Caroline and I had a good conversation about math at Prairie Creek today and she asked me to write a reflection about what we're doing, what we could be doing and what this year's MCAs scores mean in the context of progressive education.
First, we do not discount these scores.
Lest, I sound defensive, I'll start by saying that I was surprised by the MCA scores and am looking forward to examining them in depth to find out how we can help individual students and where we can boost instruction to address weaknesses. I believe that if progressive education works, students should be able to show their competence in a variety of settings, one of which is standardized tests. Patterns in test results, even in a small cohort such as ours, can indicate areas we need to address in our curriculum. One such area that we've addressed in the past two years was basic computation -- we'd been spending a lot of time on math concepts and had not stressed basic facts and algorithms. We sought to balance these two poles by starting our math work with "foundation math." This focusses on the skills that are roughly akin to decoding and literal comprehension in reading. You can't enjoy a story unless you can read the words. Similarly, the beauty of the relationship between fractions, decimals and percents will be lost on you if you have to spend all of your time figuring out what 100 divided into four groups is. I believe that one of the areas we will be weaker in is estimation and another will be data handling. Although we use the latter regularly, we don't teach graphs and graphing vocabulary discretely. Both are opportunities for our curriculum development. As we receive more data, we'll be able to see more clearly "what happened" and address it accordingly.
The Challenge of Small Cohorts
I cringe whenever the MCA summaries come out as percentages -- even when we are well above the state average. At Prairie Creek, our current grade level cohort is 20 students. That means that each student represents five percentage points. Our performance seems incredibly erratic when viewed from a percentage standpoint. In fact, our results are from such a small cohort that most shifts from year to year are not considered statistically significant; our sample is simply not big enough. Add to that the fact that the state does not track the scores of a cohort and their improvement as they age (following a group of third graders and then looking at their scores as fourth graders). Instead, growth is measured by comparing this year's fourth graders to last year's fourth graders. It is somewhat valid to do this statewide but, with only 20 kids, our cohorts can vary dramatically from year to year. For example, many of last year's fourth graders struggled on the MCAs. Many fewer of them struggled this year. In fact, we scored above the state average for fifth graders meeting or exceeding proficiency. That growth is not captured in the way the state reports scores. Incidentally, many states have addressed this issue and have moved to following cohorts to assess growth and determine accountability for schools. It's more complicated in terms of record keeping but much more valid in my opinion.
What Does the Test Measure?
Overall, I find the MCAs, especially the reading portion, to be a fair test. They are untimed and they attempt to go beyond the multiple choice format and look more deeply at students' abilities. The reading portion really does measure students' ability to comprehend texts and write about them. The math portion assesses students' mastery of content more than pure problem solving ability. Vocabulary and familiarity with certain ways of presenting problems affect children's scores. For example, a student may be asked to take some data and create a pictograph of it. If a child does not know what a pictograph is, he does not receive credit -- even if he can represent the same data in an equally appropriate way such as a bar graph. A child's ability to comprehend data and create meaningful representations of it is not being tested, rather it is the child's vocabulary being examined. Should kids know the word "pictograph"? Sure. And we do work with pictographs occasionally, but we do not spend time drilling the word or how to make them. The word is not as important as the idea, which is intuitive for most children and does not warrant a lot of time in direct instruction (unless you make the decision to teach to the test).
Curriculum Bias in Test Design
I taught the Everyday Math program for years when I lived in Connecticut. I find it to be a solid curriculum that is well thought out, if uninspiring. The curriculum is ubiquitous and I believe that it is the foundation for the math MCA. The "In and Out" table is an example. It's a way of presenting a functional relationship and several lessons in the fourth grade Every Day curriculum teach it explicitly. However, it is not a standard mathematical representation the way a division sign or exponent are. On the test, a student would be given an "In and Out" table with two columns of numbers headed "in" and "out" and no further explanation. The question may ask for a missing number in the table. Many children, when looking at such a table for the first time, need support to figure out that the same thing happens to the "in" number to produce the "out" number. Children who have worked many "in and out" tables (i.e. those whose districts use the Everyday Math Curriculum) have a distinct advantage over those who don't. Similarly, vocabulary and concepts focussed on in Everyday Math seem to be emphasized in the MCA.
So, Why Don't We Use Everyday Math?
As I said in the last paragraph, Everyday Math is a solid curriculum but it does not spark passion and curiosity in most children. It is not progressive. I was pushed in my thinking about math curriculum a few years ago by a speech I heard on NPR. The author (whose name I now forget, although I'll continue to look through my notebooks) was talking about how we need to as a country move beyond creating children who are competent in computation and create a nation of kids who are passionate about math. He believed that the number of kids (and adults) who feel comfortable stating "I hate math" or "I'm not good at math" was a real indictment of our approach to mathematics instruction. I paraphrase, "Nobody says, 'I hate Shakespeare, I'm no good at it.' Instead, some people appreciate it at an amateur level and some at a professional. We could do the same with mathematics." We redesigned the fourth and fifth grade curriciulum so that computational math was taught discretely daily for about 10 minutes. Students worked to master basic facts and algorithms and build their foundations during this time. Then we moved into "Exploration Math" where students worked on a concept or challenge of some sort. We find many of these in the work of Marilyn Burns, our theme projects, and various other "challenge" math programs. These lessons are filled with "Wows!" and "Cool!" Children debate their ideas and sometimes even jump up and down as they figure something out. No, Pascal's Triangle isn't on the fourth grade MCA, but an awful lot of my kids sited it as their favorite math topic for the year. Descriptions of various explorations about can be found in previous blog entries:
The Home/School Connection
We have very strong literacy scores on the MCAs. This is in part because of the test design but it is also because literacy is stressed in many of your homes. You are readers. You read to your children. You go to the library. You talk together at the dinner table. Progressive education is truly a partnership because when learning is authentic it is not confined to the classroom -- it happens all the time and a child has many teachers. It is more difficult to be as supportive of numeracy in the home (hmmm...perhaps I'll do a workshop in the fall). Every curriculum night I stress how important it is for children to master their basic math facts. It is a rote skill that needs frequent work at home. Students, especially in a mixed age classroom, master the facts at different rates and it is not an efficient use of our time to address math facts as a class (although we do work on them independently in foundation math and children are required to be able to recall facts at a certain speed before they learn multi-digit mulitplication or long division). However, students often report that they don't work on facts at home. It could be that our focus on intrinsic motivation makes pushing to learn math facts at home less of a priority. Perhaps if we had "math fact tests" on Friday, parents would push students to work on them because "it counted." I know homes are busy and that, without a deadline or a tangible reason, this kind of skill work is hard to do. We've tried a variety of approaches and we'll continue to do so. Beyond basic facts (the equivalent of phonetics in reading) there are so many other things we can do to support numeracy -- cook together (for older kids, make them use only a 1/3 cup measure to make a recipe), make budgets, wonder together (how many of you would it take to reach the top of that tree? How many marshmallows would it take to make a s'more the size of your sandbox? How much is infinity plus one? (an especially good one for bed time)) I would love to strengthen our partnership with families in developing numeracy the same way we work together to create passionate readers.
Last Words
I'll close, although I still have some reflections on the impact of the multiage classroom on math scores (our fifth graders often report how much more confident they feel as mathematicians compared to their year as a fourth grader -- and their scores historically reflect that confidence), the math recovery program being implemented in Australia, the best way to develop mathematical thinking in all students... Know that we're still striving to improve our approach to mathematics and to create students who are competent and passionate about math, something which, sadly, the MCA cannot measure. I look forward to continuing this conversation informally via e-mail ([email protected]) and formally in a committee this coming fall. I hope you'll join me.