she will be able to focus on the goals and succeed. On the contrary, teaching for a test does not take into account the individual approach of teaching. There is no test that can measure the abilities of all pupils in a proper way.