We reached a new milestone in the MLCS pilot this week. Students actually wanted to use algebra to solve a problem when they weren't required to.
Upon hearing that, I immediately thought: I can retire now.
But seriously, I have never had an experience teaching developmental math where students wanted to use algebra. In a traditional curriculum, we force it on them promising a world of brain calisthenics unmet by any other means. They balk because they don't like it and find it useless. Whether it's useful for everyone is a matter of debate, but there are many folks like myself who will use algebra to solve problems in real life because we know how to. I see it as a tool in my tool belt along with pictures, graphs, numerical approaches, and technology. And now, so do my students.
This week began with us solving all kinds of equations physically with manipulatives and then in written form. Students did not like using manipulatives at first but we forged ahead, telling them, "this is a broccoli moment. You may not like it but you need it." And it did help. They understood why we subtract something from both sides or add it. They could see why we don't divide off 2x on both sides when finishing a problem that ends with 2x = -16. They saw that adding 3x to both sides is not the same adding 3. But our goal in the course is not algebraic manipulation for the sake of it. We want to solve big, interesting problems and let the algebra come along for the ride. And it does.
So once we had established how and why we solve equations like we do, we got back to the task at hand: solving a problem and determining if we should write an equation or use numbers or a table or graph. And that's when we heard something very satisfying for a math teacher: "Can we use algebra to solve this? It would be so much easier."
Why yes, you can.
We started with a fun problem setup that was completely real: you only have so much money (a $20 bill) and are heading to a restaurant for dinner with your friends. How much of something can you get for your money? We started simple with just the food items and then built all the way up to incorporating tax and tip. Not calculating tax and tip, but solving a problem like this: you have $20, you want to figure out how much of an item you can buy knowing its price and that after you purchase it, you'll add on 7% tax and a 20% tip. That's not a trivial problem. And we told them: solve it with just numbers and then with algebra. And along the way, we asked them to determine which method they preferred with each problem and why. When we finally got to this scenario (with tax and tip), students could not see their way out of it numerically. But they could with algebra. This is the equation they built and then solved:
1.20[1.07(1.50 + 5 + 0.25w)]=20
And without question, the solving was the easy part. It was creating that monster that was hard. By the time we got to the solving, they were breathing a sigh of relief.
But the best part was that they were completely engaged and really thinking. One student said, "my brain hurts but I like this." Heather said it's like they're sweating on the inside, meaning really working their brain. I used to say to my algebra classes that algebra will do that but often it doesn't. It just frustrates students and reduces their motivation to enjoy and succeed at math. When you tell them they can solve a problem however it makes sense and not force a method on them, they'll gravitate towards the things (like algebra) that we know can be quicker and more useful. It's just human nature: we like choice, we don't like mandates.
We ended the week with more problem solving that resulted in equation building, graphs, looking at linearity vs. other types of graphs. And then out of nowhere today, we saw a connection and had them build a rational function. We explored fitting data with the best type of function and how the graph doesn't tell you the whole story about data. But a table and an equation together with it can tell you everything you need to know. Again, algebra came out of nowhere (seemingly to them but we had planned as such). We had a situation that we wanted to understand more about and doing the calculations to be able to make the graph was getting time consuming. So we went to Excel. They've started to see the power of Excel. We talked them through programming the columns to do the calculations we wanted, filling in cells, and then graphing. To do that programming, they have to understand the calculations at hand to generalize them.
It's amazing how motivating something can be when it has an immediate point. I love math just for its own sake, for the beauty of it. But most of my students don't and need more to keep them interested and willing to work. Just having a real reason to do something that isn't contrived can go a long way to keeping them in the game and moving forward.
Altogether, a good week.
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