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 h8NFc4V08382; Tue, 23 Sep 2003 11:38:04 -0400 (EDT) Date: Tue, 23 Sep 2003 11:38:04 -0400 (EDT) Message-Id: <GFEDLAFJLLJNFPGOLNFBKEOGCMAA.jbennker@ticon.net> 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: "Jonathan Bennker" <jbennker@ticon.net> To: Multiple recipients of list <nifl-technology@literacy.nifl.gov> Subject: [NIFL-TECHNOLOGY:3032] RE: Special Ed High School Students in math or GED or what? X-Listprocessor-Version: 6.0c -- ListProcessor by Anastasios Kotsikonas X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2910.0) Content-Transfer-Encoding: 7bit Content-Type: text/plain; Status: O Content-Length: 26399 Lines: 536 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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