Return-Path: <nifl-technology@literacy.nifl.gov> Received: from literacy (localhost [127.0.0.1]) by literacy.nifl.gov (8.10.2/8.10.2) with SMTP id h8NFltV09851; Tue, 23 Sep 2003 11:47:55 -0400 (EDT) Date: Tue, 23 Sep 2003 11:47:55 -0400 (EDT) Message-Id: <sf701777.077@nmail.epcc.edu> Errors-To: listowner@literacy.nifl.gov Reply-To: nifl-technology@literacy.nifl.gov Originator: nifl-technology@literacy.nifl.gov Sender: nifl-technology@literacy.nifl.gov Precedence: bulk From: "Andres Muro" <AndresM@epcc.edu> To: Multiple recipients of list <nifl-technology@literacy.nifl.gov> Subject: [NIFL-TECHNOLOGY:3033] RE: Special Ed High School Students in X-Listprocessor-Version: 6.0c -- ListProcessor by Anastasios Kotsikonas Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=US-ASCII X-Mailer: Novell GroupWise 5.5.5 Status: O Content-Length: 26853 Lines: 543 Hi, I did not originate the heading. I was responding to someone that asked a question. Since the person asking did it under a certain heading, I responded with the same heading so that he would be able to find the answer. Note that I am answering to you now using the subject heading that you used to write to me. My response to you is not about the subject heading, nor was your message to me, Andres >>> jbennker@ticon.net 09/23/03 09:37AM >>> When replying or responding, please use a correct subject heading so we know what your post is about. Thanks. -----Original Message----- From: nifl-technology@nifl.gov [mailto:nifl-technology@nifl.gov]On Behalf Of Andres Muro Sent: Tuesday, September 23, 2003 9:49 AM To: Multiple recipients of list Subject: [NIFL-TECHNOLOGY:3030] RE: Special Ed High School Students in www.Escolar.com is an Argentinian website w/ hs school level stuff. Our Spanish GED students love it, especially the math stuff. Andres >>> mohno@carlosrosario.org 09/23/03 04:28AM >>> Hello: Does anyone know of some internet websites that do Pre-GED and GED items in Spanish? Thank you. -- Mary Ohno Carlos Rosario School Washington, DC ---------- Original Message ---------------------------------- From: "Nixon S. Griffis" <ngriffis@bellsouth.net> Reply-To: nifl-technology@nifl.gov Date: Mon, 22 Sep 2003 19:49:00 -0400 (EDT) >I have a few suggestions about your posting. > >1. My high school has remediation after school with mentor one on one >tutors. Mentor tutoring is a great concept. The high end kids help the low >end kids. It is a win win situation because the high end kids anchor the >information they are teaching even that much more firmly. This also takes >some stress off the teacher. A Mentor-Tutor mini-course for your mentors is >probably a good idea to give them some basic tools. > >2. "They may be able to do a particular type of problem, but really do not >understand it." >I came across this piece somewhere and believe that it will help all basic >math teacher teach their studewnts understanding rather than memorization: > > >Notes from ?Knowing and Teaching Elementary Mathematics? by Liping Ma > >upper stories, but it is the foundation that supports them and makes all the >stories (branches) cohere. The appearance and development of new mathematics >should not he regarded as a denial of fundamental mathematics. In contrast, >it should lead us to an ever better understanding of elementary mathematics, >of its powerful potentiality, as well as of the conceptual seeds for the >advanced branches. > > > > > > >PROFOUND UNDERSTANDING OF FUNDAMENTAL >MATHEMATICS > >Indeed, it is the mathematical substance of elementary mathematics that >allows a coherent understanding of it. However, the understanding of >elementary mathematics is not always coherent. From a procedural >perspective, arithmetic algorithms have little or no connection with other >topics, and are isolated from one another. Taking the four topics studied as >an example, subtraction with regrouping has nothing to do with multidigit >multiplication, nor with division by fractions, nor with area and perimeter >of a rectangle. >Figure 5.1 illustrates a typical procedural understanding of the four >topics. The letters S, M, D, and G represent the four topics: subtraction >with regrouping, multidigit multiplication, division with fractions, and t >hp geometry topic (calculation of perimeter and area). The rectangles >represent procedural knowledge of these topics. The ovals represent other >procedural knowledge related to these topics. The trapezoids underneath the >rectangles represent pseudoconceptual understanding of each topic. The >dotted outlines represent missing items. Note that the understandings of the >different topics are not connected. >In Fig. 5.1 the four topics are essentially independent and few elements are >included in each knowledge package.' Pseudoconceptual explanations for >algorithms are a feature of understanding that is only procedural. Some >teachers invented arbitrary explanations. Some simply verbalized the >algorithm. Yet even inventing or citing a pseudoconceptual explanation >requires familiarity with the algorithm. Teachers who could barely early out >an algorithm tended not to be able to explain it or connect it wish other >procedures, as seen in some responses to the division by fractions and >geometry topics. With isolated and underdeveloped knowledge packages > > > >FIG. 5.1. Teachers' procedural knowledge of the four topics. >The mathematical understanding of a teacher with a procedural perspective is >fragmentary. >>From a conceptual perspective, however, the four topics are connected, >related by the mathematical concepts they share. For example, the concept of >place value underlies the algorithms for subtraction with regrouping and >multidigit multiplication. The concept of place value, then, becomes a >connection between the two topics. The concept of inverse operations >contributes to the rationale for subtraction with regrouping as well as to >the explanation of the meaning of division by fractions. Thus the concept of >inverse operations connects subtraction with regrouping and division by >fractions. Some concepts, such as the meaning of multiplication, are shared >by three of the four topics. Some, such as the three basic laws, are shared >by all four topics. Figure 5.2 illustrates how mathematical topics are >related from a conceptual perspective. >Although not all the concepts shared by the four topics are included, Fig. >5.2 illustrates how relations among the four topics make them into a >network. Some items are not directly related to all four topics. However, >their diverse associations overlap and interlace. The three basic laws >appeared in the Chinese teachers' discussions of all four topics. > In contrast to the procedural view of the four topics illustrated in Fig. >5.1, Fig. 5.3 illustrates a conceptual understanding of the four topics. The >four rectangles at the top of Fig. 5.3 represent the four topics. The >ellipses > d >represent the knowledge pieces in the knowledge packages. White ellipses >represent procedural topics, light gray ones represent conceptual topics, > >and in China. What caused the coherence of the Chinese teachers' knowledge, >in fact, is the mathematical substance of their knowledge. >A CROSS?TOPIC PICTURE OF THE CHINESE TEACHERS' KNOWLEDGE: WHAT IS ITS >MATHEMATICAL SUBSTANCE? >Let us take a bird's eye view of the Chinese teachers' responses to the >interview questions. It will reveal that their discussions shared some >interesting features that permeated their mathematical knowledge and were >rarely, if ever, found in the U.S. teachers' responses. > >To Find the Mathematical Rationale of an Algorithm >During their interviews, the Chinese teachers often cited an old saying to >introduce further discussion of an algorithm: "Know how, and also know why." >In adopting this saying, which encourages people to discover a reason behind >an action, the teachers gave it a new and specific meaning?to know how to >carry out an algorithm and to know why it makes sense mathematically. >Arithmetic contains various algorithms?in fact it is often thought that >knowing arithmetic means being skillful in using these algorithms. From the >Chinese teachers' perspective, however, to know a set of rules for solving a >problem in a finite number of steps is far from enough?one should also know >why the sequence of steps in the computation makes sense. For the algorithm >of subtraction with regrouping, while most U.S. teachers were satisfied with >the pseudoexplanation of "borrowing," the Chinese teachers explained that >the rationale of the computation is "decomposing a higher value unit."' For >the topic of multidigit multiplication, while most of the U.S. teachers were >content with the rule of "lining up with the number by which you >multiplied," the Chinese teachers explored the concepts of place value and >place value system to explain why the partial products aren't lined up in >multiplication as addends are in addition. For the calculation of division >by fractions for which the U.S. teachers used "invert and multiply," the >Chinese teachers referred to "dividing by > > > >'In teaching, Chinese teachers tend to use mathematical terms in their >verbal explanations. Terms such as addend, sum, minuend, subtrahend, >difference, multiplicand, multiplier, product, partial product, dividend, >divisor, quotient, inverse operation, and composing and decomposing, are >frequently used. For example, Chinese teachers do not express the additive >version of the commutative law as "The order in which you add two numbers >doesn't matter." Instead, they say "When we add two addends, if we exchange >their places in the sentence, the sum will remain the same." > >TEACHERS' SUBJECT MATTER KNOWLEDGE 109 >a number is equivalent to multiplying by its reciprocal" as the rationale >for this seemingly arbitrary algorithm. >The predilection to ask "Why does it make sense?" is the first stepping >stone to conceptual understanding of mathematics. Exploring the mathematical >reasons underlying algorithms, moreover, led the Chinese teachers to more >important ideas of the discipline. For example, the rationale for >subtraction with regrouping, "decomposing a higher value unit," is connected >with the idea of "composing a higher value unit," which is the rationale for >addition with carrying. A further investigation of composing and decomposing >a higher value unit, then, may lead to the idea of the "rate of composing >and decomposing a higher value unit," which is a basic idea of number >representation. Similarly, the concept of place value is connected with >deeper ideas, such as place value system and basic unit of a number. >Exploring the "why" underlying the "how" leads step by step to the basic >ideas at the core of mathematics. > >To justify an Explanation with a Symbolic Derivation >Verbal explanation of a mathematical reason underlying an algorithm, >however, seemed to be necessary but not sufficient for the Chinese teachers. >As displayed in the previous chapters, after giving an explanation the >Chinese teachers tended to justify it with a symbolic derivation. For >example, in the case of multidigit multiplication, some of the U.S. teachers >explained that the problem 123 x 645 can be separated into three "small >problems"; 123 x 600, 123 x 40, and 124 x 5. The partial products, then, are >73800, 4920, and 615, instead of 738, 492, and 615. Compared with most U.S. >teachers' emphasis on "lining up," this explanation is conceptual. However, >the Chinese teachers gave explanations that were even more rigorous. First, >they tended to point out that the distributive laws is the rationale >underlying the algorithm. Then, as described in chapter 2, they showed how >it could be derived from the distributive law in order to > > > >In the Chinese mathematics curriculum, the additive versions of commutative >and associative laws are first introduced in third grade. The commutative, >associative, and distributive laws of multiplication are introduced in >fourth grade. They are introduced as alternatives to the standard method. >For example, the textbook says of the commutative law of addition, "When two >numbers are added, if the locations of the addends are exchanged, the sum >remains the same. This is called the commutative law of addition. If the >letters a and b represent two arbitrary addends, we can write the >commutative law of addition as: "+ b= L+ (r. The method we learned of >checking a sum by exchanging the order of addends is drawn from this law" >(Beijing, Tianjin, Shanghai, and Zhejiang Associate Group for Elementary >Mathematics Teaching Material Composing, 1989, pp. 82?83). The textbook >illustrates how the two laws can be used as "a way for fast computation." >For example, students learn that a faster way of solving 258 + 791 + 642 is >to transform it into (258 + 642) + 791, a faster way of solving 1646 ? 248 ? >152 is to transform it into 1646 ? (248 + 152). > >123 x 645 = 123 x (600 + 40 + 5) > =123x600+123x40+123x5 > = 73800 + 4920 + 615 > =78720+615 > = 79335 >For the topic of division by fractions, the Chinese teachers' symbolic >representations were even more sophisticated. They drew on concepts that >"students had learned" to prove the equivalence of 14 = 2 and 14 x 2/1 in >various ways. The following is one proof based on the relationship between a >fraction and a division (z = 1 = 2): > >A proof drawing on the rule of "maintaining the value of a quotient" is: >3 _. 1 3 2 / 1 1 >14Y ? (14 X 1) . (2 X 1) >=(14x 2/1)/1 >1 3/4 X 2/1 >4 1 >= 3' z > >Moreover, as illustrated in chapter 3, the Chinese teachers used >mathematical sentences to illustrate various nonstandard ways to solve the >problem 14 = 2, as well as to derive these solutions. Symbolic >representations are widely used in Chinese teachers' classrooms. As Tr. Li >reported, her first grade students used mathematical sentences to describe >their own way of regrouping: 34 ? 6 = 34 ? 4 ? 2 = 30 ? 2 = 28. Other >Chinese teachers in this study also referred to similar incidents. >Researchers have found that elementary students in the United States often >view the equal sign as a "do?something signal" (see e.g., Kieran, 1990, p. >100). This reminds me of a discussion I had with a U.S. elementary teacher. >I asked her why she accepted student work like " 3 + 3 x 4 = 12 > >algorithm. One teacher showed six ways of lining up the partial products. >For the division with fractions topic the Chinese teachers demonstrated at >least four ways to prove the standard algorithm and three alternative >methods of computation. >For all the arithmetic topics, the Chinese teachers indicated that although >a standard algorithm may be used in all cases, it may not be the best method >for every case. Applying an algorithm and its various versions flexibly >allows one to get the best solution for a given case. For example, the >Chinese teachers pointed out that there are several ways to compute 14 = z. >Using decimals, the distributive law, or other mathematical ideas, all the >alternatives were faster and easier than the standard algorithm. Being able >to calculate in multiple ways means that one has transcended the formality >of an algorithm and reached the essence of the numerical operations?the >underlying mathematical ideas and principles. The reason that one problem >can be solved in multiple ways is that mathematics does not consist of >isolated rules, but connected ideas. Being able to and tending to solve a >problem in more than one way, therefore, reveals the ability and the >predilection to make connections between and among mathematical areas and >topics. >Approaching a topic in various ways, making arguments for various solutions, >comparing the solutions and finding a best one, in fact, is a constant force >in the development of mathematics. An advanced operation or advanced branch >in mathematics usually offers a more sophisticated way to solve problems. >Multiplication, for example, is a more sophisticated operation than addition >for solving some problems. Some algebraic methods of solving problems are >more sophisticated than arithmetic ones. When a problem is solved in >multiple ways, it serves as a tie connecting several pieces of mathematical >knowledge. How the Chinese teachers view the four basic arithmetical >operations shows how they manage to unify the whole field of elementary >mathematics. > >Relationships Among the Four Basic Operations: The "Road System" Connecting >the Field of Elementary Mathematics > > >Arithmetic, "the art of calculation," consists of numerical operations. The >U.S. teachers and the Chinese teachers, however, seemed to view these >operations differently. The U.S. teachers tended to focus on the particular >algorithm associated with an operation, for example, the algorithm for >subtraction with regrouping, the algorithm for multidigit multiplication, >and the algorithm for division by fractions. The Chinese teachers, on the >other hand, were more interested in the operations themselves and their >relationships. In particular, they were interested in faster and easier ways >to do > >TEACHERS' SUBJECT MATTER KNOWLEDGE ..,. >a given computation, how the meanings of the four operations are connected, >and how the meaning and the relationships of the operations are represented >across subsets of numbers?whole numbers, fractions, and decimals. >When they teach subtraction with decomposing a higher value unit, Chinese >teachers start from addition with composing a higher value unit. When they >discussed the "lining?up rule" in multidigit multiplication, they compared >it with the lining?up rule in multidigit addition. In representing the >meaning of division they described how division models are derived from the >meaning of multiplication. The teachers also noted how the introduction of a >new set of numbers?fractions?brings new features to arithmetical operations >that had previously been restricted to whole numbers. In their discussions >of the relationship between the perimeter and area of a rectangle, the >Chinese teachers again connected the interview topic with arithmetic >operations. >In the Chinese teachers' discussions two kinds of relationships that connect >the four basic operations were apparent. One might be called "derived >operation." For example, multiplication is an operation derived from the >operation of addition. It solves certain kinds of complicated addition >problems in a easier way. The other relationship is inverse operation. The >term "inverse operation" was never mentioned by the U.S. teachers, but was >very often used by the Chinese teachers. Subtraction is the inverse of >addition, and division is the inverse of multiplication. These two kinds of >relationships tightly connect the four operations. Because all the topics of >elementary mathematics are related to the four operations, understanding of >the relationships among the four operations, then, becomes a road system >that connects all of elementary mathematics .4 With this road system, one >can go anywhere in the domain. > > > > >KNOWLEDGE PACKAGES AND THEIR KEY PIECES: >UNDERSTANDING LONGITUDINAL COHERENCE >IN LEARNING > >Another feature of Chinese teachers' knowledge not found among U.S. teachers >is their well?developed "knowledge packages." The four features discussed >above concern teachers' understanding of the field of elementary >mathematics. In contrast, the knowledge packages reveal the teachers' > > > >"Although the four interview questions did not provide room for discussion >of the relationship between addition and multiplication, Chinese teachers >actually consider it a very important concept in their everyday teaching. >"The two kinds of relationships among, the four basic operations, indeed, >apply to all advanced operations in the discipline of mathematics as well. >The "road system" of elementary mathematics, therefore, epitomizes the "road >system" of the whole discipline. > >understanding of the longitudinal process of opening up and cultivating such >a field in students' minds. Arithmetic, as an intellectual field, was >created and cultivated by human beings. Teaching and learning arithmetic, >creating conditions in which young humans can rebuild this field in their >minds, is the concern of elementary mathematics teachers. Psychologists have >devoted themselves to study how students learn mathematics. Mathematics >teachers have their own theory about learning mathematics. >The three knowledge package models derived from the Chinese teachers' >discussion of subtraction with regrouping, multidigit multiplication, and >division by fractions share a similar structure. They all have a sequence in >the center, and a "circle" of linked topics connected to the topics in the >sequence. The sequence in the subtraction package goes from the topic of >addition and subtraction within 10, to addition and subtraction within 20, >to subtraction with regrouping of numbers between 20 and 100, then to >subtraction of large numbers with regrouping. The sequence in the >multiplication package includes multiplication by one?digit numbers, >multiplication by two?digit numbers, and multiplication by three?digit >numbers. The sequence in the package of the meaning of division by fractions >goes from meaning of addition, to meaning of multiplication with whole >numbers, to meaning of multiplication with fractions, to meaning of division >with fractions. The teachers believe that these sequences are the main paths >through which knowledge and skill about the three topics develop. >Such linear sequences, however, do not develop alone, but are supported by >other topics. In the subtraction package, for example, "addition and >subtraction within 10" is related to three other topics: the composition of >10, composing and decomposing a higher value unit, and addition and >subtraction as inverse operations. "Subtraction with regrouping of numbers >between 20 and 100," the topic raised in interviews, was also supported by >five items: composition of numbers within 10, the rate of composing a higher >value unit, composing and decomposing a higher value unit, addition and >subtraction as inverse operations, and subtraction without regrouping. At >the same time, an item in the circle may be related to several pieces in the >package. For example, "composing and decomposing a higher value unit" and >"addition and subtraction as inverse operations" are both related to four >other pieces. With the support from these topics, the development of the >central sequences becomes more mathematically significant and conceptually >enriched. >The teachers do not consider all of the items to have the same status. Each >package contains "key" pieces that "weigh" more than other members. Some of >the key pieces are located in the linear sequence and some are in the >"circle." The teachers gave several reasons why they considered a certain >piece of knowledge to be a "key" piece. They pay particular attention to the >first occasion when a concept or skill is introduced. For example, the topic >of "addition and subtraction within 20" is considered to be such > >TEACHERS' SUBJECT MATTER KNOWLEDGE 115 >a case for learning subtraction with regrouping. The topic of >"multiplication by two?digit numbers" was considered an important step in >learning multidigit multiplication. The Chinese teachers believe that if >students learn a concept thoroughly the first time it is introduced, one >"will get twice the result with half the effort in later learning." >Otherwise, one "will get half the result with twice the effort." >Another kind of key piece in a knowledge package is a "concept knot." For >example, in addressing the meaning of division by fractions, the Chinese >teachers referred to the meaning of multiplication with fractions. They >think it ties together five important concepts related to the meaning of >division by fractions: meaning of multiplication, models of division by >whole numbers, concept of a fraction, concept of a whole, and the meaning of >multiplication with whole numbers. A thorough understanding of the meaning >of multiplication with fractions, then, will allow students to easily reach >an understanding of the meaning of division by fractions. On the other hand, >the teachers also believe that exploring the meaning of division by >fractions is a good opportunity for revisiting, and deepening understanding >of these five concepts. >In the knowledge packages, procedural topics and conceptual topics were >interwoven. The teachers who had a conceptual understanding of the topic and >intended to promote students' conceptual learning did not ignore procedural >knowledge at all. In fact, from their perspective, a conceptual >understanding is never separate from the corresponding procedures where >understanding "lives." >The Chinese teachers also think that it is very important for a teacher to >know the entire field of elementary mathematics as well as the whole process >of learning it. Tr. Mao said: > > >As a mathematics teacher one needs to know the location of each piece of >knowledge in the whole mathematical system, its relation with previous >knowledge. For example, this year I am teaching fourth graders. When I open >the textbook I should know how the topics in it are connected to the >knowledge taught in the first, second, and third grades. When I teach >three?digit multiplication I know that my students have learned the >multiplication table, one?digit multiplication within 100, and >multiplication with a two?digit multiplier. Since they have learned how to >multiply with a two?digit multiplier, when teaching multiplication with a >three?digit multiplier I just let them explore on their own. I first give >them several problems with a two?digit multiplier. Then I present a problem >with a three?digit multiplier, and have students think about how to solve >it. We have multiplied by a digit at the ones place and a digit at the tens >place, now we are going to multiply by a digit at the hundreds place, what >can we do, where are we going to put the product, and why? Let them think >about it. Then the problem will be solved easily. I will have them, instead >of myself, explain the rationale. On the other hand, 1 have to know what >knowledge will be built on what 1 am teaching today (italics added). > > >-----Original Message----- >From: nifl-technology@nifl.gov [mailto:nifl-technology@nifl.gov]On >Behalf Of Jonathan Bennker >Sent: Monday, September 22, 2003 9:36 AM >To: Multiple recipients of list >Subject: [NIFL-TECHNOLOGY:3026] Special Ed High School Students in >mainstreamed math > > >Problem: Providing support for high school special ed students in >mainstreamed math courses such as algebra, geometry, or trig. > >I am looking for ways to address the above problem. Does anybody know of >any successful programs or have ideas as to what could work? I have seen >special ed students come to a resource room for help. It seems all that can >be done is a band-aid approach. They may be able to do a particular type of >problem, but really do not understand it. Therefore, they cannot apply the >skill to more complex problems. Also, the students seem to start the course >without prerequisite skills. > >Any thoughts would be appreciated. > >Thanks, > >Jonathan Bennker >jbennker@ticon.net >262-472-9699 > > > > > -- -- Mary Kiyoko Ohno Computer Lab Teacher Carlos Rosario International Public Charter School 1724 Kalorama NW #300 Washington, DC 20009 Phone: (202) 234-6522 Fax (202) 234-6563 --
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