Subtraction with Regrouping
Subtraction with Regrouping (Pedagogical Content Knowledge Project)
- By: Thanh Tran, Elizabeth Taing, Casey Emery (Hauser), Paris Shahabi (Pompa) (UCIrvine, August 2009)
Cognition and Learning Background: Subtraction with Regrouping
Subtraction is a major mathematical concept that is introduced as early as Kindergarten. Subtraction with regrouping is part of the second grade Mathematics Standard for the state of California. By the end of second grade, students are required to know how to find the difference of two whole numbers up to three digits long, and use mental arithmetic to find the difference between two two-digit numbers (Number Sense 2.2, 2.3).
Subtraction with regrouping requires students to keep track of mathematical processes mentally and to move and manipulate numbers across place value. Learning regrouping conceptually requires twice as much effort for students to learn (Ohlsson, S., et al, 1992, p. 443).
Students are often confused about their understanding in regrouping with subtraction. The academic language addressed in learning the concept of regrouping is conceptually misleading.
The term “regroup” has little meaning for students. An easier term teachers might want to substitute for regroup is “trade”. Using the term trade has more significant meaning to students. For example, saying ten ones are traded for a ten conceptually makes more sense to young students. Research has also shown that when given directions from teachers, students can successfully understand the procedure of using tens and ones places and place values. However, students fail to understand this procedure when asked to work independently. (Van de Walle, 2004).
Cognitive Obstacles and Common Misconceptions
There is a large emphasis on teaching students to correct methods of subtraction with regrouping. Students need to understand and be able to demonstrate certain strategies that involve place value and meanings of numbers (Fuson, K. C., el at., 1997, p. 131).
According to Haylock and Cockburn (1987), learning math requires a connective model in addressing symbols, language, pictures/images, and context/concrete experience. Each category is related and intertwined. It is important for the teacher to address the appropriate mathematical language and have students be able to concretely visualize the concept with manipulatives or images, and then have them relate it to their own experiences. Without addressing one component, the student will not be able to conceptualize the mathematical concept. Therefore, teachers need to plan strategically and incorporate all components so that the students can explore, analyze and understand the mathematical concept as it relates to them. Subtraction with regrouping requires the student to learn the language of regrouping by using manipulatives to visually demonstrate place values and then having them manipulate the manipulatives in borrowing tens and adding ones. From there, students will learn and use inventive ways in solving problems as they relate it to their own experiences. In the book, Teaching Children Mathematics, the authors present a case study in which a 2nd grade teacher, shows her student how to use ones and tens interchangeable to develop mathematical reasoning by using a real life example of her Aunt Mary’s candy business. The students were able to understand the flexibility of addition and subtraction by coming up with inventive ways of packing and distributing the candy (Whitenack et al, 2001).
Initially, students have learned major mathematical concepts through the use of manipulatives (i.e. base ten blocks, unifix cubes, number lines, etc.). Ideally, the overall goal for students is to eventually be able to recognize place value without counting and recognize or construct two digit quantities using tens and ones (Fuson, K. C., et al., 1997, p. 134).
In some Asian countries such as China, students are taught to focus on making “tens” with numbers as a support for learning mathematical concepts such as addition and subtraction. For example, when saying the number 53, students in China say “five then three”. Slight irregularities fall for digits that begin with the number 1. They do not say the one. For instance, the number 12 is said “ten two” (Fuson, K. C., el at., 1997, p. 141).A common misconception students have when regrouping with subtraction is that they subtract the smaller number from the larger number. Due to the nature of subtraction, students “add one column and subtract the others” or try to use another incorrect strategy, resulting in a violation of some mathematical solution (Leinhardt, G., 1987, p. 227)
When breaking down the theoretical background of subtraction and regrouping, it is evident that students have some mental structures that help them use specific math procedures to solve the problems. As suggested by Hiebert et al •1996) “Children might acquire procedures by (a) inventing new procedures, either by creating them or by adapting known procedures to solve new problems; or (b) adopting procedures that are demonstrated by other” (Hiebert, J., Wearne, D., 1996, p. 252). Therefore, teachers need to gear their instruction towards conceptual understanding rather than procedural understanding or memorization techniques.
Berman and Freiderwitzer (1981) state “the vocabulary used by the teacher to describe the algorithmic procedure should be able to express a mathematical concept verbally before they are expected to use it in writing” (p. 109). Teachers need to align the use of academic language with the proper materials as a tool to increase mathematical understanding. It is important for teachers to help students bridge the gap between understanding the physical act or manipulating objects, using proper academic language, and the notion of abstract thinking in order to create an overall understanding of mathematical concepts.
Pedagogical Tools and Strategies
Subtraction with Regrouping Teaching Methods
A lack of understanding underlying concepts in mathematics may cause students to have difficulties learning the concept of subtraction with regrouping. Research-based recommendations have suggest teachers to use strategies to improve students’ knowledge of addition, place value, part-whole thinking and using visual representations to increase their understanding of subtraction with regrouping.
Flores (2009) has suggested teachers to show concrete representational abstract mathematics fact instruction by having students follow a series of steps. These steps include:
1) Discover the sign
2) Read the problem
3) Answer or draw and check
4) Write the answer
Teachers should follow an instructional sequence that involves three phases. The first phase suggests teachers to use manipulatives to demonstrate concept with guided and independent practice. The second phase encourages teachers to uses the same model that involved illustration of the mathematical process by using pictures to represent numbers. The third phase encourages students to use a mnemonic strategy to learn the process and procedure for operation (Flores, 2009).
The traditional method of teaching addition followed by subtraction proves to be more difficult for students, requiring more instructional time, or even failure to acquire group achievement. Research by Wiles, Romberg and Moser (1973) suggests that the most effective sequence to teach subtraction is to use an integrated sequence. According to Wiles, Romberg, and Moser (1973), an integrated sequence may be more effective because:
1. Instruction to this point in the experience of these children had emphasized the introduction and development of addition and subtraction together.
2. The regrouping associated with the addition algorithm as carrying is the reverse of the regrouping associated with subtraction as borrowing.
3. Addition and subtraction of whole numbers are both counting processes.
4. Addition and subtraction of a number as mathematical operations are inverses.
5. Expanded notation forms the basis of the development for both algorithms
6. The common mechanical characteristics of “begin at the right” and vertical form suggest a parallel development of the algorithms.
Other research has suggested teachers to teach the place values of ones and tens. Students who understand the relationship between place values can transfer this to learning subtraction with regrouping (Flores, 2009, p.232). Furthermore, Thompson (2000) has suggested that having students identify what number represents which value gives them a better mathematical image. For example, in the number 575, the 7 represents 70, rather than confusing the student with place value academic language and saying ‘the 7 is in the tens column’ (Thompson, 2000) To further understand subtraction with regrouping, students should use manipulatives, pictures, words, numerals to communicate their thinking to others (Campbell, Rowan & Suarez, 1998). One way to use manipulatives is to encourage students to substitute and use representation of both physical materials, such as base 10 blocks, and written numbers for a given problem. This strategy helps students develop solutions using both representations.(Hiebert & Wearner, 1996)
Base Ten Concept
A pedagogical tool that teachers can use to help build the concept of regrouping with subtraction is implementing academic language into their mathematical lessons. Using a visual representation with the academic language may provide the students with a concrete representation of understanding multi-digit numbers and regrouping. The examples below are taken from Houghton Mifflin’s (2008) Math Expressions 2nd grade eglossary.
Curricula and Technological Resources
Teachers can use technological resources to supplement their lessons on subtraction with regrouping. Math games can be used as a method to build a foundation of understanding mathematical concepts. The curricula and technological resources listed below are provided as a tool for promoting and supporting student learning. These resources include websites for teachers as well as online games and activities for students. These educational mathematical games can help students develop routine skills in understanding place values, mental arithmetic and regrouping in relation to subtraction. Each site provides varies ways in showing how regrouping occurs using variables, diagrams and mathematical expressions.
1. Illuminations: http://illuminations.nctm.org/
2. Mind Institute Research: http://www.mindresearch.net/
3. National Library Of Virtual Manipulatives: http://nlvm.usu.edu/
4. National Council of Teachers of Mathematics (NCTM): http://my.nctm.org/eresources/school_level.asp?lv=1
1. Funbrain: http://www.funbrain.com
2. Kidsites: http://www.kidsites.com/sites-edu/math.htm
3. Dositey: http://www.dositey.com/2008/addsub/sub2digr.html
From Class Brain:
Berman, B. & Friederwitzer, F.J. (1981). A Diagnostic Prescriptive Approach to Remediation of Regrouping Errors. The Elementary School Journal, 82 (2), 109-115.
Summary: This article points out common errors in addition and subtraction regrouping that students encounter. The authors suggest instructional strategies for teachers to approach and interact with their students to learn the concept of regrouping.
Campbell, P., Rowan, T., & Suarez, A. (1998) What Criteria for Student-Invented Algorithm. The Teaching and Learning of Algorithms in School Mathematics, 49-55
Summary: This article suggests teachers to use different teaching and learning styles, such as using manipulatives, to help students further understand subtraction with regrouping.
Flores, M. (2009). Teaching Subtraction with Regrouping to Students Experiencing Difficulty in Mathematics. Preventing School Failure. 53(3), 145-152.
Summary: This article focuses on different concrete ways to address mathematical instruction.
Fuson, K.C., Wearne, D., Hiebert, J.C., Murray, H.G., Human, P.G., et.al. (2009). Children’s Conceptual Structures for Multidigit Numbers and Methods of Multidigit Addition and Subtraction. Journal for Research in Mathematics Education. 28(2), 130-162.
Summary: This article focuses on understanding multi digit addition and subtraction problems. The authors suggest using conceptual models for 2-digit numbers.
Haylock, D., & Cockburn, A. (1989) Understanding Early Years Mathematics, New York: Sage Publications
Summary: This book is designed to help teachers develop a greater understanding of mathematics and addresses the connective model of learning mathematics.
Hiebert, J., & Wearne, D. (1996). Instruction, Understanding, and Skill in Multidigit Addition and Subtraction. Cognition and Instruction,14, 251-283.
Summary: This article discusses children’s understanding of multidigit numbers compared to their computational skills. They suggest strengthening conceptual understanding of mathematical knowledge.
Houghton-Mifflin (2008). Education place: Grade 2: eglossary. Retrieved July 25, 2009, from http://www.eduplace.com/kids/mthexp/g2/#
Summary: This website is a supplement to Houghton-Mifflin's Math Expressions textbooks. It provides a visual glossary and math expressions of mathematical terms.
Leinhardt, G. (1987). Development of an Expert Explanation: An Analysis of a Sequence of Subtraction. Cognition and Instruction. 4(4), p. 225-282.
Summary: This article provides insight and misconceptions in teaching subtraction. It also discusses inventive strategies that teachers can use to correct these misconceptions
Ohlsson, S., et al. (1992). The Cognitive Complexity of Learning and Doing Arithmetic. Journal for Research in Mathematics Education. 23(5), 441-467.
Summary: This article explains the term regrouping and its importance in mathematics.
Thompson, I. (2000). Teaching place value in the UK: time for a reappraisal, Educational Review, 52(3), 291-298.
Summary: This article discusses the importance of teaching place values to students.
Van de Walle, J.A. (2004). Elementary and Middle School Mathematics: Teaching Developmentally. Boston: Pearson Education, Inc.
Summary: This textbook contains reflections on years of practical classroom experience that has shaped mathematics curriculum in elementary school.
Whitenack, J., Knipping, N., Novinger, S., & Underwood, G. (2001) Second graders circumvent addition and subtraction difficulties. Teaching Children Mathematics. 8(4), 228-233
Summary: This case study shares a strategy that one teacher used to promote mathematical reasoning and flexibility with tens and ones addition and subtraction.
Wiles, C.A., Romberg, T.A., & Moser, J.M. (1973). The Relative Effectiveness of Two Different Instructional Sequences Designed to Teach the Addition and Subtraction Algorithms. Journal for Research in Mathematics Education. 4 (4), p. 251-262.
Summary: This article discusses effective sequences to teach subtraction.