## Sunday, May 29, 2005

### An example of what we could do for K-12

My younger son is a pretty bright kid but his handwriting is terrible. (That is likely genetically determined.) He has had some trouble on his recent math homework, not because he doesn't understand the concepts but because he can't read his own handwriting. So he makes a mistake on a subsequent step (a 3 looks strikingly like a 2) because he bases his judgment not on his recollection of how he came up with the 3, but rather on his reading of the number on his hand written page which is full of erasure marks (it sure looks like a 2). The error follows naturally from the misstep.

I would not have come up with this tutorial if not for my son and watching him do his work. But, of course, I had this idea of outreach from Higher Ed to K-12 on my brain and that was an additional motive, certainly. It's kind of strange to spend almost a full day of a long weekend designing content like this. But this stuff serves as therapy for me and illustrates the idea that it is possible to make intellectual contributions to the lower grades. Clearly this is other than a derivative of what we teach in Higher Ed. Yet the making of this tutorial was an intellectual challenge (I'll explain why in a bit) and so could be viewed as a mini research project.

Here is one other remark before describing the tutorial itself about how it differs from the paper and pencil approach. The tutorial is long on interactivity. The student gets feedback at every step and can't go further without getting the preceding step correct. In the paper and pencil world, unless there is someone else staring over the student's shoulder while the student is doing the work the student gets no feedback at all until the problem has been turned in for evaluation. The feedback elements encourage the student to try things - perhaps deliberately getting something wrong, just to see what happens. The tutorial also emphasizes the recursive structure that is the essence of long division. Within a given cycle, the arithmetic is done off to the side and should be done mostly in the student's head. The results of that arithmetic are then recorded in the long division area. The breaking apart the recording from the arithmetic gives a cleaner look to the result. I hope it also allows the student to think a little more about what is going on, since the student doesn't have to worry about writing down the results.

To use the tutorial, Macros must be enabled. (Tools menu, then Macros, then Security, then Intermediate.) Once the workbook is open there will be two spreadsheets: the Student Version and the Instructor's Manual. These are essentially the same. The Student Version worksheet is Protected (but the password is blank so it is a simple matter to unprotect it). The protection prevents the students from clicking on cells and seeing the content (or entering different values into those cells). Also, there are some cells that are hidden from the students' view. The Instructor Manual allows all the formulas to be inspected by clicking on the cell and looking at the formula bar. Further, the Macros associated with buttons can be found by right clicking on the particular button and by right clicking on the spinner buttons one can see the cells they control.

The tutorial accommodates dividends with up to 6 digits and divisors with 1 or 2 digits. Some values are in place when the spreadsheet is open. It is straightforward to put in your own values. Put the dividend into cell B3 and then hit Enter (Return). Put the divisor in cell B4 and again hit Enter. Then push the Set button. This initializes the values in the rest of the spreadsheet.

You can now start the process by looking at Step 1. If the answer to the question is yes, you are done. If not you can proceed to steps 2 through 4 in sequence. At each step you must choose an answer and then evaluate. If your response is wrong, choose another answer and evaluate again. There is some feedback given to guide the student to the right answer. After a while they'll see that getting the answer right the first time is the fastest way to get through the problem. Once you do get the right answer, you can proceed to the next step. When you have completed Step 4, press the Record and Update button. Now you begin again with Step 1, but with a new and smaller dividend.

The first time try it just to see what it does. The next time through, think about what it takes to make something like this. There are certain "tricks of the trade" (for example if you don't want a value to be visible have the font color the same as the background color, in this case white on white) that can get you so far. Algorithmically, the arithmetic part is not that hard. (And indeed part of the idea is to convince the students that it is not that hard.) It is only a little harder to figure out algorithmically what the full long division layout should look like (all the intermediate dividends and subtrahends). The real difficulty is making the stuff appear at the right time. This is particularly difficult for the quotient area where two quotients of the same length will nonetheless be generated with a different number of iterations, depending on how many zeroes they have in other than the first and last digit.

So to do this correctly, one has to come up with some counting algorithms. That counting is not so straightforward. And that represents the intellectual challenge in making this sort of thing. In general, if one designs an interactive exercise of this sort there are elements of conditional response and if the entire decision tree is fairly long or if there are many possibilities at each branch of the tree, the entirety can become quite complex. This, I believe, is why you don't see so much interactive material of this sort, but rather flatter stuff that is mostly straight presentation. The flatter stuff is intellectually easier to produce. But it is far less compelling to use.

Does this tutorial work for the fifth grader? I don't know, but I'm interested in finding out. Of course there are the access issues. But what about the pedagogic issues? Would this help students who did have access to Excel on a computer? And if so which type of students would benefit?

Let me assume for a minute that the students would benefit. And let me assume further that as a consequence the schools would like to have this particular tutorial and many others like it. Could we in Higher Ed come up with a program where tutorials of this sort were either the main product or a significant by-product? I can envision an online network that would be aimed at supporting teachers who used this type of tutorial content. I can imagine both peer teachers and students and faculty in Higher Ed participating. But I'm having a harder time envisioning the structure inside of Higher Ed that would sustain this.

And here is my fear. At the outset I made it quite clear that this tutorial is not a faculty member's research. And though I didn't elaborate on the point, I hope it is clear that because this is just an Excel file, it is fairly accessible. If schools have computers, they likely have Excel. Further, the content is not my view of how the schools should teach math. It is a computer approach to what they in the schools are already doing. I believe that content designed that way has a chance. Content that is part of a faculty member's research project or part of their college level instruction that is then refitted for K-12 has a much lower chance of working. It is not impossible, but there are a lot of constraints that make replicating the college environment difficult in the K-12 setting. The concern, then, is not just that we try these other less likely to succeed approaches, but also that these other approaches end up crowding out what might work.