## Students’ errors and misconceptions

Very often students struggle with mathematics because they have certain misconceptions about the nature of some operations or geometrical objects. To a great extent, these errors can be attributed to poor instruction and the failure of teachers to identify the most common mistakes that students can make. Overall, educators should find ways of helping such learners make progress. Teachers should keep in mind that children can easily come to the wrong conclusions about mathematics, and become averse to this subject. The most important task is to avoid such pitfalls.

It is possible to provide several examples of such misconceptions. In particular, very often students misunderstand the nature of some basic operations such as division, multiplication, addition, or subtraction. Later, they encounter significant difficulties when it is necessary to work on different tasks. For instance, they cannot understand the concept of a denominator (Gonio & Nillas, 2002, p. 1).

In particular, they often forget about the lowest common denominator when adding or subtracting fractions (Gonio & Nillas, 2002, p. 1). They can also make mistakes when comparing two or more fractions. To a great extent, such errors can take its origins from the poor use of instructional methods. The students, who make such errors, often do not know what the term division means and how it can be applied in different situations. Apart from that, learners often cannot see the difference between different types of quadrangles. For instance, they may find it difficult to identify a distinction between rhombuses and squares.

In many cases, teachers believe that learners can easily comprehend the nature of every mathematical operation and apply this knowledge to other areas of mathematics. Furthermore, a misconception that emerged at an early stage of mathematical education can have profound implications for the later academic performance of a student. It should be kept in mind that these learners may struggle not only with mathematics but with other disciplines such as physics or chemistry. This is why this problem requires the close attention of educators.

The task of a teacher is to identify the most common misconceptions and develop remedial exercises that can help learners get rid of their errors (Bamberger, Oberdorf & Schultz-Ferrell, 2010). Provided that it is not done, students will find it very difficult to study more complex fields of mathematics such as combinatorics, probability theory, or calculus. This is one of the main risks that should be considered.

Sometimes, it is necessary to give students preliminary tests that prompt learners to display their skills and knowledge of mathematics. Such tests can be completed at the very beginning of the academic year. If there are some errors, teachers should discuss this problem with learners. In this way, teachers can dramatically improve the later academic performance of these students.

These cases indicate that misconceptions about mathematics can pose a significant challenge to both teachers and students. In particular, learners can come to the belief that mathematics is too complicated. As a result, they will be unwilling to study this subject in the future. In turn, teachers can believe that students are unable to cope with mathematical problems. As it has been said before, mathematical misconceptions can affect a student’s performance in other subjects. This is the main difficulty that educators should be aware of while designing their lessons.

## Responses to the guided questions

### Observation and description

The lesson that I observed took place in the fifth-grade classroom. 24 students were both males and females. It should be noted that there were 12 desks in the room. Moreover, the desks were placed near one another so that learners could work in groups. Overall, the students in the class were able to understand and apply mathematical notions, and I did not notice conceptual errors. There were some computational errors, for example at the time when they were doing long division. It should be noted that some of the students experienced difficulties when they had to work five-digit numbers. In my opinion, these mistakes can be attributed to inattentiveness, rather than a lack of understanding. I cannot say that learners had misconceptions about mathematical concepts.

### Analysis, exploration, and reasoning

During the lesson, I was able to observe a set of different techniques. At the very beginning, the teacher provided examples of division that could be relevant to the daily life of students. In particular, they were presented with a picture of a pie divided into six equal parts. This introduction was necessary since in this way the teacher made the topic of the lesson less abstract and more engaging. Later, the teacher proceeded to explain such an operation as long division and offered examples of this exercise. Moreover, learners were asked to speak about how such problems could be solved.

These activities can be regarded as an example of direct instruction. However, I can also point out that much attention was paid to such a technique as cooperative learning groups when learners were allowed to work jointly on different tasks. On the whole, the strategy chosen by the teacher was aimed at showing how division algorithms could be applied and developing the skills of students. The learners were supposed to see how the knowledge of division could be used for solving mathematical problems.

### Connection to other effective teaching practices

On the whole, the strategies observed during this lesson can be applied to other areas of arithmetic and mathematics in general. For example, such a strategy can be employed when it is necessary to teach such a concept as fractions and their addition or subtraction. Moreover, this approach can be beneficial when an educator intends to introduce such a topic as equations and ways of solving them. The strategies used by the teacher are based on the idea that learners should have more autonomy during the lesson. This is why learners were able to work in groups.

### Evaluation

On the whole, the instructional method chosen by the teacher enabled the students to gain a better idea of division and the tasks that involve this operation. While evaluating the effectiveness of the instructional methods, I focus on such a criterion as the type of errors that students committed. As has been said before, there are conceptual errors that imply that a learner does not understand the logic of mathematical operations.

In contrast, there are computational mistakes that occur in those cases when a student does not take into account a certain integer, fraction, and decimal mark. Certainly, such errors are also significant, but they do prevent a learner from understanding new mathematical topics. Students committed only rare computational mistakes, but not conceptual ones. Therefore, one can argue that the instructional methods were quite effective.

## Recommendations

It is possible to offer several suggestions that can enhance students learning. In particular, one can give assignments that can suit learners with more developed mathematical students. As far as I could observe some of the students were more willing to do more challenging tasks. This approach is beneficial because it can engage learners who are more interested in mathematics than others. Secondly, it might be possible to encourage students to do tasks that connect mathematical knowledge to other fields of science, especially physics.

For example, learners may be asked to compare the speed of a train with the speed of a rocket. In my view, this strategy is beneficial because learners should be able to see the connections between abstract operations such as divisions and their real-life applications (Bassarear, 2007, p. 196). This is the main rationale for the interdisciplinary approach. These are the main recommendations that can be made. It seems that they can make lessons much more engaging.

### Personal meaning

The observation of this lesson has been of great value to me. The strategies that I have observed will help me to increase students’ understanding of mathematical concepts. Moreover, I will be able to ensure that students can avoid both conceptual and computational errors. By relying on various examples, I will try to explain various mathematical topics and prompt students to apply this knowledge in different situations. Additionally, this observation has been of great value to me because I could see how a teacher should interact with learners. The techniques that I observed will be of great value to me.

## Reference List

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). *Math Misconceptions: **From Misunderstanding to Deep Understanding. *New York: Heinemann.

Bassarear, T. (2007). Mathematics for Elementary School Teachers. Boston: Cengage Learning.

Gonio, B., & Nillas, L. (2002). *Students’ Misconceptions in Basic Math Skills*. Bloonington: Illinois Wesleyan University.