I wonder if I could write my arithmetic book starting with the decimal numbers, but it does seem to me that explaining the meaning of decimal numbers and how to compute with them requires knowing how to compute with whole numbers since computations with decimal number multiplication and division, for example, require knowing how to multiply and divide whole numbers.
Maybe I could intertwine these two chapters. I see that this is possible, but whether that is good for teaching or not is another question.
But here's another good question: To use full decimal number exponential notation requires justifying what negative integers and what 10^(-1), 10^(-2), 10^(-3), etc. mean and why they mean that. So we either would have to introduce integers before decimal numbers or use names such as tenths, hundredths, thousandths, etc. (or we could shorten them to 10ths, 100ths, 1000ths, etc.) or metric prefixes before we later introduce integers and then exponential notation for the decimal part of decimal numbers. Or we could use 1/10, 1/100, 1/1000, etc. (or 1/10^1, 1/10^2, 1/10^3, etc.) before introducing integers and negative exponential notation.
Of course, I do not plan to make computation as the only goal or only focus of the book; I want these computations to make sense conceptually and logically. That is, I want the students to see what the operations mean so that they can learn to encode real world situations into the language of mathematics and so that they can see what the operations mean. Saying that the operations are functions and stopping there is baffling to students; that approach is essentially nothing but a fancy way to say that the operations are merely arbitrary rules and things we do in arithmetic, and whatever they learn about the operations they cannot use outside the classroom. Any approach that leads to students thinking about mathematics in this way is an utter waste of time for both the students and the teacher. And I want the students to see why the operations do what we want them to do-- that is, why the operations solve the problems we want them to solve.
On 9/19/2010 at 11:08 pm, Alain Schremmer wrote:
> I find it a bit difficult to respond to Groves in my > usual bottom > posting manner since his text is spread out between > two lists and > private correspondence. So, once again, I am going to > top post. > > (1) Fractions. > > ?Mathematically, they are just NxN\0. What has been > known since the > Greeks, sort of more or less, is that what is really > useful is the > quotient set by the equivalence relation (a,b) > ~(c.d) iff a*d = b*c. > It is also interesting that this same construction > gives Z: (a,b) ~ > (c.d) iff a+d = b+c as any accountant will tell you. > It is known as > the Grothendieck construction. This of course is not > what most people > would want to discuss in a developmental arithmetic > course. Still, it > links Q and Z. > > ?In the real world, fractions have no use except in > contrived > questions within exams students have to take. And the > price of course > is that the rationals are usually introduced under > the table. > > ? But what I had in mind when I asked "Are fractions > really > necessary?" is that they do not lead nor support any > of the > arithmetic/algebra that is of the essence for adults. > The book on > which I am working may or may not have an appendix on > fractions but > that is it. What I will be developing and using are > of course > decimals and that for a number of reasons. > - They come built-in in the development of place > e notation, > exponential notation and the metric system. > - The ordering is completely visible. While > e comparing two fractions > makes for a good exercise, it is settled much more > immediately by > doing the divisions than by reducing to common > denominator. > - They are the natural environment (With all due > e respect to Mackey) > for this most important notion: approximation > - They are what engineers call the real real > l numbers. > - They prepare for polynomials > - etc > > --- And it is interesting to note that all these > proponents of > "applications" spend very little time, if any, on > fractions of the > form x/2^n. > > --- It is also interesting to note that educators > generally prefers > to use as examples "x number of aaaa students in a > class of y > students", or number of pizza pieces etc which is at > best very hard > to do correctly and at worst completely misleading. > Note that the > definition I have often given here, that a "quarter" > is "what it > takes 4 to get a dollar" has none of these defects. > > --- To justify the study of fractions by the need to > present ratios > and proportions is a bit surprising inasmuch as the > latter are > another historic remnant from the Greeks who invented > them because > they could not make their peace with numbers others > than whole > numbers. As for percentages, there are two aspects to > them. One is > just that: x% of y is x*(y/100) as in 4 cents per > dollar. The other > is percentage problems which are much better dealt > with in algebra. > > (2) On a more personal basis, in his response > (elsewhere) to Marsh, > Groves wrote: > > > I wonder where [schremmer] gets that idea [that if > I continue the > > book with this organization, I may make much > further progress in > > escaping this mindset than most do but that I still > won't escape > > that entirely.] when he hasn't seen the text of the > book. > > But maybe he can see something in that organization > that tells him > > this fact whereas I don't see it. > > > I am afraid so. I hope that I didn't/won't hurt his > feelings but his > proposed table of contents is quite standard. For > instance: > --- Chapter 2: "is whole number arithmetic. Meanings > of whole > numbers, comparing whole numbers, and the four > operations on whole > numbers." > --- Then, "Chapter 3 is decimal number arithmetic. > Meanings of > decimal numbers, comparing decimal numbers, and the > four operations > on decimal numbers." > > But any re-thinking of arithmetic necessarily forces > a very different > table of contents. Think, for instance, of the table > of contents that > would be forced by the construction of the decimal > from scratch or > just about. Or think of the table of contents that > would be forced by > the immediate introduction of the integers. > > (Again, keep in mind that we are not dealing with > children but with > adults.) > > See for instance > http://www.freemathtexts.org/IntegratedA2DC/Math4Learn > ing/2- > AccountingForMoney.pdf > which appeared in the Spring 2006 issue of the AMATYC > Review. > > By the way, even though, after all I have said for > the last several > years about collaboration, it should go without > saying, I would like > to say that any one wishing to work on such a text > would be more than > welcome. Of course, since I will upload the chapters > as I go and > since they will be under a GNU Free Document License, > there is no > need to collaborate and anyone will be able to just > take the stuff > and modify it to your heart's content. > > Still, this is an invitation to Groves with whom I > have many ideas in > common and Marsh with whom it seems I have model > theory and > mathematical linguistics in common, but of course not > limited to them. > > One possibility might be to begin by discussing, > analyzing and > perhaps comparing detailed tables of contents, > possibly based on 18 > 80-minutes classes / 27 55-minutes classes so as to > remain aware of > the time limitation. > > Regards > --schremmer > > > On Sep 18, 2010, at 9:40 AM, Jonathan Groves wrote: > > > On 9/17/2010 at 12:04 pm, Alain Schremmer wrote (in > part): > > > >> Are fractions really necessary--other than code > for > >> division? If so, > >> for what? (Aside from appeasing ignorant > >> administrations.) > > > > Alain and others, > > > > You may recall that my post on the table of > contents for my book got > > copied over to Math-Teach by accident. Bill Marsh > on Math-Teach > > had written to me, and his response is found in the > post > > > > > http://mathforum.org/kb/message.jspa?messageID=7208301 > &tstart=0. > > > > My response to him can be found at > > > > > http://mathforum.org/kb/message.jspa?messageID=7208551 > &tstart=0. > > > > I had explained to him that what I had planned so > far for the > > book is only undergoing development, not a final > plan, and in > > fact will almost certainly need to change anyway > and that my reasons > > for thinking this were motivated by what Alain had > said. Actually, > > I was wondering about that development even before > Alain had said > > something, but he had encouraged me to think about > other issues > > as well in deciding what goes into the book. > > > > Robert Hansen had replied to me last night, and I > just wrote him > > a reply. Because the comments I had said are of > interest on this > > list, I will include my reply below (since the post > has not appeared > > yet, there is no link I can give): > > > > > > > > On 9/17/2010 at 9:31 pm, Robert Hansen wrote: > > > >> Jonathan wrote... > >> > >> "Are fractions really necessary--other than code > for > >> division? If so, for what?" > >> > >> How can you not include fractions? They are the > basis > >> of ratios, percentages and all other rational > >> expressions. Can you be more specific by what you > >> mean when you say "not include fractions"? Is this > >> primarily a book for remediation? > > > > > > Robert, > > > > First, that was a question Alain Schremmer asked, > and I think it is a > > good question to ask and to consider. He does at > least make a good > > point in that perhaps we beat fraction arithmetic > to death too > > much, that > > many support a heavy emphasis of fractions in math > courses simply > > because > > of conventional wisdom or mere tradition. If there > is indeed a good > > reason for fraction arithmetic receiving the > emphasis it does, I don't > > see that reason. And here I'm talking about the > general student, not > > those interested in STEM careers or other > specialized careers where a > > strong knowledge of how to perform fraction > arithmetic is essential. > > I don't see that many careers where a strong > knowledge of fraction > > arithmetic is essential. > > > > Adding and subtracting fractional measurements > seems necessary to > > a lot of people but only because some people need > that and only > > because > > the customary system of measurement uses fractions. > The metric system > > does not, and if America goes completely metric > someday--and I'm sure > > it will happen, but when that will happen is > anyone's guess--then > > all our measurements will be expressed in decimal > form. I really > > do wonder how many people will truly need to know > how to add or > > subtract measurements expressed as fractions, and > much of that need > > will disappear when we go metric. I also wonder > how many people > > can even read rulers for themselves well enough to > express lengths > > as fractions. I would guess that relatively few > know how to do that. > > > > Yes, I agree that fractions are the basis of ratios > and percentages, > > but not any significant knowledge of how to perform > fraction > > arithmetic > > is required. We will need, of course, to know how > to take a > > fractional > > or percentage amount of a quantity and need to know > that nx to ny > > is the > > same ratio as x to y, but even that last statement > can be proven by > > noting > > that multiplying the dividend and divisor of a > division by a common > > number > > does not change the quotient. > > > > I would not argue that fractions be omitted > entirely because they > > do appear > > in the real world. But an understanding of what > fractions mean in > > terms of > > partial units and in terms of division (which is > actually a more > > general > > interpretation of the "parts of a whole" idea) and > what it means to > > take > > a fractional amount of a quantity are the most > important ideas. Many > > students don't get even that. And if they don't, > then all this > > fraction > > arithmetic makes no sense to them. And I have > observed huge > > numbers of > > students who are greatly confused enough about > fraction arithmetic > > that > > they can't learn even to do fraction arithmetic > correctly. Many > > others > > who do learn it quickly forget it, especially if > they had struggled > > to learn it or that their work clearly shows that > they "get" it > > somewhat but that their understanding of how to do > the fraction > > arithmetic is still shakey. > > > > Of course, education is not really career training > or at least it > > shouldn't be reduced to nothing but career > training, so we can > > rightfully justify teaching some ideas in > mathematics if they at > > least help students learn to reason mathematically, > even if those > > ideas are ideas they will never use later on. So > if fraction > > arithmetic is indeed not going to be very useful > for students later > > on in terms of the mathematics they will use, then > can we really > > justify placing the emphasis on fraction arithmetic > that we do in > > terms of what I just said at the beginning of this > paragraph? > > Perhaps so, but there are probably better choices > we could make to > > accomplish such goals. That is, we can help > students learn to > > make sense of mathematics and learn to reason > mathematically > > via other ways. > > > > Even if we can truly justify the emphasis with > fraction arithmetic > > in these courses, we cannot justify the current > ways that we teach > > it because it ends up being a huge waste of time > for everyone and > > ends up becoming a major headache for the teacher > to see huge > > numbers of students not getting it. > > > > I do plan to keep the material on fraction > arithmetic in my book, > > regardless of the answers to these questions. > Those who will > > need it later on and can learn to reason > mathematically can learn > > the fraction arithmetic they need from the book. > And I will > > include it also because many teachers will feel the > need to teach it > > and because some students will be interested in > learning fraction > > arithmetic, especially if I motivate it by the fact > that nonzero > > terminating decimal numbers are not closed under > division and > > also by the fact that fractions are generalizations > of terminating > > decimal numbers in that fractions allow us to give > the sizes > > of partial units where the unit is split into any > number of equal- > > sized > > parts rather than a number of equal-sized parts > that is a power of 10. > > > > In short, Alain Schremmer has raised a good > question that is worth > > considering, and that has motivated me to ponder it > further. It has > > reminded me that we should not justify that this > idea here or that > > idea there is essential to a math curriculum or > should receive the > > amount of emphasis it typically does just because > lots of people > > say so. > > > > > > Jonathan Groves > > > ****************************************************** > **************** > > ****** > > * To post to the list: email email@example.com > * > > * To unsubscribe, email the message "unsubscribe > mathedcc" to > > firstname.lastname@example.org * > > * Archives at > http://mathforum.org/kb/forum.jspa?forumID=184 * > > > ****************************************************** > **************** > > ****** > > ****************************************************** > ********************** > * To post to the list: email email@example.com * > * To unsubscribe, email the message "unsubscribe > mathedcc" to firstname.lastname@example.org * > * Archives at > http://mathforum.org/kb/forum.jspa?forumID=184 * > ****************************************************** > **********************