Experiecing School Mathematics - Ch 2

Two aspects discussed in chapter 2 raised enough curiousity within me that warranted an entry here on my blog: test design in standardized testing and mixed vs. ability grouping. In Newfoundland CRT tests at the junior high level are designed with 62% of the test closed constructed repsonses, exclusively multiple choice items. The items cover all three main stands of procedural, conceptual, and problem solving. The problem we've found in the past is that stem of a lot of these questions are so long in text and worded so poorly that many students cannot even understand what it is there're asked to find. In many instances too a lot of these questions required more workings, more thought, more understanding of mathematics that some open-ended questions would (which were worth three times the multiple choice item). A lot of weaker students would end up getting most of these questions wrong because a) they couldn't process all the reading of the question and b) most students have the impression that multiple choice questions should not require a lot of time and work to find the correct answer. Here's a questions from the CRT test in 2005.

Josie noticed a rainwater barrel read 18 L at 2:00pm. At 3:00pm it read 14 L and was leaking water at a constant rate. Josie got back at 3:30pm with a 5 L bucket to catch the water until she could fix the leak. There was 12 L left in the barrel then. How long, in minutes, will Josie have to fix the leak if she works until her bucket fills up?

(A) 30
(B) 60
(C) 75
(D) 125

Some will argue it is a perfectly fine question and perhaps it is. But for one mark in a test I think is perhaps a stretch. I remember looking at this quesiton in particular the following year in my first year of teaching. I gave it to my class on a chapter quiz to see how they would handle it. I recall there being a lot of questions from students not understanding the premise of the question. Perhaps it's a language problem, perhaps it's a lack of ability on the student's part to think critically. That's just one example, perhaps there's a lot more out there. My point in all of this is that if the standardized tests are to reamin then their design styles need to become more open response in nature. Students will only put down their workings on the page in a coherent manner if they know there's a chance they can earn marks from it. Of course it's about the money - it's a lot cheaper to put a response form through a solution feeder to spit out the results than itis to hire hundreds of teachers to mark the papers. But, really how can we discover the falsehoods students have in their mathematical thinking if an evaluator will never see their work on the question. It would be interesting to have a pilot CRT with exclusively open response items and then have follow-up commentary and research done on how much better or worse the students would perform, in addition to antedotal evidence on their views about writing such a test. We definitely need to rethink how we assess our own students' thinking. I've already been a fan of oral defenses, mathemtical modelling presentations, and other alternate forms of assessment to provide my students with the opportunity to show how much and what they understand about mathematics through alternate means from pencil and paper.

The other idea raised by Boaler here that I've often debated with myself is how our math classes are organized. Amber Hill used a system of teaching their students in ability grouping, or sets, whereby the different sets were taught similar content, but the higher sets were generally taught at a faster pace and covered more difficult material. At Phoenix Park the students were taught in mixed abilit classes through all three years less three weeks before the national exams when they were place in target groups for the test preparation. Here in Newfoundland we have mixed ability grouping for the first ten of thirtenn years of the students schooling. During times in my practice I wished for ability grouping because I seen so many of my brighter students bored and dragged down by the pace of our classes. I seen so many weaker students frustrated by the seemingly fast pace I was "covering the curriculum" Yes I too was one of those obsessed with covering the curriculum, teaching with the text. Perhaps, there in lies my and my classes problems! But in any regard I felt I could not provide the help the weaker needed and coul not meet the expectations of my enriched students. The job of teaching to the "middle, normal" students is frivilous. I was frustrated with what was happening in my math classes, feeling helpless in an attempt to save the class and save myself. If only I could have them all separated accoridng to ability I though so many times. Unfortunately I never gave enough attention to what would happen to those int he bottom "barrels" Boaler says "the set in which students are placed has significant implication for their attainment some years later." Powerful statement. And so from that I asked, "Should I really be deciding which opportunities a student should have access to later int heiur life right now when they are merely 13 or 14 years old." If we decided to initiate such a program whereby a student was put in the lowest set in grade 7 and as a result could not apply to any university or college program because they would not have the pre-requisites met further in their secondary schooling. I seriously found myself begininng to realize that we need to stop the practice of cutting the legs off from under our children in their early teenage years. We're basically throwing those who don't fit the norm, who don't comply, who don't understand the way we teach, into courses that bring them to a dead end, that close more doors in their lives than are opened.

If only we would lay down our texts and answer keys, then maybe we could develop a system that allows the weakest of the weak and the strongest of the strong to co-exist in the same classroom. The teachers at Phoenix Park did and it worked. Why can't we?