It's puzzling to me that that responses to my post titled "INTRO: Should we teach computation?" do not appear with that post as responses. Because of that, and because I would like any interested people to be able to follow the discussion, I am posting my reply to Dr. Manly as this separate post titled "Replying to Dr. Manly".
HERE IS DR. MANLY'S RESPONSE TO MY ORIGINAL POST:
Charles seems to be trying to spark a discussion on this list and I applaud him for that. However, in doing so, he has overstated our differences.
His first sentence states: "Conventional wisdom says that the GED math teacher should teach problem-solving and NOT computation."
He then goes on to suggest that my remark that "I suggest you start with word problems" is the same as saying that I would not teach computation.
I have never said that one should NOT teach computation in a GED class, but I will always say that one needs to teach MORE than computation in a GED class. After all, the GED test is not a test of computation skills.
Charles knows (and quotes elsewhere) that I am one of the authors of a paper, "The Components of Numeracy" which stresses that context, content, and cognitive/affective aspects are critical. With that in mind, from the first class session, I would strive to engage learners with meaningful contexts where knowing the facts and procedures of arithmetic would be helpful in solving problems. Computation is thus a tool, not an end in itself. (quoting Lynda Ginsburg)
It would seem that the disagreement between Charles and me is more style than substance. I would not start a GED class with everyone reviewing naked multiplication facts.
HERE IS MY REPLY TO DR. MANLY:
Yes, Dr. Manly speaks truth in saying that I am trying to spark a discussion here. And I am pleased to see that others (e.g., Mark Schwartz) share that motivation!
Dr. Manly is also providing correct guidance in saying that whatever our disagreements may be, they are more of style than of substance. Just as Dr. Manly did NOT say that we should not teach computation, I did NOT say, nor did Dr. Manly say, that we should start "with everyone reviewing naked multiplication facts" (Dr. Manly's phrase).
What I did say is that SOME students to be identified through assessment (NOT "everyone") would probably benefit from, (quoting myself), "a specialized CAI application that would allow them to practice the facts they don't recall automatically in the context of the times-table that they themselves create." When students discover multiplication facts in the context of constructing a times-table, those facts are not "naked" but are clothed in interrelated connections.
As shown by the work of Robert Siegler (Carnegie-Mellon) and others, practice for times-table fact automaticity can and should be understood, not as necessarily rote memorization, but as a competition among the self-initiated strategies of each student. E.g., see, the early paper by Siegler (1988): Strategy choice procedures and the development of multiplication skill (Journal of Experimental Psychology: General, 1988, Vol. 117, No. 3, 258-275.) Also see subsequent publications archived at webpage titled "Robert Siegler's Publications" --
MY EXPERIENCE AND TEACHING STYLE: ASSESSMENT
My experience, I am sure, is much more limited than Dr. Manly's. I have only taught adult students in tuition-free GED settings where the students are either employed or highly motivated to seek employment. The specific motivation of such students, if you ask them and I do ask them, is to get their GED certificates. When they indicate by their presence in my class that their focus is on math, I then briefly assess the students individually as to basic computational skills, including facility with times-table facts. (Often some earlier assessment is done by the GED program, e.g., CASAS.)
Perhaps it's a matter of teaching style, but I cannot imagine starting an adult numeracy class -- or any student(s) entering any math class that has no prerequisites -- with no assessment. (And I certainly do not mean to imply that Dr. Manly would not assess incoming students!) In my limited experience, adult numeracy students vary enormously respecting every conceivable scale or criterion, excepting income and the single unifying fact that none of them has graduated from high school. Also, my experience indicates that if students' individual needs are not specifically addressed in the first session we may very likely never see them again for a second session! And how should we do that -- address individual needs -- if our approach includes no individual assessment?
THE FIRST SESSION OF AN ADULT NUMERACY CLASS
Dr. Manly would, beginning with the first class, "engage learners with meaningful contexts where knowing the facts and procedures of arithmetic would be helpful in solving problems." What I propose for the first class session -- "putting students to work constructing times-tables" -- may seem at first to be devoid of "meaningful contexts." However, my observation and experience is that students find what they are seeking in constructing times-tables, namely, the meaning of number. The context is triumph over the opacity of numbers, over the seeming meaninglessness of numbers unless we hide them behind a wall of concrete objects. In constructing times-tables, students learn to see themselves as masters of numbers, confident users of numbers, rather than as fearful victims whose only hope is to run from something so darkly mysterious.
None of this should be taken as implying that there should not be, along with the construction and study of times-tables, concrete examples and applications. For instance, as we start the times-tables construction, we can put a penny (or a dollar?) into a glass, agreeing that each time that we reach a benchmark that proves our success with our efforts, we'll double the amount in the glass. When we reach 1 X 2 = 2, we can double our 1 penny to 2 pennies. We square the table out to 2 X 2 = 4, pausing to double our pennies by putting 2 more pennies into the glass at that point. Perhaps we will then choose to run our table out to the right again, limiting ourselves for the time being to the rows for 0, 1 and 2. We will thus find ourselves at 2 X 3 = 6 and at 2 X 4 = 8. Bingo! We double the number of pennies in the glass to 8. Now we can square out our table so that it shows all the facts through 4 X 4 = 16. Bingo again! We double the number of pennies in the glass to 16. (Maybe dollars would be more dramatic, but our ABE budgets are always strained.)
WHAT ARE THE TIMES-TABLE FACTS?
The great Gauss said something like that "mathematics is the queen of the sciences and arithmetic is the queen of mathematics." Some contend that Gauss wasn't really talking about arithmetic, but was rather referring to something much loftier, what mathematicians call 'number theory'. I contend that times-tables are, in fact, results in number theory! But that need not scare us away from teaching the times-tables to ABE students: mathematical thinking is natural to the human brain and all students of math ARE mathematicians!
IN SUM, OR SHOULD WE SAY, "IN PRODUCT"?
Obviously, as indicated in my report on my survey, I would not start students to work on the times-tables if they already know well their times-table facts or if they are in need of remedial work or tutoring to keep up with the group that is constructing times-tables. The reason I propose this starting approach is that, in my limited experience, many adult numeracy students are not as described by Dr. Manly (October 1991 issue of GED Items), that is, having the computation skills and yet being unable "to apply computation skills to real-life situations." Yes, they do have more computation skills than they realize, but no, their main problem is deeper than an inability to apply such skills. The main problem is primarily affective, but it is nonetheless real and must be addressed. Many, although not all, ABE students themselves clamor for help specifically with computation!
Quoting from my report, as sent to Dr. Manly and Mark Schwartz, as well as many others:
The presumption that we as teachers are somehow imposing the goal of learning or improving facility with times-tables on adult numeracy students -- whose goals, after all, do not necessarily conform to numeracy goals as set forth by researchers in international conferences -- is ungrounded. In our experience, students are very well aware of their deficiency (or lack of same, as individually applicable) respecting basic times-table facts and, if deficient, are usually highly motivated to change that situation. This reality, and seeking to respond to it, need not imply that times-table learning is necessarily divorced from the learning or study of problem-solving. The appropriate teaching question is whether to build on students' motivations respecting times-table learning through a specific constructivist learning environment, or not. This question may be answered differently depending on circumstances and teaching style. An advantage of the focused constructivist approach is that students who understand that they have accomplished a facility perhaps exceeding that of most high school graduates will walk tall when leaving the intervention to go on into further GED math learning. The project associated with the survey proposes turning a negative self-perception into an opportunity for what has been termed "self-efficacy" by way of creating a learning environment designed to result in a "mastery experience" (Comings, Parrella, & Soricone, 2000):
Adult education programs should provide the following experiences to their participants as a means to build self-efficacy[:]
Mastery experiences allow an adult to be successful in learning and to have authentic evidence of that success. This does not mean that instruction should be designed to produce only easy and constant success. Adults must also experience overcoming failure and eventually achieving success through a sustained effort. Instruction should help them develop this insight. Some programs take care to provide regular recognition of progress and celebrations of achievement. Others make sure that instruction provides opportunities for success early in program participation. These efforts provide learners with opportunities to experience success.