This week, I’m supposed to teach my students how to solve a system of equations by elimination. It’s the hardest of the four ways for solving systems (graphing, substitution, guess-and-checking), and I’m not entirely sure everyone in the class gets the first three. The main point of the unit is to determine exactly where two or more linear relationships meet, if they ever do. The situation could be realistic (two cars running a race, two trains on a schedule, cell phone pricing plans) or already abstracted (simultaneous equations), but I’m hoping they can find this meeting point because it’s really interesting. Is it relevant to their daily lives? Perhaps. Do I care? Not necessarily.

Really, I don’t always want the things that people learn to be within the realm of possibility, because I’m trying to prepare them for the improbable as well.

One missing piece of the Common Core State Standards debate is that, while I’m glad to teach students higher order math that puts them on the pathway to do well in high school algebra (and trickles into geometry, trig, and hopefully calculus / statistics), I can’t, and sometimes don’t want to, *always* do what the standards are asking me to do.

After nine years, I’m still trying to find ways to push my teaching – and their learning – further, and I believe I’ve earned the right to say what works and what doesn’t with any group of students. I often find it’s a waste to try and create a week’s worth of lesson plans when things are bound to change from day to day. I know *what* they have to learn and I have a given window of time I’d like to finish the learning by, but I’m also trying to maximize student learning if possible, meaning that I’ll have to come up with six ways of explaining something and hoping the students catch at least two of those.

That’s the thing with math generally. People are taught that we ought to teach math only one way, and, if not taught this way, they’ll never get it. I’m happy just knowing that, out of the hundreds of ways of doing a problem, they at least caught on to one. Because I *really* don’t have all the answers. I have mine. The next person might have a better way. I have no ego or qualms mentioning this all the time.

At some point, however, I’d like to get them confident enough to not have to depend on whether I know it or not, but that I gave them some tools for them to do it themselves. Because maybe the math we’re doing isn’t necessarily relevant, but it could be useful for the things we haven’t yet used, or necessary for the things we haven’t yet needed.

## Comments 4

The unit on systems always ties me into knots philosophically. Students don’t like it, don’t find it relevant, and don’t retain it. Our geometry teacher finds himself re-teaching systems every year.

I wonder if we give ourselves a wrong goal when teaching systems. I am not at all sure the purpose is to figure out where two lines meet or where the common solution is. I wonder if there are bigger principles that we’re glossing over that are really at the heart of the systems unit.

* You can substitute one quantity for an equal quantity in an equation, and it can shed light on the situation further.

* You can subtract one equation from another equation – it leaves the equality in balance and can either simplify or put the situation in context.

* Graphing two equations can give you information on how the equations are related.

* Systematic guessing-and-checking is perfectly valid, though optimization strategies help (using a computer, for example)

The bigger principles are universal and useful in more situations than just solving a system. If a student really understands the meaning of equations and functions, they can be versatile problem solvers in more situations than just solving a system by substitution.

>It’s the hardest of the four ways for solving systems

Actually, there are other methods than the three (we don’t consider “guessing” ever appropriate for a mathematics classroom). By solving systems with arithmetic, not algebra, such problems can be moved from high school to 6th grade or so and students can spot the underlying patterns of numbers, which are elementary school concepts. In one such arithmetic approach, Singapore attempts to teach solving systems with bar diagrams, but they certainly didn’t invent the notion of using visual aids, and besides, Singapore shoehorns students into the limited goal of solving problems rather than the broader, more useful goal of recognizing patterns.

http://fivetriangles.blogspot.com/2014/02/143-bicycle-ride.html

This bicycle ride problem can certainly be solved with algebra, but as you say, you can be left “not entirely sure everyone in the class gets” it. A more important “teachable moment” can come from students’ recognizing the change to the total time that occurs when the slowing point is shifted a fixed amount. Algebra never elicits this; it can only be shown with an arithmetic approach. Yet when an arithmetic sequence is discerned, not only can solutions subsequently be found, but a broader conceptual notion of systems is gained. Consequently, when algebra comes along, its basis is already established. That makes the transition to algebraic methods come easily.

You are asking great questions (in my opinion)! We should be reflecting on what we are teaching and its relevancy. If we have been in the classroom for several years, we have lots to learn but we also develop a sense of what the kids need to know to be successful.

My advice with regard to the math – ask the students how they would make what they are learning relevant (IOW: ask them to put themselves in your shoes). Reach out to the business and science worlds and see how these equations are used (and this way of solving problems, too).

Good Luck!

I recently used this blog post in a workshop, and I thought I would share with you some of the comments educators wrote about it:

– Very cool

– Time is the variable, learning is the constant.

– The CCSM don’t tell you what to teach, they tell you what students should know at the end of the year. See a beautiful paragraph on page 5 about topic A and topic B.

– So how do you make decisions about how long to spend on the mechanics of the CCSS so that students can demonstrate what they “know.”

– Who says there is only one way to teach topics?

– I believe each math concept has a relevance even for a historical perspective — where the ideas came from.

– @MeaganRhodes: I recently attended the NCTM conference and in every session heard math teachers muttering, “now that explanation of the formula makes sense to me.” I think you are doing the right thing!

– If you teach it “that way” and they don’t get it, why is that still THE way?

– Placing the topic within a context will help motivate students to want to learn the method.

– Learning for the sake of learning is enough.

– I think when you consider the mathematical practices, Common Core does support your way of thinking.

– I completely agree, I feel that way about the standards too at times!