Topic (Pedagogical Content Knowledge Project)
- by Leticia Acosta, Boram Lee, Grace Rosa, Haley Shimizu, Dorothy Tzeng (UCIrvine, August 2010)
Cognition and Learning Background: Rethinking Regrouping
Regrouping seems to be particularly challenging for many students, both conceptually and procedurally. We chose this topic as our PCK project in order to seek out more effective instructional strategies, activities, and curricula in the hope of helping students more easily, and more deeply, understand the skills involved in solving mathematical problems which require regrouping.
What makes regrouping so challenging? Two of the main reasons are:
- It is an abstract skill.
- What does it really mean to regroup? How can we expect students to be able to regroup when the term itself has no real meaning for them?
- It is a complex skill.
- Regrouping requires students to keep track of several mathematical processes all at once and to move/manipulate numbers across place values.
- According to Ohlsson, et al. (1992), learning regrouping conceptually requires approximately twice as much effort as any other condition.
Van de Walle (2004) noted that children are often able to disguise their lack of understanding in the area of regrouping by following directions, using the tens and ones places in prescribed ways, and using the language of place value. This is evident by what we have seen in the classroom – students can successfully complete the procedure, using the appropriate academic language, when the teacher is guiding them, but when left to work independently they are unable to produce the same results.
Addressing Academic Language
According to Van de Walle (2004), the terms carrying and borrowing are obsolete and conceptually misleading. The word regroup also has little meaning for young children. Van de Walle argues that a preferable term is trade: ten ones are traded for a ten; a hundred is traded for 10 tens. He also notes that if you develop traditional algorithms in a problem-solving manner, there is no reason why students should not be expected to understand them. Using a problem-solving manner causes students to create their own understanding of the procedure, which in turn will create personal meaning behind the academic language.
Berman & Freiderwitzer (1981) claim that the use of manipulatives (modeling) encourages the development of oral language skills in relation to mathematics. The authors aver that children should be able to express a mathematical concept verbally before they are expected to use it in writing. In addition, they state that the vocabulary used by the teacher to describe the algorithmic procedure should directly indicate what is actually occurring when manipulatives are used. A student may not inherently be able to make a specific association between the physical act of manipulating objects, the proper language, and the abstract notation.
Regrouping with CRA
The Concrete-Representational-Abstract (CRA) sequence is an intervention program geared towards students with mathematics learning disability. This program enhances student learning and retention of conceptual knowledge through three specific steps. When teaching regrouping using CRA, the teacher will move from concrete models of regrouping (10 rods and units), to picture diagrams, and finally to an abstract understanding of regrouping using mathematics symbols to enhance fluency of basic computation skills (Flores, 2009). The goal of moving from concrete counters to more abstract depictions for teaching regrouping is for students to be able to move away from using concrete representations, such as counters, and be able to use various materials to represent tens and one without counters (Carpenter, Franke, Jacobs, Fennema & Empson, 1998).
Research conducted by Mercer and Miller (1993) shows that students being instructed in basic computation knowledge using the CRA sequence outperformed students taught with traditional curriculum. Students using CRA were also able to transfer basic computation understanding to solve simple word problems. According to Miller and Mercer (1993), using CRA instruction has shown to be effective in increasing mathematics computation ability for remedial students. Research implies that reliance on CRA instruction may greatly improve retention and fluency of mathematics facts. Implementation of the CRA sequence for teaching regrouping may be beneficial for all students.
Cognitive Obstacles and Common Misconceptions
(Berman & Freiderwitzer, 1981)
Where do these misunderstandings come from?
- Lack of experience with manipulation of concrete objects
- Inability to make specific association between physical act of manipulating objects, academic language, and abstract notation
- Vocabulary employed by the teacher may not have indicated what was actually occurring as materials were manipulated
- No consideration was given to the preferred learning style of the child. The student may have required one or more additional stimuli than offered by the teacher
- Premature transition may have been made to paper-and-pencil activities when the student required further active manipulation of materials
- Student may demonstrate rote learning of the algorithm without actual understanding of what is being done
(Fuson & Briars, 1990)
- Preaddition/Presubtraction error: Columns left blank or filled in with random numbers; preaddition errors can include subtracting of numbers rather than adding, and vice versa
- Column addition/subtraction error: Addition/subtraction problems approached column by column (In addition, the sum of each column, disregarding place value, e.g., 28+36=514; In subtraction, the smaller number in each column is subtracted, e.g., 36-28=12)
- Trading error: A partially successful attempt to trade (carry, borrow)
- Fact error: Incorrect adding or subtracting in a column
- Alignment error: Failure to align multidigit numbers according to their place value
(Miller & Mercer, 1997)
An educational factor that contributes to student misconception and errors in regrouping can be attributed to poor curriculum and instruction. Basal mathematics programs that are frequently used in elementary schools do not promote conceptual understanding and problem solving of the mathematical concepts due to its superficial coverage of many different concepts and skills in a short amount of time. It is difficult for students to grasp the concept of regrouping because of the common misconceptions that are listed above. For such an abstract concept like regrouping, students should be taught first at a concrete level and gradually advance to make connections at the representational and abstract level of understanding. The Concrete-Representational-Abstract (CRA) Instructional Approach is a great method than can be used to promote student retention of conceptual knowledge. This gradual progression of learning may help students reduce misconceptions and errors that are so frequently made.
Pedagogical Tools and Strategies
Promising Instructional Approaches
Take into account the Instructional Sequence (Wiles, Romberg, & Moser, 1973)
- Traditional sequence of addition followed by subtraction vs. Integrated presentation of addition and subtraction tasks
- Addition algorithm is easier to learn than the subtraction algorithm. Regrouping is a major difficulty for both operations, but poses more of a problem for subtraction than addition.
- Studies show that children experienced more difficulty with the traditional sequence than the integrated presentation
- It is suggested that addition and subtraction algorithms be introduced and developed together because:
- addition and subtraction algorithms are related in terms of behaviors and I in the relationships in the mathematics involved
- addition and subtraction of whole numbers are both counting processes
- addition and subtraction of a number as mathematical operations are inverses
- expanded notation forms the basis of the development for both algorithms
- the common mechanical characteristics of “begin at the right” and vertical form suggest a parallel development of the algorithms
Diagnostic-Prescriptive Approach (Berman & Friederwitzer, 1981)
- “Most promising” approach
- Suggested to be used as a routine part of math instruction, because this approach permits the teacher to discover or correct student errors before they become ingrained over time
- Observe students’ thinking processes/strategies
- Address misconceptions/Diagnose errors
- Listen to student explain how example was done
- Have student display a problem with manipulatives (observe each step of algorithm and listen to oral explanation)
- Ask student to show the problem as a picture, and then explain picture
- Prescribe corrective activities
- Diagrams/Place value charts
- Use of structured materials and manipulative models
- Students write algorithm in conjunction w/ their manipulation of materials
Concrete Representational Abstract Approach (Flores, 2009, 2010; American Institute for Research, 2004 & Thompson, P. W. 1992)
Integrating a Concrete-Representational-Abstract (CRA) method is an alternative way to transition students from a concrete representations of addition and subtraction to a more abstract concept that involves regrouping. The CRA process is a scaffolded approach that has shown to improve mathematical competence with struggling students. The three-step instructional strategy involves transitioning from one stage of understanding to another (i.e. concrete understanding -> representational understanding -> abstract understanding ). The following will represent an overview of the instructional process:
1. Concrete Understanding Stage:
a. A teacher begins instruction by modeling a specified mathematical concept using concrete materials such as base ten blocks, colored chips, pattern blocks fraction bars, geometric figures or other forms of manipulatives.
2. Representational Understanding Stage:
a. The teacher transitions students from their concrete understanding of a mathematical concept by modeling the same concept from the concrete stage but using less tangible materials to represent the content. For example, the following representational tools may be used: illustrations of circles, dots, or tallies.
3. Abstract Understanding Stage:
a. This is the desired stage of achievement by the student. At this stage the teacher will use the representations to transition students to understand the symbols involved in the mathematical content the students are learning (+, -, =, ÷, x). The teacher will model how to use the symbols first and then have the students practice.
CRA- with regrouping example
• The following figure represents how to teach regrouping at the Concrete level, which involves the teacher modeling and guiding how to represent subtraction with regrouping using manipulatives. Later, this process will be practiced independently by the students.
a. Inserting and labeling the ones column and the tens column and translating the subtraction problems, such as “32 – 15 = means 3 tens and 2 ones minus 1 ten and 5 ones equals how many”
b. Using base-10 manipulative blocks to represent the minuend (a problem such as “32 – 15” was represented with 3 tens blocks and 2 ones blocks)
c. Solving the problem by regrouping the manipulatives in the tens place (removing 1 tens block from the minuend and adding 10 ones blocks to the existing 2 ones blocks within the minuend) and subtracting the appropriate number of blocks based on the subtrahend. After three lessons with the student’s achieving 80% accuracy or better, instruction would progressed to the representational level.
• This figure examples how to transition from the concrete usage of blocks to a more representational version of regrouping using tally marks.
• The same process would be used as above except that this time the manipulatives will be represented by illustrations (tally marks).
• The ones are represented as small vertical tallies, written on a horizontal line, and the tens are represented using long vertical lines. Regrouping in the tens place is represented by circling one of the long vertical lines and adding 10 small tallies to the horizontal line. Subtraction in the ones place is represented by circling the appropriate number of lines or tallies, based on the subtrahend.
• After the student achieves at least three lessons with 80% accuracy on independent lesson tasks, the DRAW (Discover the sign, Read the problem, Answer or draw and check, Write your answer) strategy would be introduced and modeled by the teacher. Once the student can independently use the strategy to solve problems with at least 80% accuracy and they could recite the steps accurately the could move to the abstract page.
The next phase of instruction is the abstract level, in which the students are encouraged to answer problems from memory, rather than use drawings, but they can use the DRAW strategy. The last phase of instruction involves fluency activities in which the student are given 2 minutes to complete a sheet containing subtraction problems.
Curricula and Technological Resources
Curricula and Technological Resources
Teachers can use technological resources to supplement their classroom lessons on addition and subtraction using regrouping. Teachers can use the following technological tools, which include online math games and activities for students, as well as websites for teachers. Berlin and White argue that technology can also be used in the assessment of students’ mathematical processing (1995), therefore the following tools are not only beneficial for providing students with the necessary practice to master the skill of regrouping, but they can also be used to assess the students. Garofalo, Drier, Harper, Timmerman, and Shockey also argue that in order for teachers to teach mathematics with technology effectively, they need to be comfortable with the technology themselves (2000). Therefore, teachers should take the time to familiarize themselves with the following tools in order to use them effectively in their classroom and to their full potential.
- The K–12 Mathematics Curriculum Center http://www2.edc.org/mcc/about/default.asp
- Elementary Education Resources: Mathematics • http://www.pitt.edu/~poole/eledmath.html
- Apples For The Teacher • http://www.apples4theteacher.com/math.html
- Math Goodies • http://www.mathgoodies.com/lessons/
- TeAchnology • http://www.teach-nology.com/teachers/lesson_plans/math/
- Teacher’s Place • http://www.mathforum.org/teachers/
- Abc Teach • http://www.abcteach.com/
- Ed Helper • http://www.edhelper.com
- National Library of Virtual Manipulativs • http://nlvm.usu.edu/
- Fun Brain • http://www.funbrain.com/
- Cool Math 4 Kids • http://www.coolmath4kids.com/
- Math Playground • http://www.mathplayground.com/
- Project Interactive • http://www.shodor.org/interactivate/
- Brain Pop Jr. • http://www.brainpopjr.com/
- Primary Games • http://www.primarygames.com/math.htm
- Gamequarium • http://www.gamequarium.com/math.htm
- A Plus Math • http://www.aplusmath.com/games/index.html
- Learn 4 Good • http://www.learn4good.com/kids/index.htm
The following image comes from Cool Math 4 Kids:
Following images from Brain Pop Jr.:
Berlin, D. F., White, A. L. (1995). Using Technology in Assessing Integrated Science and Mathematics Learning. Journal of Science Education and Technology. 4 (1), 47-56.
Berman, B. & Friederwitzer, F.J. (1981). A Diagnostic Prescriptive Approach to Remediation of Regrouping Errors. The Elementary School Journal, 82 (2), 109-115.
Bruner, J. (1966). Toward a Theory of Instruction. Cambridge: Belknap Press.
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction, 29(1), 3-20.
Dositey Corporation. 1998-2005. “Adding 2-Digit Numbers with Regrouping: Mini Lesson.” Dositey. Retrieved August 10, 2006, from the World Wide Web: http://www.dositey.com/addsub/add2rbasic.html
Dositey Corporation. 1998-2005. “Addition with Regrouping 1 - Step-By-Step Exercise.” Dositey. Retrieved August 10, 2006, from the World Wide Web: http://www.dositey.com/addsub/add2rbasic.html
Dositey Corporation. 1998-2005. “Subtracting 2-digit numbers with regrouping Mini-lesson.” Dositey. Retrieved August 10, 2006, from the World Wide Web: http://www.dositey.com/addsub/add2rbasic.html
Dositey Corporation. 1998-2005. “Subtraction with Regrouping Step-by-step Exercises.” Dositey. Retrieved August 10, 2006, from the World Wide Web: http://www.dositey.com/addsub/add2rbasic.html
Flores, M. M. (2009). Teaching subtraction with regrouping to students experiencing difficulty in mathematics. Preventing School Failure 53(3), p. 145-52.
Flores, M. M. (2010). Using the concrete-representational-abstract sequence to teach subtraction with regrouping to students at risk for failure. Remedial and Special Education 31(3), p. 195-207.
Fuson, K.C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education 21(3), 180-206.
Garofalo, J., Drier, H., Harper, S., Timmerman, M.A., & Shockey, T. (2000). Promoting appropriate uses of technology in mathematics teacher preparation. Contemporary Issues in Technology and Teacher Education [Online serial], 1 (1).
Harcourt School Publishers E-lab. 2006. “Subtraction of Three-Digit Numbers. Harcourt School Publishers.” Retrieved August 10, 2006, from the World Wide Web: http://www.hbschool.com/activity/elab2004/gr3/4.html
Hooks, Candace. 1996-2006. “To Regroup or Not to Regroup?.” Lesson Plans Page. Retrieved August 10, 2006, from the World Wide Web: http://www.lessonplanspage.com/MathAddition-RegroupOrNot23.htm
Leinhardt, Gaea (1987). Development of an Expert Explanation: An Analysis of a Sequence of Subtraction Lessons. Cognition and Instruction 4(4), p. 225-282.
Miller, S. P., & Mercer, C. D. (1993). Using data to learn about concrete–semiconcrete–abstract instruction for students with math disabilities. Journal of Learning Disabilities Research and Practice, 8, 89–96.
Miller, S. P., & Mercer, C. D. (1997). Educational aspects of mathematical disabilities. Journal of Learning Disabilities 30(1), 47-56.
National Council of Teachers of Mathematics (NCTM). 2000-2006. “Ten Frame.” Illuminations. Retrieved August 10, 2006, from the World Wide Web: http://illuminations.nctm.org/ActivityDetail.aspx?ID=75
Ohlsson, S., et al. (1992). The Cognitive Complexity of Learning and Doing Arithmetic. Journal for Research in Mathematics Education. 23(5), 441-467.
Piaget, J. & Inhelder, B. (1969). The Psychology of the Child. New York: Basic Books.
Rivera, D.M. Smith, D.D. (1987). Influence of Modeling on Acquisition and Generalization of Computation Skills: A Summary of Research Findings from Three Sites. Learning Disability Quarterly. 10 (1), p. 69-80.
The Acess Center, (2004). Conrete – Representational – Abstract Instructional Approach Summary Report. Doing What Works. Retrieved from, http://dww.ed.gov.
Thompson, P. W. (1992). Notations, conventions, and constraints: Contributions of effective uses of concrete materials in elementary mathematics. Journal for Research in Mathematics Education 23, 123–47.
Utah University. 1999-2006. “Base Blocks.” National Library of Virtual Manipulatives. Retrieved August 10, 2006, from the World Wide Web: http://nlvm.usu.edu/en/nav/frames_asid_209_g_1_t_1.html?open=activities
Utah University. 1999-2006. “Chip Abacus.” National Library of Virtual Manipulatives. Retrieved August 10, 2006, from the World Wide Web: http://nlvm.usu.edu/en/nav/frames_asid_209_g_1_t_1.html?open=activities
Van de Walle, J.A. (2004). Elementary and Middle School Mathematics: Teaching Developmentally. Boston: Pearson Education, Inc.
Vygotsky, L.S. (1978). Mind in Society: The Development of Higher Level Psychological Processes. Cambridge: Harvard University Press.
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.