Comments

How homogenous are your (Smartest Tractor) classes?

One of the challenges I face teaching college math is that I have students in College Algebra for for whom the material is pretty much review of high school math that they just need a little bit of a refresher on. I also may have students in the same class who struggled through basic math and basic algebra. Plus the students somewhere in between the extremes.

If we had more students, I would be tempted to have a sequence of classes that covered the same material we now cover in two semester (3 credits per semester) over three semesters (2 credits per semester, but still meet three times a week). I think this might be especially helpful for those elementary education majors who do not need to go on to calculus, but really should have down the algebra concepts and everything that comes before.

-- MattGoff - 01 Feb 2006

"For the first few examples, students watch and listen my mental dialogue when I solve the task."

This is what I did in my math and computer science courses. This "mental dialogue" process was good because it forced me to slow down and explain all of my thinking and assumptions which might not happen if I consolidated it down into a simple handout. I would also start with a problem I had never seen before or have a student make up or pick out a problem. This is important because so many explanations in math and computer science are the result of a lot of thinking and false starts. In a neat book solution, you never see that mental thinking process. You just see a perfect solution. I think it's good for students to be lead through this process, not by semi-clueless students in a group discovery process, but by someone who knows the way.

I might explain two different approaches to the same problem to make sure the students know that they are the same. I might also explain what happens when they make common mistakes. I might make the mistake and continue with the problem to show them what happens. I would show them how they can check their work.

There is no substitute for good direct instruction.

-- SteveH - 01 Feb 2006

THIS IS FANTASTIC!!!

THANKS SO MUCH FOR DOING THIS!!!

(AND THANKS FOR WRITING IT UP IN SUCH A BEAUTIFUL FORM)

I'm going to add this to my 'armory' - my list of Handouts I take with me everywhere.

I should figure out how to make a nicer version of my Gambill handout.

-- CatherineJohnson - 01 Feb 2006

WOW

THAT IS INCREDIBLE

I'M PRINTING THIS OUT AND MAILING IT TO OUR MATH DEPARTMENT & PRINCIPAL

I think I'll just go ahead and formally 'out' myself while I'm at it.

THANK YOU SO MUCH

and: do let us know how the kids do -- tell us what level of improvement you see --

THANK YOU!

I'm logging this into the book-style index, and into Recommended Reading AND Math Supplements

-- CatherineJohnson - 01 Feb 2006

OK, I've got this logged in Recommended Reading & in the Book-style index]

-- CatherineJohnson - 01 Feb 2006

I think this might be especially helpful for those elementary education majors who do not need to go on to calculus, but really should have down the algebra concepts and everything that comes before

That sounds great to me.

-- CatherineJohnson - 01 Feb 2006

I might explain two different approaches to the same problem to make sure the students know that they are the same.

I think this is incredibly important.

I don't think a person needs to wait patiently for the moment when he's achieved 'expertise' to defrag his domain knowledge.

I would have had far less fragmented knowledge if I'd had more teaching like this - and I'm now, constantly, showing myself two ways of doing problems when I can tell I'm confused or that I need this in some way

For instance, I have MAJOR overlearning of ratios & proportions. I've based my entire life on ratios & proportions; practically every real-life problem I've dealt with I've figured out some way to make it into a ratio & proportion.

I'd never been taught ratio as 'rates' beyond the obvious mph you use in everyday life.

So now, every time Saxon gives me a ratio problem, I do it as a ratio problem and also as a rates problem.

I need the practice, but I also need the connection.

-- CatherineJohnson - 01 Feb 2006

Very good idea, too, starting with a problem you haven't done.

From what math people have told me, it's difficult to disaggregate your knowledge (I'm discovering that's true of me with writing - I had no idea!)

When I think about how to teach Christopher to write my mind instantly goes to......'sit down and write' or 'I just sit down and write' or 'I make a lot of lists and it comes to me' or 'I just write stuff, and then I revise a whole lot.'

I had NO idea how 'unified' my domain knowledge of how to write had become over the years.

I'm now saving individual drafts, so I can take a look and see what happens each go-round.....

but writing this comment makes me think it would be a very good idea for me to write a short essay in front of students. (My neighbor and I have already talked about having me do this for our two kids this summer.)

-- CatherineJohnson - 01 Feb 2006

SmartestTractor's technique is very similiar to the technique that DI uses for students at these grade levels -- model, lead, test, retest. One additional thing that DI does to keep the students maximally engaged is during the lead portion, every student is working the problems step at a time at their desk (and some at the board -- the students having the most problems are the ones working the problem at theboard).

During this time the teacher is walking around the room checking each student's progress and assessing each child's performance. The techer uses this immediate feedback to determine how much practice is necessary and to correct problem areas (gaps in knowledge). The feedback determines the pacing of the class -- when to move on and when to do more practice.

As long as the quizes come back with about 90% of the problems correctly answered and there is sufficient distributed (determined by biweekly, monthly, and/or quaterly cummulative tests that should also be coming back at about 90% corect) you have mastery learning.

-- KDeRosa - 01 Feb 2006

I think it is important to distinguish two different types of problems.

First are problems where you are given a formula or expression. In this situation, in every case, there should be clear instructions to say what the goal is, e.g. solve for x; substitute the following values and compute the numerical result; convert the fraction to a decimal; reduce the faction; add the fractions and reduce the result, factor out a common monomial, etc.

These instructions describe individual procedures (or sequences of procedures) that should each be taught to mastery.

Second are word or story problems. These are much harder in general because you are not generally told which procedures will be necessary to answer the question. These are the ones where it is useful for the teacher to explain their reasoning process and where there can be more than one way to solve the problem.

-- SusanJ - 01 Feb 2006

Martin Kozloff has written an excellent (and highly detailed) article on incorporating DI techniques into instructional design. (Word Doc)

-- KDeRosa - 01 Feb 2006

Here's another. (Word Doc).

-- KDeRosa - 01 Feb 2006

Reading these documents it's easy to see why teachers prefer constructivist spiraling curricula over DI curricula -- a lot of thought goes into designing an effective DI curricula. it's a lot easier to teach lessons in a haphazard, almost random, constructivist manner that spirals around so that there's no need for clear instuction and mastery. IF you don't care about mastery, then you don't care about how much practice the students are getting. In fact, you don't you don't care that not all the students are "getting it." All the better for some not to get it because now you have a grade distribution. Better to be a tour guide than to take on the responsibility of actual teaching.

-- KDeRosa - 01 Feb 2006

I guess I was thinking about Smartest's teaching method subconsciously -- I got to wondering whether peer grading is really as good an idea as having the teacher grade the daily quiz. I have my doubts. I think the teacher's being the grader would be more motivating (and I was a very underachieving, low-executive-function sort of kid so I know whereof I speak),

-- CarolynJohnston - 01 Feb 2006

How homogenous are your (Smartest Tractor) classes?

3 of 31 are on IEP (LD or MID) 3 of 31 have some serious gaps in their knowledge. 5 ESL students, all have acquired English to at least the forth grade level.

sequence of classes

The sequence is "suggested" to us by the board.

I might make the mistake and continue with the problem to show them what happens.

I do it a few times every lesson. Kids love correcting the errors. Checks are required by the textbook. Negative intergers and negative coefficients messed them up the other day. Today's mean was 87% and the median was 95, so we moved on to our unit review.

I got to wondering whether peer grading is really as good an idea as having the teacher grade the daily quiz.

The "initial screening" (marking) is done by the students, but I collect them to record their marks and go through them to check for "correcting errors" (missing negative symbols, etc.). Dramatic mark differences are dealt with immediately.

-- SmartestTractor - 01 Feb 2006

The "initial screening" (marking) is done by the students, but I collect them to record their marks and go through them to check for "correcting errors" (missing negative symbols, etc.). Dramatic mark differences are dealt with immediately.

OK, that's cool. I was thinking of a peer grading situation at my son's elementary school, in which none of the kids had the frontal lobe function to do the job. Also, those teacher-eyes on the paper matter a lot.

-- CarolynJohnston - 01 Feb 2006

SmartestTractor's technique is very similiar to the technique that DI uses for students at these grade levels -- model, lead, test, retest.

Ken - where is this stuff for older kids?

Is it in the big Doug Carnine book?

(I have my copy now, but haven't read at all...)

-- CatherineJohnson - 02 Feb 2006

good grief

you already posted the documents

i MUST go do my KUMON before I pass out on my keyboard

-- CatherineJohnson - 02 Feb 2006

Gambill has a kind of 'wave grading,' where she grades the first tests done, then those kids grade the next wave down....I think she's buys recording at that point. (I THINK maybe she's recording the first wave of quiz grades, while that wave of kids is grading the next wave.)

I do recall that anyone who didn't pass the quiz THEN has their homework checked.

Otherwise, if the kids can do the quiz, she assumes they've done the homework (which to me sounds reasonable.)

Kids who flunked the quiz because they didn't do the homework then have detention and do the homework 'supervised.'

IIRC

-- CatherineJohnson - 02 Feb 2006

model, lead, test, retest

-- CatherineJohnson - 02 Feb 2006