# Division

Pedagogical Content Knowledge Project

- by Michelle Blanco, Nicole Engelmann, Lorin Hayward, and Christen Hesketh, UCIrvine, August 2008

## Cognition and Learning Background: Elementary Algebra

### Division in Whole Numbers

- Let
*a*and*b*be whole numbers with*b ≠ 0*. Then*a ÷ b= c*, if and only if,*a = b ∙ c*for a unique whole number*c*(Long & Duane, 2006)

#### Conceptual Models

The division of two whole numbers, a ÷ b, b ≠ 0, has three main conceptual models (Long & Duane, 2006):

**Partition Model**- Also known as: Division by Sharing and Partitive Division
- Example 1
- Can be easily understood with the assistance of physical objects

**Repeated-Subtraction Model**- Also known as: Division by Grouping and Measurement Division
- Example 2

**Missing-Factor Model**- Also know as: Opposite of Multiplication
- Example 3

#### Division by Zero

Division by zero is undefined, meaning it is an operation for which you cannot find an answer, so it is disallowed. Thinking about division in relation to multiplication will help in understanding this concept (Long & Duane, 2006)

- 15 ÷ 3 = 5 because 3 • 5 = 15
- 15 ÷ 0 = x would mean that 0 • X = 15, but since no value would work for x, since 0 times any number is 0, division by zero does not work.

#### Division with Remainders

“More often than not, division does not result in a simple whole number.” Depending on the context of the division problem, remainders can have 5 different effects on division answers (Van de Walle, 2007, p. 154). Below are examples from Van de Walle (2007) to illustrate the different remainder scenarios:

- The remainder remains as a quantity left over
- Example: You have 30 pieces of candy to share fairly with 7 children. How many pieces of candy will each child receive?
- Answer: 4 pieces of candy and 2 left over (left over)

- Example: You have 30 pieces of candy to share fairly with 7 children. How many pieces of candy will each child receive?
- The remainder is partitioned into fractions
- Example: Each jar holds 8 ounces of liquid. If there are 46 ounces in the pitcher, how many jars will that be?
- Answer: 5 and 6/8 jars (partitioned as a fraction)

- Example: Each jar holds 8 ounces of liquid. If there are 46 ounces in the pitcher, how many jars will that be?
- The remainder is discarded, leaving a smaller whole number answer
- Example: The rope is 25 feet long. How many 7-foot jump ropes can be made?
- Answer: 3 jump ropes (discarded)

- Example: The rope is 25 feet long. How many 7-foot jump ropes can be made?
- The remainder can “force” the answer to the next highest whole number
- Example: The ferry can hold 8 cars. How many trips will it have to make to carry 25 cars across the river?
- Answer: 4 trips (forced to the next whole number)

- Example: The ferry can hold 8 cars. How many trips will it have to make to carry 25 cars across the river?
- The answer is rounded to the nearest whole number for an approximate result
- Example: Six children are planning to share a bag of 50 pieces of bubble gum. About how many pieces will each child get?
- Answer: About 8 pieces for each child (rounded, approximate result)

- Example: Six children are planning to share a bag of 50 pieces of bubble gum. About how many pieces will each child get?

## Cognitive Obstacles and Common Misconceptions

It is the educational consensus that division is the most challenging to teach of the four arithmetic operations due to the computational complexities of the standard long division algorithm. Many students are developmentally ready to conceptually learn division long before they are ready to internalize the standard division algorithm (Slesnick, 1982). Therefore, there are many commonly held misconceptions and errors related to division.

#### Error types

- There are two types of errors:
*systematic errors*and*slips*. Systematic errors occur when a flawed application, rule or algorithm is consistently applied, whereas slips are due to unsystematic careless errors (Lee, 2007).

#### Place-Value Conceptions

- It is widely accepted that systematic errors are due mostly to faulty or weak conceptual understanding of place value systems.Consequently, this lack of conceptual understanding of place value in terms of the division algorithm leads students to mere parroting of procedural computations. As a result, learners and teachers mistakenly resort to nonsensical mnemonic remainder phrases such as “Dirty Monkey Smells Bad” in order to retain the procedural order of “Divide-Multiply-Subtract-Bring it down.” Hence students fail to “make sense” of divisions and systematic errors as well as slips persist (Lee, 2007).

#### Relational Thinking: Multiplication and Division

- Multiplication and division are often taught as two separate educational entities; consequently, students often fail to see the relationship between multiplication and division. This lack of relational thinking inhibits students from developing flexible connections related to division and multiplication. In turn, this affects students’ appropriation of effective division retrieval strategies (i.e. 20 divided by 5 can be solved more readily if a student knows to relate 5 x 4 = 20 to such division problem) (Van de Walle, 2007).
- Another conceptual hurdle for students is their ability to conceive of a group as a single entity while also understanding that a group contains a given number of objects. For example, when asked,
*“How many cookies on 3 plates of 5 cookies each?”*students have difficulty conceptualizing the problem as four sets of eight (Van de Walle, 2007).

#### Division Notation

- Division problems can be represented a number different ways. For example, the quotient 20 divided by 4 can be written as either 20 ÷ 4, 20/4, or 4)20. Traditionally, in school the 4)20 is used for paper to pencil computation and certain misconceptions arise from students prior experience with left-right order of numerals. Therefore, students often incorrectly read such problems as 4 divided by 20 (Van de Walle, 2007).
- Students’ division notation misconceptions are further compounded by adults’ use of the phrase “four goes into twenty.” Van de Walle (2007) states, “This phrase carries little meaning about division, especially in connection with fair-sharing or partitioning content. The ‘goes into’ (or ‘guzinta’) terminology is simply engrained in adult parlance and has not been in textbooks for years” (p. 154).
- A misconception on the part of many educators is the tendency to think that large numbers are too burdensome for students to handle. That is, teachers are more likely to pose 6÷3 than say 45÷8 because it is thought smaller numbers are more manageable. However, research indicates that an understanding of products or quotients is not affected by the size of the numbers as long as the numbers are “within the grasp” of the students. In other words, students have the capabilities to work with larger numbers in division problems using whatever counting strategies they have at their disposal, including invented computational strategies (Van de Walle, 2007, p. 154).

#### Division with Remainders

- As previously mentioned, remainders are more common than whole number division and affect the interpretation of answer significantly. Unfortunately, many students only conceive of remainders in their decontextualized notation (i.e. “R 3”) and, subsequently, never authentically conceptualize the implications of such “leftover” numbers in and outside of the classroom (Van de Walle, 2007). For example, in a study conducted by Li (2001), it was found that “many middle school students solved a DWR [Division with remainders] problem by directly applying division procedure without interpretation of their calculation results based on the question being asked” (p. 502). In essences, if these conceptual failures with remainders are not remedied in elementary school, these gaps in understanding will persist well into middle school and beyond—able to compute the answers, but unable to understand or the interpret the division answers.

#### Seemingly Intuitive Properties of Division

- The commutative property is not intuitively obvious for most students (i.e. 3 x 8 is the same as 8 x 3) thus allowing for failures in student cognition in regards to flexibly conceptualizing division (Van de Walle, 2007).
- Zero and 1 as factors often cause difficulty for students. Consequently, when students divide with these numbers errors can occur frequently if students don’t conceive of
*zero groups of 3*as being zero or*one group of 3*as being only 3.

## Pedagogical Tools and Strategies

The following are four key pedagogical strategies that can be used to teach division. Each has the intention of developing students understanding of the division operation and is not just standard algorithm. The use of models and tangible manipulatives is encouraged.

### Student Invented Strategies

- Children can invent their own methods for solving multi-digit division problems without learning the conventional algorithms that would typically be taught. Utilizing existing mathematical schemes, they are quite literally engaging in active construction of new arithmetic (Caliandro, 2000, p.420).
- Problems should increase slowly in level of difficulty and complexity as students develop familiarity with the math concept (Caliandro, 2000, p.420). Pose problems and allow for time to brainstorm, use scratch paper, or manipulatives as necessary.
- As students increase their confidence in attempting new problem solving strategies, push students toward critique of their own work. Understanding the reasoning behind their approach, checking that their answer is reasonable, and discussing confidently with classmates and teachers are desired outcomes (Caliandro, 2000).
- “Children’s first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic” (Caliandro, 2000, p. 424). The procedures that are developed are undoubtedly meaningful to the student and flow from of their deeper mathematical understanding (Caliandro, 2000, p. 424).
- Unlike traditional algorithms, this approach is valuable for students of ALL mathematical abilities (Caliandro, 2000, p. 424).
- Below-average students will continue to make the same errors repeatedly with algorithms “like machines that cannot be unprogrammed. Their thinking remains blocked and paralyzed by programming” (Caliandro, 2000, p. 424).

#### Using the student invented strategies method:

### High Level Chunking

From a list of techniques used in a study, the informal strategy of ‘High Level Chunking’ emerged as the most successful for those who used it, than any other strategy. Students were presented with two challenge division problems and then given the opportunity to utilize the method of their choice for solution. The formal and invented strategies were categorized into eight classifications ranging from ‘low-level’ inefficient, labor intensive, and elementary in complexity, to ‘high-level’ strategic, more efficient, and related to number facts (Anghileri, 2001, p.89).

- High Level Chunking was used to produce correct answers for problems either in word problem context or stand alone division 64% of the time (Anghileri, 2001, p.93).

##### What is High Level Chunking?

Students learn to simplify larger numbers into appropriate ‘chunks’ for the problem they are trying to solve.

- Example:To solve 432 ÷ 15

- Students would simplify 432 into 300, 60, 60, and 12
- These quantities are the largest easily relateable to 15 and thus easily solved individually

- “Teachers need to help pupils to structure written recoding of the intuitive approaches that the children understand, rather than teaching the algorithm. Pupils need to develop flexibility in their approaches so that the method chosen can be varied according to the numbers that are involved” (Anghileri, 2001, p. 101)
- “It appears inadvisable to have children relying on the standard algorithm, and teaching division needs to focus on a progressive development of more intuitive approaches to achieve efficiently gains without loss of understanding” (Anghileri, 2001, p.101)

### Singapore Math Models

Using Singapore Math Models is one teaching strategy for division. “It is a method which requires kids to draw rectangular boxes to represent math values (both known and unknown values). It is widely used in the teaching of kids math in primary schools (or elementary schools) in Singapore. Kids in Singapore are introduced to the method from as young as Primary One (the equivalent of Grade One).” (Hogan, 2005)

- This method has been shown to be especially successful for students who:

- a- respond well to visual stimulus and instruction
- b- have difficulty with conventional instruction and assistance methods
- c- have not yet learned or mastered algebra
- (Hogan, 2005)

#### Using the Singapore Math Model method

- Example problem solving format:

- A.-To divide 24 by 3, draw 3 boxes to represent 3 groups and put 24 as the number to represent to total for the 3 groups. For this model, there are a few ways to draw it. They are presented below.

- From the model, we arrived at the conclusion that the unknown value (?) can be calculated by dividing the total (24) by 3 boxes (which represents 3 groups. Hence, we formulate the number sentence ======> 24 ÷ 3 = ?

- When the number of groups is too large, you can draw just 1 box to represent the first group followed by an arrow and a number at the end of the arrow to represent the number of groups.

- B. Ann has 360 sweets. She packed them equally into 45 packets. How many sweets are there in each packet?

- (www.teach-kids-math-by-model-method.com)

### Multiplicative Thinking with Arrays

Additive Thinkers are limited to calculation of difference, change, and ratios, where Multiplicative Thinkers recognize these relationships with automaticity. Building Multiplicative Thinking provides the foundation on which proportional reasoning can be built, a difficult concept without the use of multiplicative part-whole strategies. The visual utilized for part-whole thinking here is arrays (Young-Loveridge, 2005, p.34).

- Concrete materials are helpful in introducing mathematics to children in early childhood and primary grades. Apparatus and/or manipulatives (e.g. bead frames or bead strings grouped in tens by color) are useful calculating aids for developing imagery (Young-Loveridge, 2005, p.35)
- Imaging is a part of diagram literacy (See Singapore Math Model). A diagram can be used to display information in a spatial layout and can reveal the structure of a problem. Diagrams can help elementary students’ bridge conceptual and procedural knowledge (Young-Loveridge, 2005, p.35)

- Two different number conceptions:
- 1- Counting-based solutions are based on the number line. “They begin with one of the numbers in the problem and involve jumping along the number line, either forwards (in the case of addition or multiplication) or backwards (in the case of subtraction or division).”
- 2- “Collections-based solutions involve the partitioning of numbers into component parts and the subsequent joining (in the case of addition or multiplication) or separating (in the case of subtraction or division) of the parts to get the answer” (Young-Loveridge, 2005, p.36).

#### Using Multiplicative Thinking with Arrays (collections based model)

“Arrays allow students to develop a deeper and more flexible understanding of multiplication/division, and to fully appreciate the two-dimensionality of the multiplicative process. Because the grids presented here are structures to show the ten-based structure of the number system, and enable various different partitioning possibilities to be shown using different coloring/shading, they provide a basis for students to image or visualize multiplicative process” (Young-Loveridge, 2005, p.38).

- Example: Ways to partition a 3x12

## Curricula and Technological Resources

Research and classroom experience has shown that learning the process of division proves to be difficult for elementary students (Hurts, 2006; Lee, 2007; OTHER SOURCES). In order to improve students’ conceptual understanding and computational fluency of division, appropriate skill-building practice is crucial. Teachers looking to provide students with engaging lessons and meaningful practice both in and out of the classroom, have several technological resources and curricula available to for their use. The following computer-based software, Internet resources and curriculum tools can provide teachers and students the support needed for improving student understanding, skill and achievement in learning and practicing division.

### Online Resources

- NCTM Illuminations: http://illuminations.nctm.org/
- This website is the resource site of the National Council of Teachers of Mathematics. This website provides a wealth of resources ranging from lesson plans, math activities and other website and technological sources. Teachers can search the lesson and activities pages for lesson ideas that are aligned with NCTM standards and support students’ acquisition of mathematical knowledge and understanding.

- Funbrain.com: http://www.funbrain.com/
- (Soccer Shootout-whole numbers)

- Gamequarium.com: http://www.gamequarium.com/division.html
- This website has links to division computer games. Teachers and students can click on the links provided and will be directed to various games that focus on division practice.

- Eduplace.com: http://www.eduplace.com/math/mthexp/
- (multiplication and division memory cards)
- This website is the home of Houghton Mifflin educational curriculum publishers. From the home site teachers can choose on the subject of mathematics and can choose between two links that correspond to two different math textbook series. The division memory cards shown below can be found on this website by clicking the link above or by choosing the CA Math Expressions textbook resource link. Teachers can choose which grade level they are targeting and various resources and teaching tools are made available on this site.

- The multiplication and division memory cards are used to help introduce and reinforce using strategies such as repeated addition and arrays to help students’ understand and connect multiplication with division.
- Accessible algorithms for division are also discussed and shown on this website under the CA Math Expressions textbook link, Grade 4 teaching tools. Two formats can be used to help students understand and procedural fluency when learning long division. One format is called expanded notation, while the other format, called the rectangle method is also shown as an alternative way to teach the traditional long division algorithm.

### Lesson Resources

- Online Lesson Plans
- http://score.kings.k12.ca.us/lessons.html
- This website contains lesson plans for K-12 mathematics. Lessons are broken down into mathematical concepts (i.e. Number Sense, Algebra and Functions, Measurement and Geometry etc.) and are aligned with CA state standards and NCTM standards.

- http://score.kings.k12.ca.us/lessons.html

### Technological Resources

- Gamequarium.com Database: http://www.gamequarium.com/division.html:
- Sum Sense Division Computer Game http://www.oswego.org/ocsd-web/games/SumSense/sumdiv.html:

- Interactive computer game where student is given number cards that have to be dragged and dropped into the correct position to create a division equation that is true.

- Mystery Picture Game http://www.dositey.com/addsub/mystery1D.htm
- UFO space ships with division equations are presented next to a number board with a hidden picture underneath. Player must click on the UFO number sentence first then click on the correct quotient in number board. If the player chose correctly, a piece of the picture is revealed.

- Divide 2-Digit Numbers Tutorial http://www.dositey.com/muldiv/divml1.htm
- Tutorial used to teach the traditional algorithm for division.

### Books and Curricula

- Elementary and Middle School Mathematics (2007) 6th Ed. By: John Van de Walle
- This book, particularly chapter 11, discusses children’s acquisition of basic facts fluency. The book recommends correlating mulitplication with division when teaching students to divide. The book also includes various activities and lesson concepts to use in the classroom.

- Making Sense: Teaching and Learning Mathematics with Understanding by Thomas P. Carpenter, James Hiebert, Elizabeth Fennema, and Karen C. Fuson (Paperback - April 21, 1997)
- This book presents a research-based framework for teaching mathematics with understanding and accessibility. The book covers several research study findings and discusses effective instructional strategies used to help student’s build mathematical understanding and competency.

- California Math Expressions Grade 4, Volume 1 (Curriculum Math Series) by Karen C. Fuson, Publisher: Houghton Mifflin
- This curriculum book, published by Houghton Mifflin, contains scripted lessons and materials to build students’ mathematical knowledge. The first unit of lessons focuses on multiplication and division.

## Annotated References

Anghileri, J. (2001). Development of Division Strategies for Year 5 Pupils in Ten English Schools. British Educational Research Journal, 27(1), 85-103.

Caliandro, CK. (2000). Children's Inventions for Multidigit Multiplication and Division. Teaching children mathematics, 6(6), 420-424.

Gregg, J. & Underwood, D. (2007). Interpreting the standard division algorithm in a “candy factory” context. Teaching Children Mathematics, 14(1), 25-31.

Hogan, B. (2005). Singapore Math: Problem-Solving Secrets from the World's Math Leader. Peterborough, New Hampshire. Crystal Springs Books.

Hurts, K. (2008). Building cognitive support for the learning of long division skills using progressive schematization: Design and empirical validation. Computers and Education, 50, 1141-1156.

Lee, J. (2007). Making sense of traditional long division algorithm. *Journal of Mathematics*, 26, 48-59.

Li, Y. (2001). Does the acquisition of mathematical knowledge make students better problem solvers? An Examination of Third Graders’ Solutions of Division-With-Remainders (DWR) Problems?. *Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*, Snowbird, UT. 501-508.

Long, Calvin T., & DeTemple Duane W. (2006). *Mathematical Reasoning for Elementary Teachers*. 4th ed. Boston: Pearson.

National Council of Teachers of Mathematics. (2008). *Illuminations*. Retrieved July 26, 2008, from: http://illuminations.nctm.orgs

Slesnick, T. (1982). Algorithmic skill vs. conceptual understanding. *Educational Studies in Mathematics*, 13(2), 143-154.

Van de Walle, J. (2007). Elementary and middle school mathematics: Teaching developmentally. Virginia: Allyn & Bacon

Young-Loveridge, J. (2005). Fostering Multiplicative Thinking: using array-based materials. Adelaide, South Australia. Australian Association of Mathematics Teachers.