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 Donovan as this separate post titled "Replying to Dr. Donovan".
HERE IS DR. DONOVAN'S RESPONSE TO MY ORIGINAL POST:
Charles seems to raise two questions:
* How do we engage students who are weak mathematically?
* How do we hone computation skills/should we hone math computation skills?
It appears that Charles teaches a class where students exhibit a wide range of math skills. For 12 years I taught similar groups, and can sympathize with the need both to engage and to create some level of competency that facilitates class interaction. How to do this?
One has to be amazingly flexible because, in addition to their math skills, the students in any particular class also represent numerous literacy levels, various backgrounds and interests, and myriad life goals.
Comings, et. al. found that persistence rates increased for students who felt that programing met their goals and that fulfilled the "six affective needs of adults" :
A sense of belonging and community clarity of purpose agency competence relevance stability
Therefore, if Charles wants to break the 3-week barrier, it seems he should create an instruction program that integrates activities designed to address these needs while forwarding mathematics understanding. Again, how to do this?
First, he must come to know his students in some way. Over the years, several of us on this list have promoted the use of a math autobiography and/or math collage about 'what math means to me,' etc. as starting points in a dialogue between teacher and student and among students (building that sense of community). From this dialogue, Charles might lead to a discussion of 'goals' both general (GED, training program entry, Community College) and math-specific (learn the times tables, learn long division, understand algebra, read a ruler....)
An assessment of student knowledge, either formal (standardized or local) or informal is, therefore, also essential. From these introductory lessons, which surface abilities with, attitudes toward, and prior practice with mathematics, Charles could then start to think about what mathematical tasks to present.
Working with an open enrollment system, I found it useful to combine the solving of non-routine problems (usually collaboratively, with solutions shared and discussed) with personal work often on more routine 'work sheets' that students could solve independently. However, I shyed away from conventional worksheets with an example at the top of the page followed by 20 examples to practice the step demonstrated -- it just makes math too abstract and disconnected.
I rather liked problems like -- "What happens if on day one we put two pennies in a jar and then double the amount we add each day? How many pennies will we add on day 2, day 5, day 10?" problems that appeared simple and could be solved using simple, repetitive computations, but could also open the door to questions of notation and algebraic expression that broadened horizons of understanding.
I also like daily practice activities such as "Number of the Day," which built that sense of stability, as well as competence and a sense of community. This type of activity also helps students build toward their goals, as you can ask that they use exponents, fractions, four operations, etc. to express the number of the day.
Projects also allowed students to focus on their individual goals within the community context. I might ask students to explore a career and the math used in that career, or to take a sample accuplacer test to see what the local community college expected them to know, and then report back to the whole class, with a poster demonstration, overheads, etc. For years, around the December holidays, we built gingerbread houses, thereby practicing geometry and measurement and number skills to produce a genuine product that could be shared with others after being displayed and photographed.
However, if Charles wants to focus on computation, I'd suggest, as I think Myrna was suggesting, that the computation be presented within meaningful contexts for students. Practice isn't something we do for 15 minutes; practice is a lifetime of repetition. Students need to develop ways of understanding operations -- is something growing or shrinking here -- is that happening in an adding/subtracting way or proportionally in a multiplying/dividing way? And they need to understand numbers -- do these operations work the same for fractions as they do for whole numbers? In addition, a good grounding in benchmarks -- for whole numbers, fractions, decimals and percents -- needs to be developed in order to facilitate estimation skills, which form the basis of most adult math practice.
For those who need to be able to perform precise conversions, for example, it will help to demonstrate the algorithms and then ask students to create 20 of their own example problems.
The motivation to be precise stems from a need: either to budget to the penny, pass a nursing exam, build a bookcase.... Find out what the need is and help the student meet that need (sounds a little like EFF, no?). The real learning is how to learn; how to question; where to seek help; how to keep track of what is known and what remains to be known.
Math class can be exciting, challenging, and, yes, engaging. To simplify your efforts, Charles, I would suggest you check out copies of the EMPower series books. They offer non-routine ways to build mathematical understanding and competency, starting with multiplication of whole numbers and moving on through benchmark fractions, decimals, and percents, geometry and measurement and algebraic thinking. This series could save a lot of time and generate some of your own creative thinking, Charles.
Rambling thoughts, I know. I hoped they help.
Tricia Donovan, Ed.D.
(And in the spirit of transparency -- a co-author of the EMPower series.)
HERE IS MY REPLY TO DR. DONOVAN:
Yes, those are the very students I am discussing: wide range of math skills.
I can recall one student who insisted on finger counting and was quick to inform me that he had been diagnosed in elementary school as with dyscalculia. He constructed a times-table and, although he still tended to fall back on finger-counting even for multiplication (a tricky business), he came to rely on consulting the times-table, that he had constructed, when doing word problems or practicing division. What amazed me most is that he was the best student in the class when it came to word problems. He just cut right through them and spit out the answer, without being able to explain how he had done it.
There was another student who demonstrated 100% automaticity from the first day, but he had terrible test anxiety and was certain that he could not understand algebra -- he resented what he thought of as 'algebra' as an unnecessary and impractical level of abstraction. His practical skills in geometry were great, and he eventually "got it" about x, y and z. He industriously applied himself to every one of the 50 types of GED problems presented in Contemporary's Top 50. His main problem, by far, was test anxiety under timed test conditions. He nit-picked problems without end, working hard to find interpretations never intended by the question designers. However, he passed the GED Math test, although his minimal score did not reflect his actual math ability.
I really don't like the GED math test because it is, despite Stech-Vaughn denial, primarily a timed test. IMO, they do it that way in order to get their correlation with senior high school students. They have to knock the pass rate down somehow, for their statistical purposes. I am 100% on the side of Walen, S. B. & Williams, S. R. (March 2002). A matter of time: Emotional responses to timed mathematics tests. Educational Studies in Mathematics, Vol. 49, pp. 361-378. DOI 10.1023/A:1020258815748 (also now available online as part of a PME Proceedings, I forget the date).
OF COURSE, I agree that we first discuss what students want before starting anything. Not only at the beginning, but at every step, we always stop to ask students what they want to do next or where they self-assess. It's just that it's exactly from such a process that I have seen that many students at the outset want help with the times-tables. It would seem to be a good idea to prepare, based on our experience, for likely possibilities -- otherwise we would never use prepared instructional units or materials at all! Also, as discussed in Mark Schwartz's comments on 'Compartmentalization', students often rely on teachers to direct them to the next 'compartment' ... where their efforts and time will be best invested. (See, "Roll and Compartmentalization" here at this discussion list.)
So, I am not at all about designing or planning an entire program, just a little piece to add to the existing library of strategies, methods and materials. Nor am I about a general critique of existing numeracy teaching practice, except to ascertain areas where new CAI applications may be useful. What my project is about is providing a concrete "opportunity for success early in program participation" (quote from Comings, et al.).
I agree, but only conditionally, about the 20-example practice sheets, because not all such sheets are useless. I wouldn't tar them all with the same brush. I think the best of them have a place in the library for use when appropriate. The thing is that we have students who are kinesthetic learners and learn with their hands more than with their ears or eyes -- so that's the main reason why, IMO, old-fashioned worksheets still have a place, IF they are well-designed.
Otherwise, I agree with all the specifics, and have used or observed most, that Dr. Donovan reviews as strategies or approaches: the class as a community, goal-setting, non-routine problems (although I haven't been able to predict which will 'work' so they may need to be pre-tested if possible), concrete examples like the 2 pennies a day, "Number of the Day", relation of math to careers and to critical tests beyond the GED, gingerbread houses, bookcases and other carpentry, teaching for quantitative thinking and estimation, students devising example problems.
So, very fine "rambling thoughts" by Dr. Donovan. I'll have to check out the EMPower series.