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Dividing by Zero

Why Students have Difficulty with Long Division



Dividing by Zero
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Often, when trying to teach children long division, I and other teachers find that many cannot even begin to understand how to divide because they have not mastered their multiplication tables.  They cover the paper with numbers trying to add up and figure, erase and get lost in all the scribbles.  When prompted and asked how many times the divisor will go into the dividend, the teacher is frustrated when trying to show them how to find the most logical number by multiplying it by the divisor to see if it is too big or small. They have no idea of what could be the quotient because they do not know what the two numbers times each other are. 

In a 1998 study of an intervention by Smith and Rivera which consisted of eight mid school students with learning disabilities who had already been taught long division, it was noted that they had not mastered multiplication.  The intervention was done over a period of about a week to bring them up from a 0% assessment in long division to adequate performance. But because they had not mastered multiplication, they had to stop and regroup and work on multiplication while learning long division, which took longer and prevented their being shown other work on problem solving.

Gordon Aubrecht in Helping Students Come to Grips with the Meaning of Division   refers to Arnold Arons (1976, 96) a proponent of confronting student misunderstandings of both mathematics and science:  "Science teachers are confronted with students in upper level classes of chemistry and physics who have hard-fixed misconceptions and do not understand the basics behind multiplication and division.  They will often search through formulae to find one that fits the problem. They have therefore no understanding of science, and the less mathematically inclined have no basis except memorization for their understanding of science…."  "The incomprehension these science students exhibit of the basic mathematical operations of multiplication and division needs to be addressed in physics classes."   Arons argues that many science students have no idea why they divide, which sabotages their understanding of physical principles.

Teachers have to communicate how to do mathematical operations to students so that they understand.  John Pais in his article Communicating Mathematics Using Heterogeneous Language and Reasoning refers to Constructivist Piaget who said "intelligence organizes the world by organizing itself."  The Constructivist approach requires that each learner actively constructs their own internal concepts into their mathematical schema. According to radical constructionist von Glaserfeld: "To put it simply, to 'understand' what someone has said or written implies no less but also no more than to have built up a conceptual structure from an exchange of language, and, in the given context, this structure is deemed to be compatible with what the speaker appears to have had in mind."  Thus the internal accommodation the learner makes will be accurate mathematically if he or she comprehends what the speaker is saying.     

Most students who are just beginning to learn division are approximately in the 4th or 5th grades, around 9 and 10 years of age.  But there are so many disparities that if the class is not homogenous-multicultural, special needs, average and gifted all together, it can be a real challenge for the teacher who cannot spend enough individual time with the students who require it.  There are many reasons students require extra help over what the teacher can explain in the classroom.  Language differences for new immigrants, transfers in from schools in different locales where there may be less emphasis on math, the inclusion of special needs students, and the gifted.  The gifted are often neglected because there is the assumption they can figure it out for themselves.  Not necessarily true, since giftedness is not always spread across the subject spectrum and one might write great essays but not catch onto math so quickly.

Tutoring by volunteers is one option if a teacher can have one assigned to her classroom. If there is time and space in the classroom, the teacher might be able to organize a group and have a volunteer come in certain days, or take them to the computer lab for practice. Another option is that while inclusion is the standard, there are still exceptions when children can be "pulled from a classroom" to be given special help by a special education instructional assistant.  Holding a parent-teacher conference would be a good option if the teacher can get the parents to cooperate by studying with the child or arranging for a part-time tutor.  There are very good technology techniques such as special mathematics programs where students can sit alone and do exercises at the computer. A group study technique might consist of organizing the children so that each group has a mixture of students, ones doing better on their test scores.  Worksheets could be distributed to each group and each child to work on each problem, and then compare their answers, agree and fill in the answer spaces.  After a certain time, the class would discuss the questions and answers.  Each cooperative learning group could give itself a name and be a team, competing to see who gets more right answers. The winning team members get to demonstrate how they solved the problem on the board.

Brian Boley in his How to Teach Mathematics to Students Having Difficulty (1999) says "Math is the only subject that shares two characteristics.  You must start at the beginning and build upon what you learned in the previous lesson."  There is no opinion where an answer is concerned; it is either right or wrong and a proof exists.  Constructivist theory stresses that knowledge is built on prior knowledge.  Thus, having a firm grasp of the basics is critical. It is important to drill the basic functions until they are 99% accurate or better.  Memorization and rote learning have lost favor in the modern public school system; however in certain mathematical learning processes they do have an important place, perhaps more so than in creative subjects. Poor process is the commonest cause of the student's not being able to do problems correctly, but fixing poor process is simplest: avoiding sloppiness, not taking shortcuts, checking the result each time. A lack of self-discipline is responsible for much inability to do arithmetic correctly. Do it the same way each time, be neat, do not take shortcuts, and check for reality and correctness.

Boley says to not teach multiple shortcuts to solve a type of problem even if it is simpler.  Students should do a problem the same way over and over until it is automatic and they remember it thoroughly. It is better to teach a slightly more complex method which will cover all problems of that type so that students will have fewer methods to remember. 

NCTM 8 (National Council of Teachers of Mathematics) says "proficient students look for both general methods and shortcuts..."  The learner should thoroughly understand the full process before starting to eliminate parts.  If the student is gifted then they can benefit from shortcuts but otherwise only a few simple ones should be shown to the class.  Each teacher has to adjust their style to the class.

Helping with Math.com gives these helpful suggestions for furthering students' learning: When teaching division to your child, introduce with idea of sharing, objects shared equally and divided. Discuss how division is separating sets while multiplication is its opposite, combining sets. Make sure child has grasp of the format and signs of division. Provide worksheets for guided practice.

Begin division problems using even numbers. After the child has grasped the basics of simple division, then you can introduce remainders. In the intervention study by Smith and Rivera the introduction of remainders in long division was a major stumbling block, after the lack of proficiency in multiplication.  The learner should get down the basics of long division by doing problems with even answers before they are given problems with remainders as the outcome because they do not as yet have the concept of fractions.

Long division or dividing multi-digit numbers has multiple methods: a combination of estimation, trial and error and multiplication. Knowing which the dividend is and which the divisor is must be fully understood.

Another method is the algorithmic which carries out the operation fully by putting in the 0 in front of the first number as in 425 / 25 = 017 r000.  This method carries out the operation fully, leaving nothing out which might help some people who can follow the exact process.  Others might find so many numbers confusing and do better without writing in all the zeros.

Aubrecht, writing about getting the idea across of what division is and when to use it in science and physics classes, teaches the concept of Whole and Package in operational determining of area.  "The implicit message of the package idea is that in division, there is the whole space which is being tiled and the number of tiles is being counted. Most understand that the relation between 'whole' and 'package' is similar to the tiling of an area by standard squares. The view of division as repeated subtraction has become less abstract."

The type of textbook used is sometimes overlooked. The layout of the book, style of writing, how easy to read the charts, examples and explanations are can make a difference. The teacher has to use whatever method he or she finds works best for their class.  Technological methods are available where teachers can show Power Point presentations and students can practice exercises on the computer.

For many reasons such as class size, its diversity, time schedule and more, not all methods work the same for all teachers.  This is where standardized testing can be of a help to the teacher to see how the students are doing and analyze and try to identify when the class seemed either to falter or pick up speed.  There are many aids to help with instilling the methods to do long division.  Each child is an active learner and, according to Constructivist theory, is going to form their unique accommodation to the knowledge which has been communicated.  It is the teacher's responsibility to ascertain where there are any missing or weak links in the knowledge chain and strengthen them.

References

Aubrecht, Gordon J. Helping Students Come to Grips With the Meaning of Division (2004). School Science and Mathematics. Vol.104 no7 pg.313-21 N.

Boley, Brian  (1999). How to teach mathematics to students having difficulty.  Retrieved  from University of Phoenix website.  http://www.oddparts.com/missions/math.htm.

Pais, John. (1998). Constructivism: Communicating mathematics using heterogeneous  language and reasoning. Retrieved from Drexel University (1994-2010) http://interactivmathvision.com/PaisPortfolio/CKMPerspective/Constructivism (1998).html   http://mathforum.org/library/topics/

Rivera, D., Smith, D.(1988). Journal of Learning Disabilities, v21 n2 p77-81 (EJ367459 Retrieved from University of Nevada Libraries Websitehttp://web.ebscohost.com.ezproxy.library.unlv.edu/ ehost/search?  vid=5&hid=109&sid=7 860a28e 03b1-42b0dc009d18b47a10%40sessionmgr114

Teaching division to your child  http://www.helpingwithmath.com/by_subject/division/div_teaching_division.html







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