Free Essay

Misconceptions of Algebra

In: Computers and Technology

Submitted By PhillyAggie
Words 797
Pages 4
Diagnostic Algebra Assessment Definitions

Equality Symbol Misconception Graphing Misconception Definition Concept of a Variable Misconception

Equality Symbol Misconception
As algebra teachers, we all know how frustrating it can be to teach a particular concept and to have a percentage of our students not get it. We try different approaches and activities but to no avail. These students just do not seem to grasp the concept. Often, we blame the students for not trying hard enough. Worse yet, others blame us for not teaching students well enough. Students often learn the equality symbol misconception when they begin learning mathematics. Rather than understanding that the equal sign indicates equivalence between the expressions on the left side and the right side of an equation, students interpret the equal sign as meaning “do something” or the sign before the answer. This problem is exacerbated by many adults solving problems in the following way: 5 × 4 + 3 = ? 5 × 4 = 20 + 3 = 23 Students may also have difficulty understanding statements like 7 = 3 + 4 or 5 = 5, since these do not involve a problem on the left and an answer on the right. Falkner presented the following problem to 6th grade classes: 8 + 4 = [] + 5 All 145 students gave the answer of 12 or 17. It can be assumed that students got 12 since 8 + 4 = 12. The 17 may be from those who continued the problem: 12 + 5 = 17. Students with this misconception may also have difficulty with the idea that adding or subtracting the same amount from both sides of an equation maintains equality. Kieran gives this example: Solve for x: 2x + 3 = 5 + x 2x + 3 – 3 = 5 + x 2x = 5 + x – x – 3 2x – x = 5 – 3 x = 2 The answer is correct, but several steps of the solution contain incorrect equations. In summary, a student with this misconception will: Assume the solution to a problem is the number after the equal sign. Treat the equal sign as a command to do something, thus fill in a blank or variable to the right of the equal sign with the completed operation from the left hand side of the equation. Not add, subtract, multiply, or divide equally from both sides of an equal sign. (back to top)

Graphing Misconception Definition
In algebra, students learn that a graph is a representation for a function. Students learn to translate between graphs, equations, and table of values. But just as the translation between equations and word problems is more difficult, students sometimes find interpreting the graph of a real world situation more difficult. Students may forget the algebraic relationships they have learned and resort to graphical misconceptions. The most common graphing misconceptions are treating a graph as a picture and slope-height confusion. An example of interpreting a graph as a picture might be a problem asking a student to draw a speed vs. time graph for a biker riding over a hill. Students with the misconception would draw the hill, and ignore that speed is asked for. Students do not look at the graph as showing speed as a function of time, but think of it more literally. An example of slope/height confusion might use the following graph:

The question could ask "Which internet company costs more per hour at 2 hours?" A student with the misconception would choose since it costs more at 2 hours. The student does not recognize that the problem asks for slope. (back to top)

Concept of a Variable Misconception
Transitioning from arithmetic to algebra can be a challenging learning experience for students. One of the key reasons is the use of letters to represent variables. Even after taking an algebra course, many students do not understand the use of letters in equations and therefore can not grasp the concept of a variable. Booth (1984) explains that letters can be interpreted as a specific known number, as multiple values instead of one, as an object, or simply ignored. An example of a typical mistake that students make when they do not understand the concept of a variable is when asked to "add 4 onto 3n," an answer of 3n4 or 7n is given. As another example of a concept of a variable misconception, when students are asked to write an equation for the following statement using the variables S and P: "There are six times as many students as professors at this university," (Clement, 1982), they interpret the variable P as meaning professor, instead of number of professors. (back to top) Return to the Diagnostic Algebra Assessment page

©2009 Boston College. All rights reserved. inTASC is affiliated with the Center for the Study of Testing, Evaluation and Educational Policy (CSTEEP) in the Lynch School of Education. Email us at

Similar Documents

Free Essay


...Title Author(s) Source Published by Working with logarithms: students' misconceptions and errors Chua Boon Liang and Eric Wood The Mathematics Educator, 8(2),53-70 Association of Mathematics Educators This document may be used for private study or research purpose only. This document or any part of it may not be duplicated and/or distributed without permission of the copyright owner. The Singapore Copyright Act applies to the use of this document. The Mathematics Educator 2005, Vol. 8. N0.2. 53-70 Working with Logarithms: Students' Misconceptions and Errors Chua Boon Liang and Eric Wood National Institute of Education, Nanyang Technological University Abstract: This study examines secondary school students' understandings and misconceptions when working with logarithms using a specially designed test instrument administered to 81 students in two Singapore schools. Questions were classified by cognitive level. The data were analysed to uncover the kinds of errors made and their possible causes. Students appear capable of doing routine calculations but less capable when answering questions which require higher levels of cognitive thinking. In addition, many errors are not due to lack of knowledge but appear to be based on over-generalisation of algebraic rules. Suggestions for practice based on these findings are provided. Introduction Anecdotal evidence from teachers and colleagues over the years has consistently confirmed that teaching logarithms in secondary......

Words: 7188 - Pages: 29

Free Essay

Lesson Plam

...Mathematics overview: Stage 7 Unit | Hours | Mastery indicators | Essential knowledge | Numbers and the number system | 9 | * Use positive integer powers and associated real roots * Apply the four operations with decimal numbers * Write a quantity as a fraction or percentage of another * Use multiplicative reasoning to interpret percentage change * Add, subtract, multiply and divide with fractions and mixed numbers * Check calculations using approximation, estimation or inverse operations * Simplify and manipulate expressions by collecting like terms * Simplify and manipulate expressions by multiplying a single term over a bracket * Substitute numbers into formulae * Solve linear equations in one unknown * Understand and use lines parallel to the axes, y = x and y = -x * Calculate surface area of cubes and cuboids * Understand and use geometric notation for labelling angles, lengths, equal lengths and parallel lines | * Know the first 6 cube numbers * Know the first 12 triangular numbers * Know the symbols =, ≠, <, >, ≤, ≥ * Know the order of operations including brackets * Know basic algebraic notation * Know that area of a rectangle = l × w * Know that area of a triangle = b × h ÷ 2 * Know that area of a parallelogram = b × h * Know that area of a trapezium = ((a + b) ÷ 2) × h * Know that volume of a cuboid = l × w × h * Know the meaning of faces, edges and vertices * Know the names of special triangles and......

Words: 4838 - Pages: 20

Premium Essay

Manipulatives In Mathematics

...Introduction In this chapter, research from multiple authors will provide supporting answers for my research question, how do games and manipulatives impact students' interest in Mathematics?, how do games and manipulatives impact students' performances in Mathematics?, and what are the benefits of using games and manipulatives when teaching fractions? Based on research thus far manipulative and games improve students’ interest and performance, while some researchers don’t see a significance difference in manipulatives increasing students interest in mathematics. (Kontaş) (2016).I found that manipulatives were proven to assist in helping students in building conceptual understanding, and eliminate misconception in mathematics. DeGeorge and...

Words: 1079 - Pages: 5

Free Essay

Math Facts

...recall facts without spending time thinking about them, counting on their fingers, using manipulatives, etc (Yermish, 2011). . In order to become a fluent reader, a person must memorize the sounds that letters make and the sounds that those letters make when combined with other letters. Knowing math facts, combinations of numbers, is just as critical to becoming fluent in math. Numbers facts are to math as the alphabet is to reading, without them a person cannot fully succeed. (Yermish, 2011 and Marquez, 2010). A “known” fact is one that is “answered automatically and correctly without counting” (Greenwald, 2011). In order for a child to achieve academically, the child must master basic facts. A child's progress with problem-solving, algebra and higher-order math concepts is negatively impacted by a lack of fact fluency. As a result, a child's overall self confidence and general academic performance will decline (Groves, 2011). In the...

Words: 1902 - Pages: 8

Premium Essay

ADHD Argumentative Essay

...Concerta help stimulate the brain. These medications shoe increases in patience as well as focusing on everyday tasks. Even though these stimulants work, there is also an argument that there are other ways to help with the disorder besides medication. Patients should still stick with the medication because it does show major progress, changes in behavior, and helps in the work and school place. Furthermore, ADHD had been researched for over 35 years. From that research, scientists have concluded that ADHD is acquired due to a chemical and structural mishap of the brain and the way it perceives things. One predominant misconception of ADHD is that many people believe that Attention-Deficit Hyperactivity Disorder is due to brain damage or injury to the brain. “ADHD children/adults often feel that flight path of a fly in the room is as important as the teachers algebra lesson.” One effect of ADHD is that the level of importance seen by both children and adults are unbalanced. For example, while sitting in class those students scrap paper will appear to be just as or more important that their actual test. Medication is shown to be the dominant source of containing ADHD, but when combined with a good diet and therapy the results are impeccable. Studies show that more patients prefer Adderall over Ritalin to contain their ADHD. They feel as if it has more of a drop-off, which means lesser side effects than what Ritalin would have. This shows that doctors should test which......

Words: 783 - Pages: 4

Free Essay

Integration of Video Games in Classrooms

...Integration of Video Games in Classrooms Abstract This literature review will explore the effective ways educational video games enhance student motivation in classrooms and the personalized approach of education through video games. It will also explore the adaptation of educational games in classrooms as well as educator’s concerns. The review is limited to the use of video games designed for educational purposes and does not discuss the educational value of the entertainment video games. The review does not discuss the different effects of video games on diverse groups; it focuses primarily elementary to college level students. Video games have been around for over thirty years; they are one of the most lucrative and prominent forms of entertainment across cultures. A total of $20.77 billion was spent in 2012 in the game industry (ESA, Essential Facts about the Computor and Video Game Industry, 2013). The Gaming Industry is changing and adapting every year to the new technological changes. Video games have become an important form of entertainment to society. “U.S. consumers play video games on… Sony PlayStation 3 and the Microsoft Xbox 360 and on personal computers (“PCs”)….on mobile video players such as the PlayStation Portable “PSP”, the Game Boy Advance and the Nintendo DS….“on-line” among multiple players who interconnect over the Internet and other proprietary online networks” (Siwek, 2010). The expanding market of video games and its influence on......

Words: 2385 - Pages: 10

Premium Essay


...that I had stepped into another world, a world of int elligence and aesthetics. I felt that it might be my destiny to probe this w orld. With self-confidence and my father's encouragement, I chose applied mathemat ics as my major in college. Thanks to the excellent faculty who guided me on  my pilgrimage across the mathematical universe, my love for mathematics con tinued to bloom. In my analysis courses, I first met the continuous function  under the definition of Cauchy. Then, my vision broadened to the Riemann in tegrable function space, which is composed of "almost" continuous functions.  With the advent of the set theory, my vision again expanded to the measurab le function under the theory of the Lebesgue Integral. In my algebra courses , I was equipped with a powerful tool- the matrix. The more knowledge I lear ned, the more useful I found that this tool, especially in numerical analysi s and optimization. When I moved into the real abstruse world of abstract al gebra, I came to realize that mathematical symbols did not have to stand for  numbers; for instance, they may represent matrices, quaternions, or transfo rmations. In fact, it...

Words: 1128 - Pages: 5

Premium Essay

My Experience In Math

...learned it by my classmate and also our instructor who helped me to understand the topic that I can understand that’s why he explained it clearly to us. I learned the simplest ways of solving any quadratic equations because our instructor teaches us and shared his way to solve any quadratic equation easily. mathematics also improves our mental ability as it teaches us logical ways of thinking. It combines really important information on the brain and learning with new evidence on the best ways to approach and learn math effectively. Many people have had negative experiences with math, and end up disliking math or failing. This class will give learners of math the information they need to become powerful math learners, it will correct any misconceptions they have about what math is, and it will teach them about their own potential to succeed and the strategies needed to approach math effectively. If you have had past negative experiences with math this will help change your relationship to one that is positive and...

Words: 1322 - Pages: 6

Free Essay


...very paper needs an introduction. In fact, the introduction is the most important part of your paper, because few of your readers will ever read beyond it. And there's not much hope that any of them will if you don't grab their attention from the start. So it's a mystery why so many papers begin with twaddle like this excerpt from a conference on high performance computing. Massively parallel computers (MPCs), characterized by their scalable architectures, are a viable platform on which to solve the so-called grand-challenge problems. These distributed-memory systems are expandable and can achieve a proportional performance increase without changing the basic architecture. In order to take full advantage of scalable hardware, the application software must also be scalable to exploit the increased computing capacity. If you find your thoughts drifting away, don't feel bad; we have evidence that the authors felt the same way - consider the near-meaninglessness of the second sentence. The real weakness, though, is that this extreme form of ``grandmothering'' has no other function than to tell you something you already know, and wouldn't be comprehensible if you didn't already know. The paragraph is entirely superfluous to any reader who knows the meanings of ``scalable'' and ``grand-challenge problems.'' Other readers, who don't know the buzzwords, are discouraged from continuing. Everybody loses. I'm not going to give you the usual advice that you fuss and fret......

Words: 1597 - Pages: 7

Free Essay

All About Science: Word Scavanger

...skills to many different situations, materials, and ideas; 2.2. Science—Candidates know, understand, and use fundamental concepts in the subject matter of science—including physical, life, and earth and space sciences—as well as concepts in science and technology, science in personal and social perspectives, the history and nature of science, the unifying concepts of science, and the inquiry processes scientists use in discovery of new knowledge to build a base for scientific and technological literacy; 2.3. Mathematics—Candidates know, understand, and use the major concepts, procedures, and reasoning processes of mathematics that define number systems and number sense, geometry, measurement, statistics and probability, and algebra in order to foster student understanding and use of patterns, quantities, and spatial relationships that can represent phenomena, solve problems, and manage data; 2.4. Social studies—Candidates know, understand, and use the major concepts and modes of inquiry from the social studies—the integrated study of history, geography, the social sciences,...

Words: 5673 - Pages: 23

Premium Essay

Understanding Student Miscoceptions

...Understanding Student Misconceptions Lorriane VanAlstyne Grand Canyon University: SED 544 June 24, 2015   I found myself sitting at a desk with my hand on my forehead, my foot tapping the floor and my pencil scribbling some dark marks on my homework sheet. I read the question again for the 5th, maybe 6th time. Maria and her friend Mary collected soda cans for a school recycling project. On Friday they collected 25 bottles and cans while on Saturday they collected 60 bottles and cans. How many bottles and cans did Maria and Mary collect in all? I then ask myself “What’s the point, why do they make us do this stuff, I can’t wait till this class is over.” Word problems are found to be very difficult for many students; especially those who have disabilities. Studies published in the article “Going Beyond the Math Wars” by Cole and Wasburn-Moses states that “Only 8% of students with disabilities scored at or above proficient on a national assessment of mathematical proficiency” (Cole & Wasburn Moses, 2010). The deficit of scores in math, for students with disabilities is caused by many reasons which include: reading, language, vocabulary, understanding, the transfer of information, and a lack of design in the curriculum. Students with disabilities struggle with math because they have difficulty reading texts, short term memories, and understanding the language within the problem. When they look at a typical word problem, students don’t just see numbers, they see lines of...

Words: 1851 - Pages: 8

Premium Essay

Factors Associated to the Problem Solving Skills of Physical Science Majors Students in Physics

...Cooperative Problem Solving in Physics A User’s Manual Why? What? How? STEP 1 Recognize the Problem What's going on? STEP 2 Describe the problem in terms of the field What does this have to do with ...... ? STEP 3 Plan a solution How do I get out of this? STEP 4 Execute the plan Let's get an answer STEP 5 Evaluate the solution Can this be true? Kenneth Heller Patricia Heller University of Minnesota With support from the National Science Foundation, University of Minnesota, and U.S. Department of Education © Kenneth & Patricia Heller, 2010 Acknowledgments In reaching this stage in this work, we gratefully acknowledge the support of the University of Minnesota, the U.S. Department of Education FIPSE program, and the National Science Foundation. This work would not have existed without the close cooperation of the University of Minnesota School of Physics and Astronomy and Department of Curriculum and Instruction. We have incorporated the suggestions of many faculty members from both Physics and Education at the University of Minnesota and other institutions that have communicated with us at workshops, meetings, and by e-mail. This work has depended on the efforts and feedback of many graduate student teaching assistants in the School of Physics and Astronomy over the years. Much of this development is directly based on the research of the graduate students in the University of Minnesota Physics Education Program:......

Words: 40122 - Pages: 161

Premium Essay

Computer - What Is It?

...------------------------------------------------- omputer From Wikipedia, the free encyclopedia For other uses, see Computer (disambiguation). "Computer system" redirects here. For other uses, see Computer system (disambiguation). "Computer technology" redirects here. For the company, see Computer Technology Limited. Computer | | A computer is a general purpose device that can be programmed to carry out a finite set of arithmetic or logical operations. Since a sequence of operations can be readily changed, the computer can solve more than one kind of problem. Conventionally, a computer consists of at least one processing element, typically a central processing unit (CPU) and some form of memory. The processing element carries out arithmetic and logic operations, and a sequencing and control unit that can change the order of operations based on stored information. Peripheral devices allow information to be retrieved from an external source, and the result of operations saved and retrieved. The first electronic digital computers were developed between 1940 and 1945 in the United Kingdom and United States. Originally they were the size of a large room, consuming as much power as several hundred modern personal computers (PCs).[1] In this era mechanical analog computers were used for military applications. Modern computers based on integrated circuits are millions to billions of times more capable than the early machines, and occupy a fraction of the......

Words: 1954 - Pages: 8

Premium Essay

Automated Grading System Review

...Journal of Information Technology Education Volume 2, 2003 An Overview of Current Research on Automated Essay Grading Salvatore Valenti, Francesca Neri and Alessandro Cucchiarelli DIIGA - Universita’ Politecnica delle Marche, Ancona, Italy Executive Summary Essays are considered by many researchers as the most useful tool to assess learning outcomes, implying the ability to recall, organize and integrate ideas, the ability to express oneself in writing and the ability to supply merely than identify interpretation and application of data. It is in the measurement of such outcomes, corresponding to the evaluation and synthesis levels of the Bloom’s (1956) taxonomy that the essay questions serve their most useful purpose. One of the difficulties of grading essays is represented by the perceived subjectivity of the grading process. Many researchers claim that the subjective nature of essay assessment leads to variation in grades awarded by different human assessors, which is perceived by students as a great source of unfairness. This issue may be faced through the adoption of automated assessment tools for essays. A system for automated assessment would at least be consistent in the way it scores essays, and enormous cost and time savings could be achieved if the system can be shown to grade essays within the range of those awarded by human assessors. This paper presents an overview of current......

Words: 7241 - Pages: 29

Premium Essay


...Why Normalization Failed to Become the Ultimate Guide for Database Designers? Marin Fotache Alexandru Ioan Cuza University of Iasi, Romania Faculty of Economics and Business Administration Dept. of Business Information Systems Blvd. Carol I, nr. 22, Iasi, 700505, Romania Tel: + 40 744 497 654, Fax: + 40 232 217 000 ABSTRACT With an impressive theoretical foundation, normalization was supposed to bring rigor and relevance into such a slippery domain as database design is. Almost every database textbook treats normalization in a certain extent, usually suggesting that the topic is so clear and consolidated that it does not deserve deeper discussions. But the reality is completely different. After more than three decades, normalization not only has lost much of its interest in the research papers, but also is still looking for practitioners to apply it effectively. Despite the vast amount of database literature, comprehensive books illustrating the application of normalization to effective real-world applications are still waited. This paper reflects the point of 1 view of an Information Systems academic who incidentally has been for almost twenty years a practitioner in developing database applications. It outlines the main weaknesses of normalization and offers some explanations about the failure of a generous framework in becoming the so much needed universal guide for database designers. Practitioners might be interested in finding out (or......

Words: 12110 - Pages: 49