I feel like I've had the same conversation many times today as we review linear equations. Students have been given an equation in Standard Form and are asked for the x and/or y intercepts. It has gone something like this:
"Mr. Cox, I don't know how to do this."
"What's it asking you for?"
"I am supposed to find the x-intercept."
"What's the equation?"
"2x + 3y = 6."
"So what do you know about all x intercepts?"
"They're on the x-axis."
"Alright, then give me an example of an x-intercept."
"That's a number, give me an x-intercept."
"Give me another."
"And one more."
"Ok, now what can you tell me about all of those x-intercepts? What do they have in common?"
"y = 0."
"So what do..."
"Oh, that's right, I let y = 0 and solve for x."
It's funny how the default is always "I don't get it." Don't let 'em fool ya. They know more than they let on.
Monday, April 5, 2010
Posted by David Cox at 2:00 PM
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I do the exact same sequence of questions. It works every time. Any suggestions on how to get students to the point where they can ask themselves the same questions before they ask you?
Actually, the x-intercept (or y-intercept) is a number, not a pair of coordinates! For example, if you use y = mx + b as the standard form of a line, you surely say that b is the y-intercept, not the y-coordinate of the y-intercept.
That's a great question. I try to make my students aware of questioning techniques. So not only do I model them as often as I can, I will ask them things like, "what question could you ask yourself right now." They need to learn to identify what they do know about something before they can figure out the things they don't know.
Are you suggesting that the y-intercept isn't a point on the coordinate plane? Is this an issue of symantics?
A good number of my students haven't even read the question before deciding they "don't get it!"
Yeah, I have some of those too.
I think Larry meant that we use only the y-coordinate when talking about y-intercepts. Typically we say the y-intercept is 5 instead of (0,5).
"What question could you ask yourself right now?" I'll try to use that kind of language more.
I love this-- thank you so much for sharing such a concrete example. As beginning teachers, my friends and I often struggled to identify why sometimes we intuitively knew what questions to ask, and why sometimes we ended up asking really dumb questions. Now that I'm observing teachers, I see many of them struggle to break down the thought process thoroughly enough to identify these questions and then frame them so they're not leading. I'd love to use this as an example in our training!
As a side note, one of my favorite questions for students was "what do you think I'm going to ask you next" and I was always surprised by how many of them could tell me something like "you're going to ask me what I already know" or "you're going to ask me where in my notes I can find the answer." Yes, so you don't really need my help at all, you just want me to hover over your desk and hold your hand a bit. Which is fine. But grow up :)
This is great. Some people think that being a great math teacher is all about knowing a ton of math and performing great lectures. Really, it's all about creating really effective feedback loops. You need to assess what the kid does or doesn't know and then provide feedback/questions to move the kid in the right direction.
Automating this (finding a way to get kids to ask themselves) would be lovely. Maybe every math teacher should put these questions on a card and laminate it for each student.
Yeah, I'm sure that's what he meant, but I am not sure that within the context of my student's problem allowing a mere number would suffice. In order to understand how to find the x or y intercepts, she has to know that it the other coordinate is "0." We can call the "b" value in y = mx + b the "y-intercept" but I'm not sure it does any thing to further their understanding. The question is why is the constant the initial condition/y-intercept?
I enjoy writing about these exchanges because it's easy to have them and let them slip away without realizing what actually happened.
I really like your question to get students to think about what you may ask next. Let 'em get in your head a bit. Nice!
Man, you hit the nail on the head. If you want good lesson plans, you've come to the wrong place. My plans are underwhelming at best. I have learned to ask the heck out of a question, though. It's a skill I can bring everyday and it doesn't depend on how much time I have on the front end.
Weighing in on the y-intercept number vs coordinate pair issue... I push hard to have my students list the x- or y-intercept as a coordinate pair. The value of "b" in y=mx+b only becomes a y-intercept when you let x=0. Otherwise, it's just a number in the equation. The idea of saying b is the y-intercept is a shortcut we use for ease.
On the questioning issue... I love the progression of questions in the example. In addition, I also force them to formalize an actual question. My response to "I don't get number 12" is "What are you really asking? Ask me a question I can answer. What exactly is confusing? Where are you stuck?" It forces them to think a little and often shortcuts the process -- I dont waste time explaining something and then see that they already had part of it done on their paper.
Paul and David, being a good math teacher is first and foremost about being a good teacher. IMO, math knowledge is an important but secondary consideration.
Students in general have long-ago figured out that they can usually wait out a teacher. "If I don't respond, then s/he will answer the question for me." In all aspects of school - curricular, discipline, or anything else - I find the two most important things in being a good teacher are:
1. Asking good questions (as exampled above), and
2. Allowing/expecting the students answer those questions.
Thanks for sharing this. This sort of exchange is what I always hope to have, what I sometimes forget have, and what all teachers need to be taught to have.
This "effective questioning" is a huge topic for me as a secondary numeracy support teacher. We have asked teachers to consider the problems students might have and then what line of questioning would help them get to their answer without giving them the answer. It's difficult but absolutely necessary to develop our students problem solving skills
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