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Negative Integers - Section A
Negative Integers (Pedagogical Content Knowledge Project)
- by Sarah Swedlund, Cassandra Hunter, Brandi Demsko, Cheryl Kurata (UC Irvine, July 2009)
Cognition and Learning Background
Negative integers refer to whole numbers that have a value less than zero. They are important in mathematics because they are the result of subtracting a larger number from a smaller number. They also represent debt, freezing temperature, below sea level heights, and distance away from a destination. The concept of negative integers is not addressed in the California Content Standards in Mathematics until grade four, allowing students to develop misconceptions. In grade four, students are supposed to learn how to identify negative integers and recognize real world uses, and then in grade five, they are expected to add with negative integers and subtract positive integers from negative integers. Finally, by sixth grade, students are required to know how to accurately solve any operation using negative integers. Although using negative integers in operations can be challenging, the concept of negative amounts can and should be introduced much earlier. It has been proven that students as young as five years of age have the cognitive capability to understand the concept of negative numbers and identify negative numbers on a number line (as seen in the video shown in lecture on 21 July 2009). Furthermore, cognitive frameworks support the idea that concepts can and should be introduced early on and then built upon in later years.
Theories of Cognition and Learning
A well known and commonly used theory of learning remains constructivism, which argues that learning is an active process that requires the student to use prior knowledge and experience to build, or construct, ideas. Constructivism upholds the beliefs that active and reflective thought are crucial for constructing knowledge, all learners will construct ideas differently based on their individual experiences and prior learning, students often make mistakes that are consistent with their personal understanding of a concept, and learners use assimilation and accommodation to incorporate new knowledge into their preexisting schema.
Keeping consistent with constructivist theory, it seems logical that negative integers be introduced early on in schools, since it will provide the basis for later, more complex usages. If students already have some correct prior knowledge about negative integers, making sense of more advanced operations with negative integers will be easier and allow for strong overall conceptional understanding. Furthermore, because our number sense is flexible based on our experiences with number, the more experiences students have with negative integers the more flexibly their number sense will develop (Fischer, 2003).
Zone of Proximal Development
The zone of proximal development is a concept developed by psychologist Lev Vygotsky to represent the difference between what a learner is capable of doing independently and what (s)he is capable of doing with help. This concept relates to scaffolding, which refers to the process in which a knowledgeable person provides aid to a learner and then eventually decreases the amount of help given until the student can complete the given tasks independently. Vygotsky suggests that students should be given the appropriate assistance necessary for learning new concepts and furthermore, that all education should be within their zone of proximal development.
The zone of proximal development supports the idea that students need to be provided with lots of help in order to gain mastery over concepts. In relation to negative numbers, students need to start with more guidance and accurate information, especially when they are introduced to new ideas. Then they should be allowed time to make the information meaningful, connecting it to their prior knowledge, and eventually achieve strong conceptual understanding. Teachers should be prepared to provides tools and resources that will scaffold the knowledge for their students.
The theory of multiple intelligences was proposed by Howard Gardner, who believed that humans possess a wide range of abilities and categorized these into eight 'intelligences.' Gardner countered the notion that performing well on a mathematics test demonstrates higher intelligence since it only measures intelligence in this specific area. He determined that there were eight distinct areas in which people could exhibit expertise and intelligence: kinesthetic (dealing with movement), interpersonal (interaction with others), linguistic (verbal and written knowledge), logical (mathematical), naturalistic (understanding nature and science), intrapersonal (dealing with self reflection and self awareness), spatial (having strong visualization capabilities and spatial awareness), and music (understanding rhythm, tones, and music). Students can exhibit a range of intelligence among these different categories, and depending on their strengths, have different ideal methods of learning.
Since all students have unique intelligences, their learning styles also differ and educators must account for this when they plan their lessons. Although negative integers most easily fit into the logic category of intelligences, activities can incorporate and include all of the others as well to accommodate each student. Below is a list of examples for how to adapt lessons to match each intelligence.
- Have students physically move on a number line to identify, compare, and compute with negative integers
- Allow students to work in groups and in partners to promote scaffolding and collaboration
- Provide the students with word problems
- Present new information verbally
- Ask the students to practice operations with negative integers
- Tell the students to use the proper mathematics symbols in their work
- Present the students with problems that relate to real life situations (such as problems involving money and debt)
- Encourage students to explore using manipulatives to make their own meaning
- Have the students explain their thinking and reasoning
- Present the material with visuals and written examples
- Create a rhyme, poem, or song that corresponds to important information that the students can use to remember key facts
Cognitive Obstacles and Common Misconceptions
- Students tend to have difficulty with the idea of negative numbers because it is hard to visualize a negative amount of something (Prather& Alibali, 2008). It is easier for children to imagine four items because they can picture things out of the real world such as four apples. It becomes difficult to imagine a negative number in the same way (Prather& Alibali, 2008). This becomes a cognitive obstacle because they way you think or visualize positive numbers differs from the way you visualize negative numbers in relation to a real world settings and applications.
- When learning mathematics and negative numbers specifically, mathematical discourse or academic language is prominently involved. Mathematical Discourse is a way of communicating and learning about math (Sfard, 2007). When the mathematical discourse is not understood, it becomes challenging to cognitively process the concept when you do not understand what the language means or represents. Mathematical discourse is learned in school and changes over time, which becomes challenging (Sfard, 2007). For example, negative, positive and integer are terms used within the mathematical discourse. If the term negative integer is not conceptually understood, then the process of working and learning about negative integers becomes challenging in regards to thinking and discovering for the student, otherwise known as a cognitive obstacle.
- Since negative numbers are not introduced in California mathematics content standards until grade 4, students often find their own information from various sources. In a study by Hativa & Cohen (1995), of the twenty-eight participants, forty-two percent claimed that they learned about negative integers from family members. As a results, some of these student held misconceptions and demonstrated poor conceptual understanding. Research also shows that an obvious increase in errors occurs as soon as negative integers are introduced into operations. (Vlassis, 2004).
- The number line only extends to a certain point (Towers & Anderson, 1998; Bruno & Martinon, 1999).
- Negative numbers are confused with zero (Hativa & Cohen, 1995).
- Negative numbers are confused with fractions (Carraher & Sheliemann, 2002).
- A bigger negative number is greater than a smaller negative number. For example, -10 > -5.
- Two negatives always equal a positive regardless of the operation performed.
- Positive and negative signs are commonly misapplied (Brunt, 1998).
Pedagogical Tools and Strategies
Examples and Non Examples
- Example of Negative Integers: -1,-2,-3,-4,-5
- A negative integer is a number less than zero.
- Non Example of Negative Integers: 1,2,3,4,5
- A number that is greater than zero is a positive integer.
- Negative Integers are opposites of positive integers.
- A negative integer is an integer less than zero.
- Negative Integer has no fractional part.
Negative Integers are in the form of whole numbers. (Van De Walle, 2007)
Negative Integers occur in real life situations. (Van De Walle, 2007)
- Negative Numbers are distant from zero.
- Negative integers are numbers measured distances to the left of zero. (Van De Walle, 2007)
Connections To Other Concepts
- Negative Integers are incorporated into the real world and real life applications. A negative integer is a whole number that has a smaller value than the number zero. Debt is a real life example that can represent negative integers. When a person is in debt, they owe money. That number that represents what is owed is a negative integer. It is a whole number that is an amount below zero, therefore that number is owed. (Van De Walle, 2007)
- Temperature is another real life application that incorporates negative integers. Both negative and positive integers describe temperature. A negative integer is used to describe an amount below zero establishing an extreme freezing temperature.
- Negative Integers relate to the idea of absolute value. A negative integer is the opposite of a positive integer. Absolute value is a change from either positive to negative or negative to positive. (Van De Walle, 2007)
Realtionship To Concepts and Other Situations
- The concept of negative integers is transferred into real life situations. One can use their knowledge of negative integers to understand and develop real life applications and processes that use negative integers. The concept of negative integers can be used with temperature. One has to be able to understand that a negative integer is a whole number below zero which can be transferred to the idea of temperature. When temperature is below zero, it means that the negative integer represented is an amount below zero which means a freezing temperature is created.
- Number Line
- Black and White chips
- Double Abacus
- Football field
- Sea Level
- Negative - an amount less than zero such as -7
- Positive - an amount greater than zero such as 10
- Integer - whole numbers, including negatives, positives, and zero
- Sign - mathematical symbol (+ or -) used to indicate whether a quantity is positive or negative
- Number Line - a straight line that has markings to represent real numbers (see example below)
- Thermometer - a scale to show temperature, which can be positive or negative
- Debt - something that is owed, often referring to money, represented by a negative amount
- +: positive, plus, addition operation
- −: negative, minus, subtraction operation
- >: greater than
- <: less than
True/False and Open Number Sentences can be used to help students see and understand the relationships between positive and negative integers and the various operations with negative numbers. These are best used with students to begin class discussions about the properties of negative integers and how they are affected when added, subtracted, multiplied, and divided by positive and negative numbers.
- 5-7 = 5+(-7)
This number sentence gets students thinking about how subtracting a number is the same as adding a negative.
- 8+10 = 8-(-10)
This introduces students to the concept of subtracting a negative and how it is the same as adding its positive counterparts.
- (-3)x(-6) = (-6)x(-3)
This number sentence helps students see that the commutative property of multiplication applies to both positive and negative integers when both numbers are positive or both numbers are negative.
- 15 / (-3) = (-15) / 3
This division number sentence shows students that when a positive integer and a negative integer are divided (or multiplied), the answer is the same, no matter which number is negative and which number is positive. (With students, the actual division sign should be used)
Note: All of these number sentences are true, but it is important to include number sentences that are false so that students are encouraged to think critically about them.
Open Number Sentences
- 5+__ = -3
This open number sentence is likely to be tricky for students because they are asked to find the number that is added to five to get a negative number. This requires them to think about how a negative number added to a positive number can sometimes be negative.
- 13-17 = __ - (-2)
An open number sentence like this is likely to be difficult for students in two ways. First, students are usually not used to seeing an unconventional number sentence like this so it takes more cognitive effort to solve. Secondly, the students have to figure out what number negative two is subtracted from to get negative four.
- -12 = __x3
This number sentence requires students to think about the multiplication properties of negative integers-- that multiplying a positive number by a negative number yields a negative result.
- 21 / (-7) = 7x__
This number sentence helps students see the relationship between multiplication and division with positive and negative integers.
Useful Contexts and Problem Situations:
- Students can use a number line to not only visually see where positive and negative integers are located, but also to help in adding or subtracting integers.
- Different colored chips can be used to also represent positive and negative integers and aid students in developing an understanding of working with these integers.
- Counting on a double abacus can help in representing addition and subtraction of negative integers.
- Real World examples can be used in the classroom to represent how negative integers are a part of every day life.
- Thermometer-shows temperature can be positive or negative
- Sea Level-elevation of land can be above or below sea level
- Football- yards can be gained and loss during a game
- Money- people can gain and spend money, or even become in debt
- Elevators-can travel both below and above ground to different floors
Problems to Promote Conceptual Understanding:
- Write the next three numbers to complete the pattern: 25, 20, 15, 10, _, _, _
- What is the function of the pattern? (What do you need to do to one number to get to the next number?): 3, -6, 12, -24, 48, -96
- Jessica has $10. She spends $7 on a card game and then wants to buy a basketball that costs $5. Will she be able to buy the basketball? Why or why not.
- State whether the following is true or false and explain your reasoning: -7 > -4
- Billy climbed 3 feet into a tree because he wanted to build a treehouse. Then he wanted to dig a tunnel underground instead. He went down a total of 5 feet from the branch he was sitting on in the tree. Show an equation that represents how deep he dug his tunnel.
- A squirrel was in a 15 foot deep hole. A dog came out and scared him. From the bottom of the hole the squirrel climbed up 27 feet into a tree. How high up is the squirrel?
- If it is -17 degrees outside and then the temperature drops 6 degrees, what temperature is it now?
Curricula and Technological Resources
There are several resources available for teachers who are looking to help their students develop a better understanding of negative integers. These resources include lesson plans and curricula, math applets available online, and math software that can be downloaded to classroom computers.
Lesson Plans and Curricula
- Positive and Negative Integers: A Card Game
-An adaptation of the card game Twenty-Five provides practice adding and subtracting positive and negative integers.
-Students work in groups of two or more and deal out all of the cards in one deck to every student. Explain to students that every black card in their pile represents a positive number, while every red card represents a negative number. For example a black three is worth +3 (three), while a red seven is work -7 (negative 7).
-Depending on the level of your students you can choose to use or discard face cards. If you choose to use them, aces have a value of 1, jacks have a value of 11, queens have a value of 12, and kinds have a value of 13.
-The first player turns over their top card and says the number on the card. The next player turns over one of their cards and adds it to the first card and then has to say the sum of the two cards aloud. For example, if the first card is a black 8 and the second card is a red 9, which has the value of -9, the player says 8+ (-9)= (-1). The game continues on from there, each player finds the sum as the cards continue to be piled on.
-The game continues until someone shows a card that when added to the stack results in a sum of exactly 25.
-This game can be both adapted for students who need more challenge, by doing subtraction, and also for students with special needs, by lowering the sum to a number less than 25.
- Multisensory Teaching: Positive and Negative Integers
-Since negative numbers are an abstract concept, this lesson focuses on that big idea that students can learn more advanced mathematical ideas by starting with ideas that we understand and extending those ideas.
-Part one focuses on the overall big idea. The examples of opposites and balance, as well as elevators, football, and money are used. A thermometer is also used to explain to students that ten below zero is colder than zero, by showing that even though it looks bigger, it stands for less. For independent practice the students circle which is the colder temperature out of two different options.
-Part two focuses on adding and subtracting. The number line is used and students are asked to compare it to the thermometer. Other concrete applications of negative numbers are introduced, such as elevators going below ground floor and money being owed. Different colored chips are also used to distinguish between positive and negative integers. These chips can be used to demonstrate how two numbers can add up to equal nothing and how negative numbers can also be added.
-Part three focuses on multiplication. This concept is taught through showing multiplication as repeated addition with the colored chips and also through real world applications. Students need to have a lot of practice with these concepts so that they are able to reach automaticity.
- The National Library of Virtual Manipulatives has a math applet on their website that uses color chips to help students understand addition with negative numbers. Students are given an expression, such as 3+(-4) and are directed to use the black (positive) and red (negative) virtual chips to represent the problem. Once students have dragged the correct numbers of chips into the oval, they use them to cancel each other out by dragging each negative chip to a positive chip. Those zero-pairs then disappear and the student is left with the chips that represent the correct answer.
This applet is useful for improving students’ understanding of addition with negative integers because it allows students to virtually manipulate the problem by having them physically represent the addition of the two numbers and then cancelling out the zero-pairs. Also, because the positives and negatives are two different colors, it is easy for students to predict what their answer will be before completing the entire sequence of actions. Furthermore, using the color chips applet helps to develop the concept that a positive and a negative cancel each other out if they have the same value.
- Funbrain has a math applet that gives students practice with adding and subtracting fractions using a number line. Students are given addition or subtraction problems with negative numbers and they then use the given number line to solve them.
This applet helps students improve their understanding of negative integers because it gives them a tool to use when solving addition and subtraction problems with negative numbers. Having the number line allows students to see why subtracting a positive number from a negative number makes a more negative number and why adding a positive number to a negative number makes it less negative. The game also includes problems that do not involve negative numbers so students are able to see the relationships between addition and subtraction with positive integers and addition and subtraction with negative integers.
- Math.com has a math applet that goes through four steps of understanding with signed integers (both positive and negative). Students are able to visually see different examples and then are given the opporunity to try out the problems on their own.
This applet can be useful because it goes through the learning process in detail. The applet begins with a first glance section, that just introduces the student to the number line and allows them to play with it and see where the positive and negative integers are located. Next there is an in depth page that goes into a lot more detail by explaining what positive and negative integers are as well as giving real life examples that students can related to. On the bottom of this page and the next page there are a few examples along with their appropriate solutions explained. Finally, step four is a workout for the students to complete on their own, which is useful because students need practice.
- The Mind Research Institute has developed math software using research in neuroscience, mathematics, and education that teaches kids important math concepts through fun, interactive games. Students are introduced to negative integers in fourth grade, which matches the California Content Standards. Each game has various levels that range from conceptual and concrete to abstract and symbolic. This allows students to gain conceptual understanding of negative integers while still moving on to more abstract operations and representations.
- Bruno, A. & Martinon, A. (1999). The teaching of numerical extensions: the case of negative numbers. International Journal of Mathematical Education in Science and Technology , 30, 789-809.
This article is focused on a unified view of the teaching of numbers, in particular negative numbers. The authors base their ideas on three dimensions of numerical knowledge: abstract, contextual and number line. Findings show the importance of previous ideas of positive numbers and how these ideas can influence knowledge of negative numbers.
- Brunt, G. (1998). Questions of Sign. Physics Education, 33, 242-249.
This article pertains to the problem that often in textbooks the appropriate usages of signs (positive or negative) are often ignored. The author focuses on mathematical operations used in physics textbooks. He states that positive and negative integers are frequently misapplied, which may be due to the fact that textbook authors and teachers are unaware of the rules involved with sign. The author outlines steps that can be taken by teachers to avoid such sign errors.
- Carraher, D., & Sheliemann, A. (2002). The transfer dilemma. Journal of the Learning Sciences, 11, 1, 1-24.
This study includes interviews of two 5th grade students learning about negative number operations. These students use their prior knowledge to try and guide their thinking and wind up gaining a good foundational understanding of negative number properties. The article argues that transfer learning cannot take credit in this instance and prior experiences and knowledge are responsible.
- Chiu, M. (2001). Using metaphors to understand and solve arithmetic problems: novices and experts working with negative numbers. Mathematical Thinking and Learning, 3, 2, 93-124.
In this study, the author asks both novices and experts to use metaphors to solve negative number problems. He finds that, although the metaphors provided are the same, the children novices used metaphors more often than the adult experts in the arithmetic, but adults used them more in explaining the arithmetic. The results show that metaphors are beneficial to learning and understanding concepts.
- Fischer, M. H. (2003, May). Cognitive Representation of Negative Numbers. Psychological Science, 14(3).
This article reports on the findings of an experiment that sought to find out how we understand negative numbers. The author concluded that in order to understand negative numbers, we refer to a mental number line in which the negative numbers are associated with the left, as in a traditional number line.
- Hativa, N., & Cohen, D. (1995). Self learning of negative number concepts by lower division elementary students through solving computer-provided numerical problems. Educational Studies in Mathematics, 28, 4, 401-431.
This article focuses on the types of negative number concepts students are able learn and understand at young ages. A computer program is used to introduce negative numbers on a number line and through the use of problem solving models. The results show that negative number instruction is beneficial for students, even if they came in with misconceptions.
- Linchevski, L., & Williams, J. (1999). Using Intuition from Everyday Life in 'Filling' the Gap in Children's Extension of Their Number Concept to Include the Negative Numbers. Education Studies in Mathematics, 39, 131-147.
This article describes two instructional strategies aimed at building students' understanding of integers--negative integers, in particular. These two instructional strategies are based in research and focus on helping the students connect outside-school knowledge with what they are learning about integers.
- Prather, Richard W. and Alibali, Martha W.(2008)'Understanding and Using Principles of Arithmetic: Operations Involving Negative Numbers',Cognitive Science: A Multidisciplinary Journal,32:2,445 — 457.
The article describes a study of the knowledge that is found between negative and positive numbers. Overall, it was found that positive numbers are easier for people to conceptualize positive numbers in comparison to negative numbers. It was found in the study that those people who were proficient with negative numbers had an easier time with representations of problems that deal with negative numbers.
- Sfard, Anna. (2007). When the Rules of Discourse Change,but Nobody Tells You: Making Sense of Mathematics Learning From a Commognotive Standpoint. Journal of The Learning Sciences, 16, 565-613.
The article describes the mathematical discourse that is a part of positive and negative numbers. The research and study shows how influential and prominent mathematical discourse is in an educational setting. The differences in discourse can affect cognitive thinking when working with concepts such as positive and negative numbers.
- Streefland, Leen.(1996). Negative Numbers: Reflections of a Learning Researcher. Journal of Mathematical Behavior, 15. 57-77.
The idea of negative numbers is presented in the article and how they are viewed in different ways and with different approaches. The research and use of problems involving negative numbers created a reflection, hypothesis and questions about negative numbers. The article discusses the ideas of how children build their ideas of negative numbers.
- Towers, J. & Anderson, A. (1998). The Wall that Stops the Outside Coming In: Exploring Infinity and Other “Difficult” Concepts with a Preschooler. Early Child Development and Care, 145, 17-29.
This study focuses on a conversation the authors had with a preschooler in an attempt to discover and re-think topics that educators deem difficult for this age group. This interview allowed the authors to gain insight into what puzzles or interests students this age pertaining to mathematics. They found that the topics that teachers often avoid, such as infinity and negative numbers were concepts that were brought up and questioned.
- Van De Walle, J. (2007) Elementary and Middle School Mathematics Teaching Developmentally. 490-499. Boston: Pearson.
Van de Walle’s book has helped to give insight on negative integers and how they are a part of real life. The book gives activities, games as well as suggestions in regards to different resources. The information presented in the book gives insight on how to approach negative integers as well as the implementation into the learning process.
- Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in 'negativity'. Learning and Instruction, 14, 469-484.
This article focuses on students' conceptual understanding of negative integers and the minus sign and their difficulties with understanding the two. The author argues that students cannot fully understand these two concepts without developing a meta-conceptual awareness of the symbols.