explains how the names of numbersfrom 10 through 99 in the Chinese language include what are essentiallythe column names (as do our whole-number multiples of 100), and she thinksthat makes Chinese-speaking students able to learn place-value conceptsmore readily. But I believe that does not follow, since however the namesof numbers are pronounced, the numeric designation of them is still a totallydifferent thing from the written word designation; e.g., "1000" versus"one thousand". It should be just as difficult for a Chinese-speaking childto learn to identify the number "11" as it is for an English-speaking child,because both, having learned the number "1" as "one", will see the number"11" as simply two "ones" together. It should not be any easier for a Chinesechild to learn to read or pronounce "11" as (the Chinese translation of)"one-ten, one" than it is for English-speaking children to see it as "eleven".And Fuson does note the detection of three problems Chinese children have:(1) learning to write a "0" when there is no mention of a particular "column"in the saying of a number (e.g., knowing that "three thousand six" is "3006"not just "36"); (2) knowing that in certain cases when you get more thannine of a given place-value, you have to convert the "extra" into a higherplace-value in order to write it (e.g., you can say "five one hundred'sand twelve ten's" but you have to write it as "620" because you [sort of]cannot write it as "5120". [I say, "sort of" because we do teach childrento write "concatenated" columns --columns that contain multi-digit numbers--when we teach them the borrowing algorithm of subtraction; we do writea "12" in the ten's column when we had two ten's and borrow 10 more.] (3)Writing numbers normally without "concatenating" them (e.g., learning towrite "five hundred twelve" as "512" instead of "50012", where the childwrites down the "500" and puts the "12" on the end of it).
The USB flashdrive contains a drive program, with three modules. (1) Classroom teachingcases. Based on language proficiency, the elementary comprehensive course isdivided into three stages—simple dialogues, paragraphs, and texts, presented inthree different cases. Each case is composed of six sections, i.e., classroomvideo, textbook interpretation, text, teaching plan, courseware, andinstructional dialogue. Among them, the “classroom video” is a live-actionrecording of classroom teaching, which is authentic, concise and intuitive; the“instructional dialogue” is a one-to-one dialogue between the advisory expertand the lecturer, exploring the underlying teaching concepts and design skillsof the exemplary classroom. (2) Course review, including six sections, i.e.,general remark, student-centered teaching, concise explanation and morepractice, gradual progress, textbook analysis, and teaching procedures. The aimis to help teachers understand the comprehensive course in all aspects, so thatthey can do well in teaching the elementary comprehensive course and build asystem of knowledge and ability for the teaching of comprehensive course. (3) Remarkson teaching procedures, including eight sections, i.e., organization of class,review and check, vocabulary teaching, grammar teaching, text teaching,comprehensive exercises, summary, and home assignment. Based on three classroomcases, the experts make a in-depth discussion and analysis of the majorteaching procedures of the comprehensive course, especially the course at theelementary level. The aim is to help teachers set the pace for the classroomand better control the steps and methods of instruction, so that they can improvetheir teaching efficiency and achieve their teaching objectives.
Case studies are stories that are used as a teaching tool to show the application of a theory or concept to real situations. Dependent on the goal they are meant to fulfill, cases can be fact-driven and deductive where there is a correct answer, or they can be context driven where multiple solutions are possible. Various disciplines have employed case studies, including humanities, social sciences, sciences, engineering, law, business, and medicine. Good cases generally have the following features: they tell a good story, are recent, include dialogue, create empathy with the main characters, are relevant to the reader, serve a teaching function, require a dilemma to be solved, and have generality.
Human beings are highly diverse and cases can provide an ideal platform for addressing diversity on many fronts. First and foremost, the content of our cases directly addresses issues relevant to an often neglected group of students. Secondly, the pedagogical approaches recommended here provide opportunities for bringing in the considerable community-based knowledge of the students. Third, the case method is an inclusive pedagogy. It is open-ended and inquiry-oriented, inviting a diversity of responses.
We now offer teaching cases with an emphasis on data, business analytics, social media, and related topics. Developed in cooperation with faculty at the University of Minnesota and elsewhere, these cases have been used in graduate level analytics courses, MBA programs, and executive education courses around the world.
If you don't teach children (or help them figure out how) to adroitlydo subtractions with minuends from 11 through 18, you will essentiallyforce them into options (1) or (2) above or something similar. Whereasif you do teach subtractions from 11 through 18, you give them the optionof using any or all three methods. Plus, if you are going to want childrento be able to see 53 as some other combination of groups besides 5 ten'sand 3 one's, although 4 ten's plus 1 ten plus 3 one's will serve, 4 ten'sand 13 one's seems a spontaneous or psychologically ready consequence ofthat, and it would be unnecessarily limiting children not to make it easyfor them to see this combination as useful in subtraction. ()
And I believe teaching involves more than just letting students (re-)inventthings for themselves. A teacher must at least lead or guide in some formor other. How math, or anything, is taught is normally crucial to how welland how efficiently it is learned. It has taken civilization thousandsof years, much ingenious creativity, and not a little fortuitous insightto develop many of the concepts and much of the knowledge it has; and childrencan not be expected to discover or invent for themselves many of thoseconcepts or much of that knowledge without adults teaching them correctly,in person or in books or other media. Intellectual and scientific discoveryis not transmitted genetically, and it is unrealistic to expect 25 yearsof an individual's biological development to recapitulate 25 centuriesof collective intellectual accomplishment without significant help. Thoughmany people can discover many things for themselves, it is virtually impossiblefor anyone to re-invent by himself enough of the significant ideas fromthe past to be competent in a given field, math being no exception. Potentiallearning is generally severely impeded without teaching. And it is possiblyimpeded even more by bad teaching, since bad teaching tends to dampen curiosityand motivation, and since wrong information, just like bad habits, maybe harder to build from than would be no information, and no habits atall. In this paper I will discuss the elements I will argue are crucialto the concept and to the teaching of place-value.
Since I have taught my own children place-value after seeing how teachersfailed to teach it, and since I havetaught classes of children some things about place-value they could understandbut had never thought of or been exposed to before, I believe the failureto learn place-value concepts lies not with children's lack of potentialfor understanding, but with the way place-value is understood by teachersand with the ways it is generally taught. It should not be surprising thatsomething which is not taught very well in general is not learned verywell in general. The research literature on place-value also shows a lackof understanding of the principle conceptual and practical aspects of learningplace-value, and of testing for the understanding of it. Researchers seemto be evaluating the results of conceptually faulty teaching and testingmethods concerning place-value. And when they find cultural or communitydifferences in the learning of place-value, they seem to focus on factorsthat seem, from a conceptual viewpoint, less likely causally relevant thanother factors. I believe that there is a betterway to teach place-value than it is usually taught, and that children wouldthen have better understanding of it earlier. Further, I believe that thisbetter way stems from an understanding of the logic of place-value itself,along with an understanding of what is easier for human beings (whetherchildren or adults) to learn.
When Iexplained about the need to practice these kinds of subtractions to oneteacher who teaches elementary gifted education, who likes math and mathematical/logicalpuzzles and problems, and who is very knowledgeable and bright herself,she said "Oh, you mean they need practice regrouping in order to subtractthese amounts." That was a natural conceptual mistake on her part, sinceyou do NOT regroup to do these subtractions. These subtractions are whatyou always end up with AFTER you regroup to subtract. If you try to regroupto subtract them, you end up with the same thing, since changing the "ten"into 10 ones still gives you 1_ as the minuend. For example, when subtracting9 from 18, if you regroup the 18 into no tens and 18 ones, you still mustsubtract 9 from those 18 ones. Nothing has been gained. ()
Our teaching notes provide you with some suggestions, but these are only suggestions, not necessarily THE right way for everyone. So feel free to adapt your approach to what fits your teaching style, your students, and the course expectations.