I like your map Maria. I do wonder if operations might fit under structures? But I will need to do some homework before I try to justify that thinking.
Yes, I agree with your second point even more. To get teachers to change, just like with students, you need to connect to what they already know and then change their schema or add enough information to it that allows them to re-evaluate their mode of operation. That's the difficult part, even when they themselves want change. It's very hard to change habits and more difficult to change habits that are embedded in such a structured environment like a school. (I remember when I came out of the classroom four years ago, and I felt bad about getting up and going to the bathroom. I felt like I was slacking because I could do that anytime I needed to)
I am working with a couple of projects now that are interesting in the sense that one forces teachers to change their instructional paradigm by teaching in a virtual environment 40% of the time and face to face 60%. It is very difficult for teachers to develop the connections between the two components and has been a source of frustration for them and the students. The other project is with teachers who have had lots of PD "done to them" over the last few years, and we have attempted to provide them choice in what they want to research and focus on. What's interesting about this second project is that while we are working with the teachers, there are state instruction teams in their schools because they haven't met their NCLB AYP goals. The state teams are mandating instructional change which mitigates any work that we try to do, because we don't have the same authority or same goals. It's interesting how such a complex system allows these kinds of dichotomous approaches to school/instructional improvement.
All of that said, I would love to open a discussion about the project or just chat with you if no one else seems to be of interest at this time. We are looking to develop a year long or approximately year long PD structure that works with teachers to develop algebraic understanding in their classroom. We would use a blended approach to the PD with some sort of F2F component supported by regularly scheduled distance conferencing. The topic of algebraic understanding is right up your interest alley. I read your early algebra article on wikipedia and think we have a very similar idea of algebra at the early years.
We haven't designed the PD but are working with our researcher to get the RFA submitted. The nice thing about these RFAs is that they don't have to be randomized control trials. Additionally, they are iterative in nature; which means that we get to test, restructure for improvement, test again, and when we are ready evaluate final efficacy based on latest improvements rather than on the whole set of data. Let me know how you want to proceed.
Maria, I'm sorry I haven't replied to this. I looked at the mindmap and find it interesting. I'm not sure i have anything to add to it, but find it interesting that statistical/regression modeling doesn't have it's own category/line. I know it falls into the different patterns/representations, but wonder if there isn't something different enough to include it as a separate item.
I have a question for you, with the new CCSSO mathematics standards creating separate topics for modeling and functions, is this going to have any impact on what you do with natural math? I see it as such a great connection to what you do that I would love to hear your thoughts on the new standards.
I'm with Kevin and others. Without reading the article I say YES. (I did read the article) I've taught gifted kids, in a spec ed pullout program, for 25 years and kids, parents and I struggle with this all the time. Many kids, not just the brightest, can move through the material more quickly. Sometimes it seems that the classroom teacher may be waiting for 100% mastery by 100% of the kids. Too bad kids can't move through math at a much more rapid pace and move from one grade curriculum to another seamlessly. Students who score in the 99th% on Fall achievement test move through the math curriculum with their classmates throughout the year and score 99%ile on the grade level Spring test. (No growth in a year? hmmmm)
We've used an Algebra program called HandsOn Equations with kids as young as 2nd grade and they love it.
An aside: There is a fascinating chapter in Gladwell's book Outliers called Rice Paddies and Math Tests. He tells some really interesting stories about the way we teach math in the US vs.the way math is taught in China Japan, Singapore, etc.
I've taught high school students who are severely emotionally impaired students, in a day treatment setting, for all 9 years of my teaching career. They come to me with a variety of achievement levels, the lowest being 2nd grade, and the highest being at grade level. I teach high school, and the average achievement level is probably upper elementary.
Since my first year, I've felt strongly that students should learn at grade level. I felt this for three reasons: it's not fun to keep failing at the same stuff every year, it's humiliating to be in high school and still learning elementary school stuff, kids have a right to learn complex ideas and be challenged by concepts like algebra and geometry.
I have had a lot of success. I teach my 2nd grade level kids the same stuff I teach my 12th grade level kids. They move through the high school state curriculum at their own pace, so no one has to wait for the rest of the class. To accommodate them, I avoid using fractions, decimals, or negative numbers when they aren't necessary to understanding the concept itself. I also encourage the use of calculators. One might expect that the kids achieving at grade level move faster through the curriculum than those that are behind, but that is absolutely not true. The pace really has more to do with work habits and attitude than previous test scores.
All of this is to say, I know that people of all ages, categories, and expectations are capable of learning material that is generally thought to belong to the older, working at grade level students. They just need the right instruction, classroom structure, and state of mind.
I am lucky to teach special ed, because I have smaller classes and can individualize more. I'm not sure how general ed teachers do it. But for me, it's possible. I've seen great things come from my students.
This is quite an eye opening article. I think educators should find new and creative ways to introduce materials to our students that will challenge them to think about the subject and learn at a much higher level. Children are very smart and creative individuals, it should be our job as educators to find ways to tap into their creative energy and steer it towards what we want them to learn.
"to tape into their creative energy and steer it towards what we want them to learn."
Yes, if I could do that I would be so happy. And the kids would be so happy.
Do you have any techniques for doing this that you've found works?
I think the key point in what you've written is that this should be our jobs. Sometimes it seems like our job descriptions are pretty basic, but you are absolutely right. It is our job to do the best we can for our students.
One thing I've found in our district, the focus since NCLB has been a 'one size fits all' curriculum with focus on struggling learners making progress on high stakes tests. I do think ALL learners have the right to learn something new everyday. With state testing over for the year, last quarter becomes a time to teach science, social studies, read for pleasure, go on fieldtrips. There seems to be no time for creativity in the general ed general ed classroom.