I've been scratching my head, rather anxious about our prospects of success. Our algebra 2 course is essentially a class about the major functions of mathematics - polynomial, rational, radical, logarithmic, exponential, and trigonometric. For each unit, we do the same set of things - graph, solve, and model.
After working with them for a couple of weeks, I developed a suspicion that they don't really understand what we're working with - specifically, the equals sign. They can somewhat reliably graph and solve basic equations, but I've seen little evidence that they understand what these actually represent.
So, I took five minutes from class yesterday and had them answer a few questions. Here are the questions with some typical responses:
- What is an equation?
- A problem that can be solved mathematically.
- Something that has an end solution through multiple applications.
- A problem you solve to get an answer.
- It is a sequence of #'s and variables, and your job is to find the right answer to it.
- Numbers or letters used to represent or solve something.
- What is a graph?
- A grid where you plot points.
- A typically square thing you draw a line on.
- Where points are plotted and to show increase and decrease.
- A graph is a visual equation.
- A graph is like another way of showing your work when you solve an equation.
While there are some glimmers of hope in there, I think this is going to be a barrier for us. Students will not be able to reach the depth of understanding that I desire unless they understand what that damn equals sign actually means. And I think this boils down to looking at operators, relations, variables, constants, and expressions. Maybe not using that language, but building on those ideas.
So, here's my plan. I'm going to students do an open sort with the following:
- Relations: =, <, >, ≤, ≥
- Operators: +, -, ÷, ×
- Variables: x, y, z
- Constants: 0, 1, 2, 3
- Expressions: 3x + 4, 5 - 1, 4ab
- Equations/inequalities: 2x ÷ 5 = 4, 2 - y < x, 5a ≥ 8b, 1 + 1 = 2
After the items have been sorted, I'll encourage student to create a hierarchy of their groups. I really want to focus on the groupings - what do things have in common, how are they different, and what do they represent.
In the end, I'm hoping that my students have a better understanding of equality as a relationship, and that an equation is simply a statement that two expressions have the same value.
I would really appreciate any feedback before I give this a go!
You are certainly hitting on one of the hardest aspects of teaching high school math. I am currently working on solving systems of equations with my algebra students and am running into some of the same problems-- particularly with substitution. It's also an issue with variables in modeling. My students can replace the "cost" in a word problem with y, but once they solve to get y= 45, they can't tie the value back to the original problem.
ReplyDeleteI know none of that is helpful, I guess I'm just commiserating. I like the idea of doing the sort and hierarchy, particularly as it forces students to reach higher levels of thinking. I hope it goes over well-- I doubt my freshmen could handle that task well.
The thing this really reminds me of is the lecture we saw by Sean Cornelly last year were he said he didn't get algebra until the end of undergrad when someone finally explained that = means is. As such, I say is A LOT in my algebra class!
Good luck, and let me know how it goes.
Yeah, I try to use "is" a lot, too, but I don't think my students really understand what that means because we haven't explicitly talked about equality as a comparison. If I can get to the point where they really get "is", I'd consider that a win.
DeleteWe'll see how it goes. I think I need to figure out how my students might sort it based on their current understanding.
I don't know how helpful this post will be, but it gives some insight to relatively recent research on students' understanding of equals and suggests some kinds of tasks that you could give to students to help diagnose misunderstandings. If you'd like me to dig a little deeper into the research, just let me know.
ReplyDeleteThanks - that post is absolutely helpful. I'll make sure to read the article you reference in the next day or two (especially since it's from my alma mater!). Hopefully that research will give me a few other ideas for activities.
DeleteI remember talking about the operational view of equality in my prep program, but I certainly did not understand just how much of a problem it can be.
Good luck! I'll be very interested to read about your findings. As a physics teacher I am constantly having to pause to re/pre-teach fundamental mathematical concepts. Almost all the kids do actually know the stuff (here in the UK and Germany) it's just that so many find it impossible, or are unwilling, to articulate and communicate - perhaps the lot of adolescence? One technique that seems to help many is to increase the frequency of meaningful use and practice. Possibly easier in physics (applied mathematics) than maths (pure)? One of the age old problems is the inability or unwillingness to use the skills and knowledge in 'unfamiliar' situations (e.g., a pupil could easily use fundamental algebra in maths and then completely stall in science; when exactly the same skill is required.) One small point to mention is that not all kids are passionate about understanding the subject, they simply want to tick the boxes and jump through the hoops. The same can be said of teachers, managers and educational 'experts'.
ReplyDeleteMuch food (healthy, low fat!) for thought here: Thinking Mathematically, J. Mason, L. Burton, K. Stacey, 2nd Ed. Pearson. ISBN 978-0-273-72891-7 (I can't remember if I showed this book during your visit?)
This site may also help some students to help themselves (UK and IB biased) www.livemaths.co.uk
Here is a book (one of three, for 11-14 year olds, UK again I'm afraid) that I found very useful in developing interest, enjoyment and skills. http://openlibrary.org/books/OL10123116M/Task_Maths_(Task_Maths) sadly, its now out of print but I'm sure it can found online somewhere (all my copies were stolen!)
Hope you're still having fun - see you again in 2 years?
Nick