Students Obstacles in Logical Thinking and Mathematics

Abstract

The first thing that comes to 13-year old student’s mind when we talk about mathematics would probably be the never-ending signs and symbols and x and y and z! If you give him a simple problem to solve, he would be the first person to get out of the place! Mathematics is a subject that requires logical reasoning abilities. Students have a very wrong attitude towards mathematics. This is something to be regretted. Who is to be blamed? This paper analyses the current situation and identifies the various causes due to which the students are literally afraid of mathematics. These causes are both psychological and genetic. “Didactic” refers to the different teaching methods adopted by the teachers or instructors. One of the major reasons as to why students hate math would be because of their instructors. Some children detest math solely because they hate the teachers. Hence, this paper provides certain remedies that can be implemented to bring about a change in the attitudes that students have towards mathematics. This article also sheds some light on how to teach mathematics differently. These remedial measures also include certain ways and means that will help the students transform algebraic expressions, simplification of complex algebraic terms and the like. All said and done, the need of the hour is change and this paper aims at making a big one at that!

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Introduction

There is a famous joke about math teachers. A little boy goes home and complains to his father, “Dad, my math teacher does not know math.” The father asks, “Why do you think so?” The boy replies, “Because she does all the easy problems in class and assigns the difficult ones as homework!” Today, it is very hard to find students who actually like the teachers. Students also do not seem to be very happy with the teaching methods adopted by their teachers. They especially hate the ‘x, y, and z’ which are always unknown! Algebra is actually a very interesting subject. It can be taught beautifully. But students hate algebra. The reasons may vary, but mostly, it is because of the teachers who handle the subject. Didactic mathematics refers to the teaching or instructing methods adopted the teachers. The teaching methods that are being followed today were introduced so many years ago. It was suitable for students from the 1980s and before. But the times are changing and it’s time for us to change, too. Children today do not deserve the same old teaching methods. We are in need of a change. A big one at that!

This article deals with the various causes or reasons that have been instrumental in shaping the attitude of students towards math. These reasons may be psychological or genetic. So, the need of the hour is a solution. How can we overcome these problems? What can bring about the expected change? Hence, this paper also sheds light on certain remedies that can help reshape the attitude that students have towards mathematics in general. The teachers should understand the psychology of the students and adopt teaching methods that suit them. The first remedial step would be to instill a positive attitude in the students toward mathematics. Then, the rest will follow. Therefore this paper aims at putting an end to the students’ regular math nightmares!

Causes

Mathematics is a subject which requires logical thinking and reasoning abilities. Majority of children who go to school have mathematics as the first item on their “Hate” list. There are so many misconceptions regarding math. Some believe that the ability to solve math problems solely lies in the genes. They think that if the father is good at it, so should be the son! It is true that heredity plays a significant role, but not entirely. There are so many reasons as to why people hate math, especially algebra. The reasons may be physical or psychological. Some of these causes are stated below.

Psychological Causes

Math Anxiety

Mathematics anxiety has been defined as feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations. Math anxiety can cause one to forget and lose one’s self-confidence (Tobias, S., 1993). Math anxiety is similar to stage fear. Lack of confidence is the main reason for this cause. This is due to unnecessary fear, for instance, fear whether he/she will do well, fear of the problem going wrong, and fear of not completing the problem on time and so on. This arises mainly due to poor teaching and poor experiences in math which gives a student a pessimistic attitude towards the subject. Rote memory also contributes to this issue. Sometimes, students tend to memorize the sums, without understanding the concepts and procedures. This practice of rote memory will not benefit them in the long run.

A strong foundation is essential for a student to be good at math. Not knowing the basics is a handicap for a student learning math in a higher level. This is caused due to lack of proper teaching and practice on the subject. A student becomes good at math only when he/she practices problems on a regular basis.

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The Attitude of Students Towards the Subject

A lot depends on the attitude that a student has towards mathematics. If a student has a negative attitude, it may be due to certain bitter incidents that might have taken place during early years. For example, having been beaten up for scoring low marks, being compared with other students or siblings who are better and so on can contribute to what we call bitter incidents. Parents and teachers play and important role here. Difficulties may also arise due to language barriers (i.e.) the inability to understand mathematical terms and meanings, and also other related words that have different meanings and expressing/verbalizing steps to solve problems.

A student’s attitude toward mathematics is not a one-dimensional construct. Just as there are different types of mathematics, there potentially are a variety of attitudes towards each type of mathematics (Leder, 1987).

The teaching and learning process has been characterized as lying on a linear continuum, with the extremes being imposition and negotiation. In their broad sociological studies, Goodlad (1983) and Stake and Easley (1978) documented that mathematics and science, as taught and learned at the elementary level, tends to hover near the imposition extreme. Evidence of this “camp” is these observations:

1. Teachers believe that elementary school mathematics is traditional arithmetic, which is comprised of basic skills and computational algorithms.
2. Teachers treat learning basic facts and skills as instructional goals “isolated” from conceptual meaning or context.
3. Teachers depend on textbooks as their curriculum guide.
4. Teachers tend to use direct instruction or demonstration, followed by paper-and pencil exercises to be done individually.
5. Teachers respond to student errors and misunderstandings by repeating their original instruction and practice routines.
6. Teachers view alternative techniques and ideas constructed by students as “undesirable behaviors to be eliminated.”

Following is another simple but significant cause for a student losing interest for the subject. A student may be good at math alone and bad at other subjects. When he is discouraged for his poor performance at other subjects and not praised for being good at math, the student may develop hatred towards math. This seemingly small reason is quite significant when it actually happens. This plays an important role in shaping the attitude of students towards math.

Lack of Encouragement

They say, “Parents are the first teachers and teachers are the second parents”. Lack of parental guidance and encouragement is one of the main reasons why a student loses interest in academics in general, let alone math. In a class room too, a teacher cannot expect all the students to possess similar standards or levels of understanding. A few students may grasp things quickly, while a few others may take some time to do the same. Problems occur when teachers do not understand this fact and fail to encourage the children who lose track of what is being taken in class. Also, when the teachers are partial to students who perform well, the rest of the children become jealous and develop hatred towards the subject. Also, it is not hard to find math teachers who have the strong opinion that a student who is bad at math is not fit to learn anything else! What they don’t realize is that small signs of encouragement like a pat on their back and a small word of appreciation will make the difference!

Lack of Interest and Love for the Subject

One should have passion and drive to the work they do. What people do not realize is that, this drive and passion, interest and joy could be shown towards learning a subject too. Unless a student has passion and love for the subject, he will not be able to give it his best shot. Even if teachers and parents guide well, a student should have the interest and passion. If the student does not show involvement, it would be difficult to teach him anything, let alone complex formulae and calculations. When the student is pressurized to learn things by his parents and teachers, he may develop a negative attitude towards the subject.

Genetic Causes

Dyscalculia

Dyscalculia is a broad term for severe difficulties in math. It includes all types of math problems ranging from inability to understand the meaning of numbers to inability to apply math principles to solve problems. Dyscalculia is one type of learning disability that can be served in special education programs. (Logsdon.A, n.d).

Dyscalculia is a lesser known disability than Dyslexia, but it can be detected from a very young age. Like Dyslexia, children suffering from Dyscalculia can be helped by taking a different approach to the way mathematics is taught to them.

Some of the symptoms of Dyscalculia can include confusion understanding the basics of mathematics, such as plus, minus, divide and multiply signs. Another of the more prominent signs of Dyscalculia is the inability to differentiate letters and numbers such as “5” and “S”, “0” and “O”.

People suffering from this disability can be affected many ways on a day to day basis. Tasks that seem simple to most people, like shopping and the ability to calculate expenditure, can pose a problem for those affected from Dyscalculia.

Mathematical skills are thought by many scientists to be implemented by the left supramarginal and angular gyri. Dyscalculia is thought to be associated with lesions on these parts of the brain located between the temporal and parietal lobes of the cerebral cortex. (Jenkins, n.d).

This problem maybe caused due to genetic disorder. A student will not understand the different languages used for the same term. For example, the student will understand when a person asks him to add two numbers but will get confused when they ask to find the product of two numbers. He may also get confused with math symbols and signs. In stead of ‘+’ he may write ‘-’ or for ‘%’ he may write ‘/’ and so on. Writing the numbers like 17, 12 etc. as 71 and 21 are some of the common defects in students suffering from dyscalculia.

A student who once suffered from dyscalculia said, “And a lot of the people I knew, they just thought I was being lazy and difficult at the same time. It made me feel dumb and like I was just not a part of the human world in a way.”( Stalworth.A, 2008).

Symptoms of Dyscalculia

The following symptoms of dyscalculia are extensive. A student may have all the symptoms or only a few of them. They are often misunderstood by teachers and parents and mislabeled as lazy’ or not trying’.

1. Misaligning numbers in columns.
2. Placing or reading numbers out of sequence.
3. Struggling with orientation of left and right.
4. Perform operations backward or out of order.
5. Confuse past and future event sequences.
6. Similar number confusion.
7. Trouble using calculators.
8. Struggle with time and direction concepts. (In keeping track of time they maybe chronically late or chronically early in an attempt to not show up late.)
9. Poor recollection of names.
10. Difficulties with name or face retrieval. ( Young, A.S. n.d).

Methods for diagnosing dyscalculia

Currently, there are three well-documented methods for diagnosing dyscalculia:

• Child’s attainment of age-appropriate mathematical skills
• Direct observation of dyscalculic tendencies
• The Dyscalculia Screener ( Michaelson.M.T. n.d)

Study: Brain Furrow May Cause Math Problem

Scientists have homed in on a brain region that leaves some people struggling with mathematics. Their research might point up better ways to teach numbers.

The study looked at people with dyscalculia – the mathematical equivalent of dyslexia. Up to 6% of children are thought to suffer from the condition; they toil with times tables and can find it tough to add small numbers even as adults.

Dyscalculics have abnormal pulses of activity in a brain furrow called the right intraparietal sulcus, find Nicolas Molko of INSERM, the French Institute of Health and Medical Research in Paris, and his colleagues. The fissure helps the mind to conjure spatial images.

It was also unusually shallow and short in the 14 women that Molko’s team scanned with functional magnetic resonance imaging. The women had a genetic condition called Turner’s syndrome, which is linked with dyscalculia.

The finding supports the idea that dyscalculics have difficulty conceiving arrangements of numbers, such as a line stretching from one to 100. “It goes very well with what has been found before,” says neuropsychologist Monica Rosselli of Florida Atlantic University in Boca Raton.

Molko hopes that brain imaging could eventually diagnose dyscalculics better than today’s cognitive tests. The finding might also inform educational schemes that encourage affected children to use different strategies to number lines, say.

But dyscalculia is probably part of a wide spectrum of math learning difficulties. Some people may have trouble keeping track of tens and units columns, others in recalling rote-learnt sums. “It’s unlikely that one brain area can explain all of the problems,” says developmental psychologist David Geary of the University of Missouri, Columbia (Pearson.H. 2003).

Psychological Effects of Dyscaluclia

Students born with genetic disorders have to face a world which is tough and brutal as they are not equipped with all the natural gifts that others are born with. They have to thrive to keep pace with those already born with gifts to do everything that a normal human being is capable of. Such is the case also with students suffering from “dyscalculia”. People possessing this disability don’t have the very basic skills to do math. For example they find it difficult to differentiate between the words “sum” and “product” and they tend to misplace the order of numbers like, they write 12 for 21. This disability has a natural insecurity that comes along with it. Such students are mocked by the other kids their age which naturally makes them feel low and incompetent, they tend to grow up to be individuals who are looked upon as social outcasts and considered “not normal” by the less humane society. They need to be given special attention without being treated and as one of a different kind. There are special schools for children suffering from this disability where they are taught to do math in a way that is simple and more comprehensive to them. With such care and encouragement, we can empower them to be normal and competent individuals just like everybody else.

Remedial Measures

What is the solution for these issues? How can we overcome these problems? What can we do to make the children love doing math? The following are certain remedies that can be adopted to bring about a change in the attitude of students towards mathematics. That in itself is the first step- trying to change their attitude!

As an ice-breaking activity, the fear that students have on any subject and algebra in particular should be removed and then learning becomes easier. All do agree to the fact that the visual impressions get deeper in the mind than just the verbal communications. Also, this point should be more accepted in the teaching process practiced for the students. Hence, methodology that would incorporate such use of visual content in algebra is welcome.

Play and Teach

A giant- sized book, around 300 pages long, with never-ending formulae and bugging calculations can put an adult off the edge, let alone 13-15 year old children! It is important to understand the psychology of the children and adopt teaching methods that suit them. The teachers and others who are responsible for framing the syllabus and adopting teaching methods must understand that only when the students enjoy doing what they do, will there be any effective results. Only when there is a good deal of involvement and interest, will there be any value to what the students study. Hence the students must be given an opportunity to enjoy themselves while learning. And one must understand what “enjoyment” is for kids. It is not just playing cricket or dressing up Barbie dolls. The teachers can adopt teaching methods that agrees with the psychology of students.

The following article appeared in The Hindu. On 20th October, 2008, HeyMath! conducted a ‘Maths trail’ at Dakshinchitra for a group of 60 students from Queenstown Secondary School (Singapore), Vidya Mandir, Chennai and Lady Andal Secondary School, Chennai. Dakshinchitra is a center for the living traditions of art, folk performing arts, craft and architecture of India with an emphasis on the traditions of South India. The trail consisted of a set of Math problems hidden in different parts of the miniature houses. The students, divided into groups and armed with maps, explored the place, examined the architecture and solved Math problems related to the artifacts that they saw. (“The Hindu”, 2008).

Thus in this case, the students do what they love to do and learn math in the process. It need not necessarily be a Math trail! There are so many other different ways by which the students can learn while playing. In fact, any learning process should be done that way. When the students actually do things by themselves, they will remember them better. The results which are derived this way would definitely be better than the results secured when the children are dumped inside the four walls of the classrooms!

Using Computer Graphics and Animations

One should remember that we are dealing with children here. Kids will always be pleased when something is presented visually than verbally. It’s their nature to understand things better when we teach them things with pictures and other visual representations. As they say, “Seeing is believing” and this includes mathematics, too! Schools can arrange for something called the audio visual room, where the students will be able to learn math formulae and calculations accompanied with pictures and sound. The technology has developed so much that people are dealing with robots, but here in schools we still have students struggling with heavy books and bags. It’s time to accept change. Students can work with CD ROMs and pen drives. Carrying them is not a big deal. They are as light as the feather. Thus the physical strain of carrying heavy bags is also removed in the process. With animated characters helping them solve math problems, the students could learn things faster and more effectively. One main drawback of classroom learning is that the students do not remember what they learnt in class after probably a week. With all the tension and anxiety, math humor is greatly needed. Young children enjoy cartoons and jokes. Cartoons may be used to introduce a concept or for class discussion. Most children will master mathematical concepts and skills more readily if they are presented first in concrete, pictorial and symbolic format. For example manipulatives are concrete objects used to teach a concept. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for young learners to understand the abstractions they represent. Students enjoy the change from lecture and books and they are more inclined to explore with manipulatives and show greater interest in classwork. (Phillips. M.C, n.d) But one should not overload the presentations with animations. Otherwise, all that the children will remember is the cartoon characters!

Create an Interst in Students

As stated earlier, a student develops hatred for a subject (especially math) due to various factors. One factor which is quite common is the inefficiency of the teachers. In most cases, the teachers who teach math do not love the subject. They do not possess the interest or involvement. They are not committed to their jobs. In that case, whatever they teach will not reach the students effectively. For example, when a teacher teaches a formula, he/she must love the formula. The teacher’s admiration and the love for it must be evident. It must be shown in their faces. This kind of enthusiasm is always contagious! So, at least 80% of the students will travel along with their teachers in this journey of teaching and learning math. The problem is that this does not happen most of the time.

• Either the teacher is not fully qualified to teach whatever he/she is supposed to.
• The teacher is qualified but does not know how to get things across to the students.

Hence, when the teachers themselves are not involved in teaching, it becomes evident that they are not passionate about math. When the students understand this fact, their level of interest comes down, too. When that happens, automatically, the students will start developing a hatred for math.

Law of Averages

The teachers should also be aware of something called the law of averages. The law of averages is nothing but the principle holding that probability will influence all occurrences in the long term. In a class room, however, the teacher should be aware that around 80% of the students are under his/her control. If the teacher is good, the students will perform well, and if the teacher is bad, the performance will reflect that. The remaining 20% of the students are beyond the teacher’s grasp. At one extreme, there is a set of students that would perform extremely well, even despite the teachers incompetence. At the other extreme is the other set of students that would perform very badly, even if the teachers are brilliant in teaching. For instance, in a class of 40, according to the law of averages, around 4 students would be extremely brilliant regardless of the teachers’ inefficiency. Around 4 other students would perform very badly despite the teachers’ efforts to make them learn well. The remaining 32 students would dance to the teachers’ tune. They are under the teachers’ grasp. Hence the teachers must be made aware of this principle to secure better results.

To do this, the teachers must spend some time and use certain methods to try and find how many people fall under the 80% category and how many under the other two extreme 10% categories. Once they have identified this, they can adopt different teaching methods to suit the needs of the students who come under the 80% category. When this is done properly, the results secured just can’t be better!

Students, who always love to play and very aggressively attracted towards games can be effectively taught through games. The methodology of integrating algebra and games could help. If not, moving a step ahead, placing aside the integration process, one can develop a methodology in similar with abacus that would help the students play and learn the subject.

The books of algebra can also be designed well so that, more pictorial representations are used to explain the logic behind every solving process as it is done in any management books. CDs with the e-book format and the use of audio and video enhancement that students can make use of it to learn can also be considered.

Here are some strategies to help students develop their use of mental imagery in problem solving.

Have students draw pictures to represent what is going on in a word problem. Students may draw actual objects from the problem (e.g., 3 shirts, a 6’ by 12’ garden plot, etc.), or they may represent objects with check marks or dots.

Engage students’ imaginations by proposing a number sentence, e.g., 6 +4 or 5(12 X 5), and having them come up with a story problem for that number sentence.

Incorporate problem solving activities using maps, diagrams, graphs, and tables to strengthen students’ use of visual/spatial materials. For example, have students calculate the distances of trips taken by students in the class, then display this information in a graph or table format.

Involve students in making predictions in situations where visualization can aid problem solving. For example, ‘If I place three green marbles and one red marble in a bag then pull one out, what color marble am I most likely to get?”

Help students practice manipulating images in their minds in order to solve a problem. For example, provide students with a variety of shapes made from connected squares, some of which can be folded to form an open box. Ask students to find the shapes which will make an open box. Students will need to visualize the anticipated results in order to solve the problem. Many may need to develop their ability to visualize by making cut-out models and actually doing the folding. (Adapted from Brumbaugh, Ashe, Ashe & Rock, 1997).

Establish that students have the necessary background skills to move ahead to formal instruction in areas of higher math. For example, students who have not mastered factoring from algebra I will have great difficulty simplifying rational expressions in algebra II.

Utilize computer software programs to help students explore areas of higher math. Programs exist for all levels and areas. Incorporate tutorial programs that are interactive and dynamic.

Set up a ‘math mentor’ for the student. This person may be a mathematics teacher, or a professional in the community who uses math in his/her work, e.g., a surveyor, an architect, a research scientist, an accountant, etc.

Use real life problem solving to help students connect concepts in higher math. For example, when students are exploring the question of how a spacecraft stays in orbit around the earth they will use formulas for gravity, geometric concepts, proportion formulas, etc. (Learning Base, n.d)

Conclusion

Parents should first accept children for what they are. Though many parents are broad minded these days, their eyes become blind and a sense of unacceptability arises when it comes to their children’s studies, especially when the child is poor in it. They are more ambitious in what the child has to do and what the child has to be in the future and never consider their interests. When a child is not good at a subject, especially in math, they send him to tuitions, pressurize him to study more, but never think of the root cause for the child’s weakness in the subject. They play an important role in helping the child to cope up with his math difficulties, for this, they have to accept the child’s weakness in the subject at the first place. A lot depends on how a child is introduced to math in his primary school. Patience is a quality that characterizes a good teacher. A teacher should be patient and tolerant in his teaching. He should adopt various methods of teaching to capture the student’s interest.

Two children with math difficulties are not the same, it is important to find what specific strengths and weaknesses a particular student has and then take the necessary measures to improve their IQ in math. A practical study for math (i.e.) didactic mathematics is a useful tool for improving a student as doing something is more interesting than listening to the teacher in class. Various didactical games can be adopted for teaching math. For example, catch the fly, number cop, etc. Experiment evidently showed that usage of didactical games improves the interests and attitude of the students and motivate them during the classes. One thing that the educators should realize is that anything is possible with love. Though didactic math will help children improve in the subject, a boost and support will increase their confidence and encourage them to do well in the subject.

References

Jenkins,S.(n.d): Learning disabilities: What is dyscalculia, Web.

Pearson.H ( 2003): Brain Furrow May Cause Maths Problem: Dyscalculia appears to cloud number images, Web.

Tobias, S. (1993). Overcoming math anxiety. (Electronic Version) New York: W. W. Norton & Company.

Goodlad, J. A Place Called School: Prospects for the Future. New York: McGraw-Hill, 1983.

Phillips. M.C (n.d): The Causes and Prevention of Math Anxiety, Web.

Hey! Math (2008): The Hindu: Young World, p. 8.

Young, A.S(n.d): Learning disabilities: What is dyscalculia?, Web.

Michaelson.M.T. (n.d): An overview of dyscalculia: methods for ascertaining and accommodating dyscalculic children in the classroom, Web.

Stalworth.A( 2008): Dyscalculia, Web.

Logsdon.A, (n.d): Dyscalculia – What is Dyscalculia, Web.

Stake, R. and Easley, J. Case Studies in Science Education. Urbana (IL): University of Illinois CIRCE, 1978.

Leder, G. “Attitudes Towards Mathematics.” In T. Romberg and D. Stewart (eds.) The Monitoring of School Mathematics. Madison (WI): Wisconsin Center for Educational Research, 1987.

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