There are a few of you out there who will know what I'm talking about when I say that for some of us, the very same aesthetic resonance that takes place for artists when they create happens to us when we do math, solve problems or puzzles, play certain kinds of games, etc.
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| courtesy xkcd.com |
That's part of one of the challenges that I outlined in my last post. In reality, it's a challenge for all schooling. How do we design our programming so that the most engaged and able students are able to push the limits of their ability while at the same time allowing those who are less mathematically inclined to move in a positive direction? I'm not sure I have the answer, but I'm at least going to try to suggest some starting points.
I'm going to start my "here's some things we should try doing in math" post with a mission statement of sorts.
I believe that Math education should allow each student to explore, understand, create, and communicate mathematically to the height of his or her potential.
It's a simple enough statement, but its execution may seem daunting. How do you determine what the height of a student's potential is? In what ways are they supposed to explore, understand, create, and communicate? Why isn't there anything in there about learning? In my mind, none of those details actually matter. What matters is the statement acts as a litmus for all the other endeavors we pursue. One only needs to ask this question: "Does this action promote each student's potential in math exploration, understanding, creation, or communication?" I'm sure you could toss in a couple other nouns in there as well, but you get the point.
On the surface this seems like a waste of mental energy. Isn't that what math curriculum and math classes are already designed to do? That may be what the intent of the design is, but it is most certainly not the prevailing result. I see disconnects between some of the practicalities of teaching the subject in our current schooling paradigm and maximizing learning for each kid. "Each" is a very key word in that mission statement above. The goal should be try to get every single student moving forward, and leaving no one behind, most particularly during the formative and foundational early years.
With that in mind, I'm going to pretend I'm the Emperor of Math Education for a day. Here's where I would start, if I had unlimited ability and resources to effect change:
Suggestion #1:
Research everything. More specifically, I would direct a great deal of energy into nailing down the specifics on how the genuinely effective teachers seem to be getting their results. I am almost certain that there are enough people out there doing an amazing job of teaching math that if we could simply collect and disseminate their wisdom efficiently, all would be right.
The lines of communication between the research community and the teaching community are threadbare and one-directional. The researchers are doing research, and some of that information is making it to teachers -- at least those who are keeping a watch out for it. Very little information is making it back. We do research every day into what works and what doesn't in our own classrooms, but rarely does that get documented and reported in a meaningful way back up the research chain. Highly effective teachers may get recognized with awards or accolades, but how often are these folks given an opportunity to show the rest of us how it's done? Not nearly often enough would be my answer.
All of that needs to change.
Suggestion #2:
Develop a plan for year-by-year remediation for students that are demonstrating weak foundational skills. There needs to be methods and practices established for students who aren't operating "at grade level." We do this for our English Language Learners who arrive at our schools from other countries or regions and don't speak the language fluently. ELL students often receive resource help and/or alternative programming to assist them in their linguistic readiness. This needs to occur across years and courses in math as well, and not just for students with identified learning disabilities. Frequent revisiting of prior learning and reinforcement need to be part of every student's experience.
The anti-testing establishment will likely have a problem with me here, because it requires that we regularly measure student progress and ability. I want to make a very fine point here: when I test it's to find out what I need to do as a teacher to help kids grow and learn. That is what I believe the purpose of all testing should be at its root. Government-issued testing also carries with it the "accountability" element we hear of in the media. What that really means is that it's an indirect test of the teachers as much as a test of the students. I'm neither ideologically for nor against this practice. It all depends on how the information is gathered and used and what impact it has on student learning.
Whether the ongoing testing occurs inside individual classrooms in isolation as designed by one teacher or across entire school jurisdictions as standardized tests, the necessity of monitoring student's ongoing progress in math is vital. Good assessment serves the learning process, and my goal here would be to identify those students who are starting a year or a course with holes in their skill-sets, steering them to resources that will fill those holes, and thus giving them a chance to be more successful at learning the incoming content. Test early and act accordingly.
Suggestion #3
Personalize the learning for students as much as possible. I recognize that this is very challenging, particularly in jr/sr high school where a single teacher can be responsible for the instruction of many dozens of pupils, often for fairly brief periods of time (perhaps a single 8-9 week term or a 16-17 week semester). A teacher might be responsible for 3 to 6 different courses or grade levels at one time, never mind the total number of students they must attend to. These, frankly, are all issues in their own right, and likely deserve careful design consideration. One cannot honestly argue that a single person responsible for the differentiated and individualized instruction of 100+ students is going to be the most likely method for maximizing each student's potential. Teamwork and collaboration here are minimally necessary.
Within the context of a single class, however, there are ways that teachers can vary learning. Here are a few that I've either encountered or tried. Note of course that the context here is a high school setting, so some suggestions may not fit in lower grade levels, and some that would be obvious at a younger grade might be missing.
- Split formative work into multiple levels of difficulty. Assign each section of work or question a "point value" based on difficulty or time needed to complete. Have students work toward completing a certain number of points or difficulty levels. More advanced students could be encouraged to do problems that challenge them more deeply while skipping the "no-brainers."
- Provide open ended tasks where there are multiple possible solutions and encourage students or groups to find as many as they can.
- Use assessments, formative work, and projects that allow students to show reasoning and work in multiple ways.
- Make use of online and flipped classroom resources (e.g. Khan Academy) to reinforce instruction or challenge more advanced learners. These resources can also be used to allow students who are capable of accelerated learning to advance more rapidly, potentially freeing them up to work on projects that are more meaningful to their learning styles.
- Model and make use of manipulatives and visual aids in small group settings for learners that are tactile, visual, and or expressive.
- Make use of small group and paired settings for students who are interpersonal learners. Pairing students who are eager to coach with those who prefer being coached is effective.
- Record lessons, vlogs, tutorials, or problem walk-thrus for students to access independently. This is especially handy when students are working at home or have missed classes.
- Focus assessment and communication on outcomes and competencies. When students are demonstrating their learning, responding only with percentages or letter grades doesn't sufficiently communicate where they are missing the goals. Simply "going over the test" after may not be sufficient. Structure grading so that it communicates how well students perform at each of the many competencies that might be tested on a particular summative assessment. Don't just grade the test. Grade the skills.
Suggestion #4:
Eliminate "failure" as a destination. Reduce size and scope of summative assessments so that they can be generated quickly and repeatedly. Allow students to be as iterative in their math learning as they are in their creative pursuits.
Suggestion #5:
Create "space" in timetabling and school structures for students to be able to complete course work outside of a "normal" time frame. For example, a student might complete a single unit from a course in a subsequent year rather than repeating an entire course that they did not fully complete.
Suggestion #6:
Once students are past a point of being as mathematically literate as they need to be in order to function in everyday society, let students and their parents decide what kind of relationship they want to have with math going forward. I can't say exactly when this should take place, but I have to think that it's somewhere around 8th or 9th grade.
Not everyone needs to do math. There I said it. Whew. Felt good, but also a little sad. By the time kids are well and truly into high school, there aren't many of them who don't have a pretty solid pulse on where math fits in their lives. They might not know what they want to be when they grow up, but they probably have a pretty good idea whether that's going to include math. If we're doing the kind of job we should be doing leading up to high school, and ensuring that as many kids as possible are reaching their mathematical potential, we'll still have plenty of students pursuing math courses. If we're going to continue to offer vocational math programs for students who lack strong academic math skills, let's tie those programs in with vocational programming or other courses relevant to these students. This is a stage where drill-and-kill really does. Make it real; make it relevant.
Suggestion #7:
Find really cool things to do in school that use Math. My class builds houses. Measurement, trigonometry, geometry, surface area, volume, and even a bit of algebra is a lot more worthwhile for both the math-inclined and the not so math inclined. Naturally, we're not going to have every high school kid build houses. So find your own houses. One way to do that is to have your math teachers working directly with your teachers in other subjects. Find the places where math fits and connect the dots. No, you won't be able to complete every math objective that way. But it'll make for a nice break from the everyday.
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Many of these suggestions can be put to use in a typical math classroom. But many of them speak to the need for structural changes in the ways that schools operate. If we're going to shake it up... let's shake it all up.
I'm going to offer one last suggestion that's related but not in quite the same vein:
Get high school kids coding.
There are plenty of gamers who want to learn how to make games. There are plenty of kids who find learning in games to be engaging. I see a fit here.
TL;DR There are a bunch of things I think that we can do make math education more effective and meaningful, leaving fewer kids behind. If you want to know what they are, you'll probably have to actually read the post.
Vive la resistance!
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