Hey there folks! Just wanted to give you a heads up that we have our first video completed and posted under the tutorials page. Please take a moment to head over there and give it a watch and leave a comment telling us what you thought. Thanks!
This post is by Jason.
Well, I figure that I should take my turn at a post. First, a bit of background on me. I am a 5th grade teacher who is interested in teach mathematics in a meaningful, relevant way to students. The interesting thing about my experience with math, is that I was never one of those students who “got it”. Rather, I was the one that would cruise through the lesson and do what I needed to get by. It’s quite surprising that a passion of mine now is teaching math.
For the past few weeks in my class we’ve been working on order of operations and expressions. I was curious to see how students would respond to making expressions from situations rather than just the standard teaching of the order of operations. So I started them off with small situations where they had to make a number sentence to show the situation. The students outdid my expectations. So, I figured that I would provide them with a more challenging context that would require some recognition of the order of operations and yet still have multiple ways to access the problem. I’ve included the situation so that you can use it as a reference.
While running a lemonade stand you’ve earned $22.48. Just as you are about to leave a person buys $2 of lemonade. Since you’ve been working in a group of 4 you decide to share the earnings equally. When you get home you pay your mom $4.68 for the materials that you’ve used. Write an expression to represent the situation above, and then simplify to find out how much money you made in all.
After I posted this problem on the screen, and answered any questions about it, the students set to work creating the expression themselves and then discussing it in their groups. As the students worked I walked around to see what they were thinking, prompting those who needed a little push with questions to scaffold their thinking, and in general finding out what they were thinking to help structure the discussion.
As the work continued there were several expressions that emerged as general patterns across the class.
(22.48 + 2) + (24.48 ÷ 4) + (☐ – 4.68) — the box represents the parentheses that are in the middle.
((22.48 + 2) – 4.68 ) ÷ 4
((22.48 ÷ 4) + 2) – 4.68
(22.48 + 2) ÷ 4 – 4.68
As those expressions were put on the board there was a sudden buzz that filled the room as each group automatically started comparing the other expressions to theirs. Our discussion started with the first one and progressed through all of them. As the students shared their ideas and methods it became clear as a class how important it was to follow the correct order of operations, and basically provided those rules themselves!
Their discussion was quite instructive as they reasoned about the different representations and what they meant. There were students getting quite invested in their ideas, and others became very interested in seeking to prove their ideas to others. This lesson also surfaced basic conventions about parentheses and even the effect of doing the wrong operation at the wrong time. It was one of those experiences where you wished that it was being recorded so that you could see the thinking and reasoning going on.
Now, I do want to be down to earth in saying that not all were at the same level, or got the same out of the lesson. However, from some data I gathered afterwards it became quite clear that all had learned quite a bit about order of operations and expressions than they had known previously. In fact, for a lot of my struggling students, this activity excited them as they saw how math could be applied!
As a class we continued to look and deepen, as well as practice, the skills of evaluating expressions both with context and without. The students all seemed to enjoy the activities and even asked to go deeper and try more challenging things.
I haven’t told this story to toot my own horn and make myself more than I am (I know I have plenty to work on as a teacher), but I do hope that it shows how much kids can learn and do mathematically and how well they can reason! Sometimes I fear with math education we expect them to not be able to get it deeply and therefore just tell and explain instead of letting them do the learning in the way that they learn math best.
Please leave a comment about a great math lesson that you had that really deepened you understanding and where a teacher let you learn in a way that really helped you see how well math could be used.
It’s funny how writing can change you. I’ve been making posts for this blog for a couple of months now, focusing on what I consider the important parts of math and math education. The ideas I’ve been writing down have been bouncing around in my head for years now, so you’d think I’d be the one in charge.
There’s nothing quite like writing to force you to really think about the words you use. The more I’ve been working on what to say here, the more I’ve decided that there is something I just must change. From now on, I’m no longer going to talk about “doing” math. Rather, I will talk about the much more active “using” math. When I can, I’ll even say what I’m using the math for.
Why the change? Partly, I realized that I don’t really know what “doing math” is. You sit down and do—what, exactly? Solve a page full of arithmetic problems? Boring. Try to solve problems in linear algebra? Why? Do the kind of crazy stunts I had to do as a graduate student in physics, involving math far beyond what we cover on this website? That was also doing math, but it was nothing like arithmetic or linear algebra. It’s too vague for a website devoted to making things clearer.
Secondly, math is useful. That’s why we have it. Students often ask “When will I use this?” Well, I say that we should be able to answer them, darn it! It’s a good question. If we change the focus from “doing” to”using,” we are focusing on what the students care about. Just about everybody has some use for some math, just like just about everybody has some use for reading. Let’s constantly keep that in mind.
This post is by James.
As part of my research for this website, I discovered a great site called “Math with Bad Drawings” by Ben Orlin. The whole thing is worth a look, but I especially like the post here, where he shows common math complaints, edited to be more accurate. I like it especially because it relates to my own essay, Math: A Victim of its Own Success, where I talk about just how very long it took people to invent math in the first place, as well as why they did it. Also, I personally cannot draw as well as he can (maybe), so I might as well outsource the drawing to him.