1 Cynthia Ashton Math Professor Beck Term Paper Due: Friday 14 December 2012 Multi-digit Multiplication: Teaching methods and student mistakes Abstract: Description of the paper. Why is it important/interesting, what are your findings? In this paper I will address the importance of conceptual understanding of mathematics regarding algorithm of multi-digit multiplication. Often times in mathematics, students tend to believe the purpose of solving a mathematical equation is to find the answer, through the procedure of the algorithm. In solving a mathematical equation, it is more valuable for the intellectual development of the students by understanding algorithms conceptually. Through my research I discovered the importance of algorithmic procedure in which is not limited in one method, but can have many different algorithms to solve the same equation. Within my research I've also found the importance of mental algorithms rather than written formal algorithm tactics, in regards to the impacts of a student s ideology of mathematical proofs. Students have a difficult time of conceptually understanding the explanations of multi-digit multiplication. Regarding to the book Knowing and Teaching Elementary Mathematics by Liping Ma, the issues regarding multi-digit multiplication is that "students often forget 'to move the numbers' over each line" (Ma, 28). Ma is referring to the "caring out the lining-up procedure algorithm, which most Americans were taught with, but don t conceptually
2 understand the reason for this method is because of place value for the digit. Ma also points out that the U.S. teaching methods vary from procedural methods to conceptual explanations, where she compares Chinese teaching methods (Ma, 51). Within this term paper I will address the importance of conceptual teaching methods regarding multi-digit multiplication. I will also touch bases of the significance of algorithms tied to Ma s thematic proofs that relate to multiplication. This topic was interesting to me because it explained why students in the U.S. struggle with math, it is because students lack of conceptual understanding because student learn algorithmic procedure to solve equations. Introduction: Issues/problem, significance, solution Recently when I was doing my multi-digit multiplication equation for my Math Concept course, I was questioning if 200 x 400 is 8000 or 80,000. Students often have an issue of understanding the concept of place value while solving a multi-digit multiplication equation. Because of this, students place numbers within the wrong column or wrong location of the place value that leads to the incorrect answer. Liping Ma addresses the issue from the styles of teaching methods between teachers of the United States and China. Teachers in the United States practice more on a procedural teaching method compared to the teachers in China practice conceptual teaching methods. These two types of methods are significant to a student s understanding in algorithms, mental computation and mathematical proofs. If students don't have a basic comprehension of the basic mathematics knowledge, students will struggle with higher levels of mathematics. Conceptual teaching is the solution for successful multi-digit multiplication, which will lead to further success for the student s mathematical career
3 and daily math problems within life itself. Method: How you went about searching info or data; impression of quality I used Laping Ma's book as a base for my term paper. Constantly referring to her terms and topics within my search on San Francisco State University's library catalog and articles & databases. Many references went into too much detail of the analysis of topics like mathematical proofs, algorithms, multi-digit multiplication, and place value. My main focus is on conceptual teaching where many articles did not reflect the subject matter of mathematics. I've decided to use the support of mathematical proofs and algorithms to further rationalize the importance of conceptual learning. Paragraph #1 The Issue Teaching multi-digit multiplication by procedural methods does not give the student proper understanding of place value and the distributive law. 70% of United States teachers said the problem was incorrect lining-up procedure, whereas the 30% concluded that the students did not understand the rational of the algorithm (Ma, 29). In the school of mathematics, an algorithm is a systematic method used for solving an equation. By following the procedure one can solve a mathematical equation that gives an output correctly or announce that it cannot be solved (Morrow, 21). In the school of mathematics algorithms gives procedure to solve an equation, yet the equation can have many different kinds of algorithm that will result in the same output. Within the issue of multi-digit multiplication Ma focuses on the algorithm of "caring out the lining-up procedure"(ma, 29). In trying to calculate:
4 27 X 24 Students seem to be forgetting to "move the numbers" (I.e. the partial products) over on each line (Ma, 28). Students carried out the equation like this: Instead of this: 27 X X Within Ma's study all the teachers agreed that it is a "problem of mathematical learning rather than a careless oversight" (Ma, 29). All teachers agreed that there was a problem with the learning comprehension for the students, which is a direct refection of the teaching methods of the teacher. Even though teachers have difficulty teaching multidigit multiplication and notices similar mistakes every year when it is taught, teachers don't seem to take much action in changing teaching methods in order to provide better comprehension of the subject matter for the students. The carrying out the lining up algorithm is taught with a procedurally directed method, refers the term "place value" as the location of the numbers (Ma, 29). The procedurally directed approach "verbalized the algorithm so it can be carried out correctly" yet by doing these teachers are not providing the understanding of the importance of the definition of true place value (Ma, 29).
5 Although teachers used other methods like using lined paper or a grid to position the zeros in the placeholder, teacher merely suggested to place the numbers correctly (Ma, 34). The term place value was not introduced to students as a mathematical concept, but as labels for columns where they should put numbers (Ma, 34). By putting artificial value into the meaning of place holder students lack an understanding of place value. With multi-digit multiplication students need to comprehend the mathematical terms in order to solve with understanding of the algorithm(s). By defining these mathematical terms students will have a conceptual understanding of these terms that will help their future academic success with mathematics. Paragraph 2: Algorithms and Multi-digit Multiplication In mathematics, algorithms are executed to announce the correct answer for the equation. For multi-digit multiplication I found three different formal written algorithms besides the "caring out the lining-up procedure" method that Ma heavily focused within her book. Nonetheless within the classroom students learn the formal written algorithmic procedures, yet mental computation strategies are just as valuable. According to Marrow, mental computation is solving mathematics equations, which are solved by a student s personal choice of strategy that is based on the numbers given (44). One can say it is the mathematical solving method that a student solves "in their head. This gives flexibility along with the required understanding of algorithm. Marrow further states the value of mental computation: Those children have powers of mental computation that
6 they develop themselves with out support. The heart of all these self-devised algorithms is that the child tries to turn a difficult calculation into an easy one: is hard, is easy; 9x17 is hard, 10x17-17 is easier. The more able children develop their ability to devise algorithms and so make life easy. Unfortunately the less able children are less capable of helping themselves and so are left with much more difficult task (46). Mental-arithmetic procedure allows students to realize the different ways of solving an equation mentally instead of limiting students to one algorithmic procedure. Formal written algorithms are helpful for visual learners. Yet others argue "teaching formal algorithms can be counter productive" (Marrow, 44). Regarding multi-digit multiplication Marrow introduces four algorithms. I will go into details with three of them. First formal written algorithm Marrow introduces is the direct modeling strategy, which is the same as Professor Beck introduced to his Math Concept course (Marrow, 152; Notes, 10/10/2012). The direct modeling, sorts the by tens and ones to create simplicity for the multi-digit multiplication equation as long as making it visual for the students (Marrow, 153). For example 28X12 is modeled out like this: will be place in the digit s proper place value outside the boxes, 20 & 8 or 2 tens and 8 ones. 12 will also be placed in their proper place value outside the boxes which will be 10 & 2 or 1 tens and 2 ones. Then one will proceed to solve the equation by multiplying 20x10= 200 and will fill it in within the box. The steps will apply to fill each box like this:
7 Finally one will use addition to add all the results like this: =336 Given to final answer to the equation, which are 336. By using the direct modeling method student can practice breaking up the number by place value, which helps them conceptually understand place value. This also makes it easier to solve the equation because the numbers are whole or end in a 0. The second formal written algorithm that Marrow introduces is the complete number strategy. The complete number strategies are based on repeated addition, which is an adding procedure by doubling the multiplicands (Marrows, 152).
8 For example the equation 28x12 will be solved like this: Complete number strategies are repeated addition or doubling (Marrow, 153). This method shows the student exactly what multiplication does, which is repeated addition. By doing this method students can visually see that 28X12 is = 336. The third algorithm Marrow introduced is the partitioning number strategy, which is spit the multiplicand or multiplier into two or more numbers and create multiple sub-problems that are easier to deal with (152).
9 For example the equation 177x3 will be solved like this: This allows students to make the digits easier by grouping them into simpler numbers. All of the three of Marrow's algorithms did not mention within Ma's issue regarding the "caring out the lining-up procedure" algorithm. Along with Marrow not mentioning
10 anything regarding place value and the distributive property. Perhaps it is because he too is procedurally explaining the rational of the algorithm and not emphasizing on the conceptual teaching. It is important that students understand the concepts by being able to identify the terms of mathematics. Regardless to say that it is important for student to learn the different ways of solving the problem in order to find our the student's preferred method by experiences more than one algorithm. Marrow does introduce the importance of mental computation, which is branch to the importance of mathematical proofs. Paragraph 3: Extent of the Problem: Mathematical proofs and importance of Multiplication Mathematical proof has a cognitive and social-cultural impact upon the student s development. Referring to the article Toward comprehensive perspectives on the learning and teaching of proof, the article refers to the historical-epistemological, cognitive, and instructional-social-cultural of mathematical proof in which directly reflects and relates to the importance of multiplication. According to the article, proofs establish the truth for a person and/or a community (Harel, 3). Mathematical algorithms give precise systematic methods that announce an output. These algorithms announce mathematical proofs. The concept of proof consecutively gives an individual assertion that who is uncertain of truth that then becomes a fact that the individual is certain of the truth (Harel, 6). The ideology of proofs, justifies the importance of mathematics and how it applies to our daily lives. This proving process removes doubts of truth of the assertion (Harel, 6). This sense making builds one's confidence in assertion but also influences others by leading to a social practice of persuading (Harel, 8). Proof gives students truths, and with
11 multiplication students can continue to exercise these truths throughout their academic career and daily life. Morrow further states: Understanding multiplication is central to knowing mathematics. The curriculum and evaluation standards for School Mathematics (National Council of Teachers of Mathematics 1989) proposed that children need to develop meaning for multiplication by creating algorithms and procedures for operations (Morrow, 151). Multiplication is considered to be the core of mathematics. According to Jerama, the times tables is considered to be the building blocks for the higher schools of mathematics (Jarema). Mental computation is directly connected to proofs, which is used in our daily lives unconsciously because they are not traditional written algorithms often used within a classroom setting. When solving a multi-digit multiplication problem students need to understand the importance of place value in order to carry the algorithmic procedure correctly this reflects the concept of proof which is also linked to mental computation. The teaching method of perceiving these algorithms impacts the student s knowledge of multiplication. Paragraph 4: Conceptual vs. Procedural i.e. solution A student should learn the mathematical algorithm that justifies and explains the output of the equation. Ma compared U.S. teaching methods to China's. Teachers in China introduced multi-digit multiplication first as distributive property as the process of transformation (Ma, 40).
12 First the teacher will compose the problem into three smaller problems like this: 127x249= 127x( ) = 127x x40+127x9 = = Then the teacher will state that the transformation accomplish by teh distributive law (Ma, 40). With the power of distributive law, 249 is broken down to 2 hundreds, 4 tens, and 9 ones. By doing this teachers are emphasizing the importance of place value. Next the teacher will rewrite the equation into columns: 127 x The three partial products are represented in the form of columns (Ma, 41). The teacher then ask students to observe the zeros then erases the zeros which then the equation will appear to be a staircase like this: 127 x Referring to Ma, the features of a mathematical argument- justification, rigorous
13 reasoning, and correct expression- were reflected throughout [her] explanation (41). The bridge of connection presents conceptual methods of teachers from China makes the algorithm into a true proof which makes it easier for the student to absorb and comprehend what multi-digit multiplication does. Going into further explanation, the teachers justify why the zeros can be erased through thorough explanation of place value. When using distributive law, this allows breaking down the numbers. Marrow stated earlier that multiplication is repeated addition. With multi-digit multiplication, it is also repeated addition but on a larger scale in which place values identifies the size of the number. For example the number 325, the 5 represents the ones place, the 2 represents the tens place, and the 3 represents the hundredths place. Also commonly can be written out as: Or verbally can be read as: 3 hundredths, 20tenths, and 5 ones. 300 can also be read at 30 tens or 300 ones. By understanding how to deal with several tens, students will be able to grasp the concept of what digit at a certain place stands for (Ma, 44-45). Students cannot get a thorough understanding of place value in one day (Ma, 45). Like any subject matter students need to be reminded of the concepts and teacher should observe student s mistakes and repeat explanation of the concepts as needed. By exposing students to explanations of what the zeros represent, students have a broader mathematical perspectives as long with developing their capacity to make a mathematical judgment (Ma, 45).
14 Discussion: What does the observations mean, summarize important findings Teachers who did not conceptual understand the meaning behind the algorithmic procedure lack understanding of the place value and distributive property, which causes to teach the procedural method that can constrain the student s growth of knowledge in math. Although the teachers know how to solve the problem teachers lacking a simple explanation of place value means students will also lack of math concepts. In my experiences of teaching English as a foreign language I often had a difficult time explaining why English grammars behaves in a certain way. When students ask me why does putting an e on a word make a long vowel I have no true explanation because I lack knowledge of the conceptual reasoning of why that the English language behaves that way. It is difficult for an adult to simplify our knowledge because knowledge is constantly being built upon, which leads an adult to easily forget the basic elements of the fundamentals. Although an individual solves the equation correctly, it doesn't necessarily mean the individual's development in mathematics is progressing. Within my research I found the importance of learning many ways of solving the problem through the use of algorithms. By knowing multiply algorithms to solve an equation teachers can use the support of the different procedure to prove the rationalization of the equation. The mental computation is equally as important as the formal written algorithms in order to apply problem solving by second nature. Mathematics is important for the cognitive and socialcultural aspect of the student's ideology of proof that connects to every day life. By mastering and memorizing multiplication students will have a solid foundation for their development of mathematics along with solving equations within everyday life. For
15 example, Susie needs 300 crackers for her block of cheese. She can buy a pack of crackers of 3 sets of 12 or a block of crackers of 5 sets of 10. Conclusion: Regarding to Liping Ma's survey more teachers teach with a procedural directed teaching method rather then a conceptual explanation of rational teaching method in the U.S. (Ma, 53). Teachers can use these two different algorithmic procedures to support the proof of one another. Teaching on an elementary level heavily depends on the teacher's knowledge of the material. Rather than being able to solve a problem just by memorizing the procedures it is more valuable to the student to memorizing the algorithm based on conceptual understanding and the algorithmic proofs of reasoning. Teachers need to approach student mistakes as an opportunity to engage students to identify characteristics of the equation and understanding mathematical concepts. Although the U.S. teachers teach conceptual material it is directed in a procedural method that lack thorough understanding for both the teacher and student. Compared to the Chinese conceptual understanding with a conceptual directed teaching, the mistake students make are identified and attached together.
16 Works Cited Ma, Liping. Knowing and Teaching Elementary Mathematics. Anniversary Edition. New York: Routledge, Print. Morrow, Lorna, and Margaret Kenney. The Teaching and Learning of Algorithms in School Mathematics. Reston: The National Council of Teachers of Mathematics, Inc., ,44-48, Print. G. Harel & L. Sowder, Toward comprehensive perspectives on the learning and teaching of proof; G. Hanna, Proofs that prove and proofs that explain. In: G. Vergnaud, J. Rogalski, and M. Artigue (Eds.), Proceedings of the 13th Meeting of the International Group for the Psychology of Mathematics Education (pp ), PME, 1989 Jerema, Susuna. "The Importance of Memorizing Times Tables." Phamtom Writers. ( ): n. page. Web. 28 Oct <