Monthly Archives: March 2014

Rational Expressions – Really Complex


Rational Expressions – Understudy


The Dreaded Worksheet

Researchers and experts have put down the concept of the “worksheet” over the last several years. At the same time, math teachers still find value in it. The argument from teachers is that kids need practice, and this is true. In order to master certain mathematical skills, like factoring quadratics or solving linear equations, to the point where it’s almost automatic, a minimum amount of practice is necessary.
The analogy most teachers use is related to either sports practices, or practicing a musical instrument at home. Both analogies are certainly valid, but without extending them fully, the meaning and purpose of the worksheet becomes lost.
Granted, when I was growing up, I did worksheets in math. And I did them because teachers assigned them. Like many teachers, I grew up in an age where students didn’t question what was going on in class – we just did it. But kids are different today, and it’s not necessarily a bad thing. Kids today want to know the purpose behind what’s being taught. They genuinely want to know, “why do I need to know this,” and “when will I use this”. While knowing the answers to these questions is important, that’s not the purpose of this post.
Let’s get back to the analogies and how they are incomplete. I’ll start with a football practice. I would equate a worksheet, which is individual, to practice your skills at home by yourself. The purpose of doing this is so that when you practice with the team, you are as valuable as everyone else out there. Team skills are just as important as individual skills. And both of these forms of practice, individual and team, prepare everyone for the next game, which would be the equivalent of the test.
So where is the missing piece of the analogy? It’s the team practice. What part of the classroom experience is equivalent to the team practice? Most classrooms don’t have one. There is no motivation for an individual student to increase one’s skills for the betterment of the entire class. And why should there be? There is no “class grade,” just grades for each student. Sure, there are group projects, but the group members change at the whim of the one creating the groups.
The analogy of playing a musical instrument in an orchestra or small ensemble falls apart here too. A musician will practice privately in order to make the group rehearsals more meaningful, allowing them to focus on listening to each other rather than individually mastering a certain passage in the piece. But where is there a “group rehearsal” component in the classroom?
It’s probably a better idea then to shift the analogy to individual sports and musical performances, like a tennis player or a solo pianist. They both practice individually without the need to prepare for a team or group situation. So what’s missing from the analogy now?
It’s easy. Both people may be professionals now, but they more than likely had instructors to learn the necessary skills, especially early on. And that’s where these kids are. They are learning a new skill every day – one that may be built on prior knowledge, but one that is nonetheless new.
If anyone learning a new skill in a sport, or in music, is not coached by someone, how could they know if they are doing anything wrong?
So that’s the key: feedback… constructive and harmless feedback. In the classroom, this is called formative assessment. The worksheets are good practice, but students should not receive a formal grade on them. Rather, their mistakes should be pointed out, without losing points, so that they can fix them and improve.
The other key is purpose, or motivation. Students have no motivation to complete a worksheet as the final goal of the lesson. There is no purpose there. Besides, who can honestly say that all students need exactly the same amount of practice?
And we can’t say that the test is the motivating factor or final goal either, because that’s a one-shot, too-far-in-the-future goal. That’s more like the championship game or a concert in front of thousands. No, there have to be interim goals to motivate kids, like early season games or a small recital, that are worth something, but do not risk everything.
Here is an idea of how to make the worksheets more meaningful: Students can complete as much of the worksheet as they feel is necessary for them to grasp the concept. Teachers then point out any mistakes, if any, and have them try a few more questions, if necessary. When both the student and teacher are satisfied with the student’s ability, the student is then presented with a random (slightly more challenging) question on the concept, maybe in the form of pulling a notecard out of a bowl. This is where all of their practice pays off. They can now receive a grade based on how they answer this question, supported by all of the practice they have done.
I will never believe that worksheets are of no value. It is in fact invaluable for students to practice their basic skills, especially since math is a subject that builds upon itself year after year, never teaching them skills that will eventually be unnecessary in their math classes.
Tennis players need skills that are automatic, like a good backswing, or the ability to project the path of the ball. Musicians, likewise, need automatic skills, like recognizing scale patterns and certain rhythms. All of this comes with individual practice, and lots of it.
Don’t get me wrong: I’m not downplaying project-based learning, even though I can admit that it does not fit this analogy at all. (Could you imagine that? I’ll let you make that extension of the analogy yourself.) I am saying that in mathematics, basic skills and practice is necessary, but the practice must have meaning and purpose in order for us to require it.
Questions from a bowl – that’s one possibility. As to where the questions come from, why not from the chapter quiz review or test review? Why not from the quiz or test itself? If the kids knew the source, they would be more motivated to practice harder and get a shot at the bowl of questions. That way, they would have a higher sense of confidence on test day.
This is just one idea. I’m sure there are others. And I’m sure that someone can take this idea and turn it into something bigger, and use it to build student confidence even more. After all, confidence by itself is one of the biggest motivators we can offer.

Rational Expressions – That’s Pretty Low


Rational Expressions – The Plot Thickens


Rational Expressions – Summer’s Off

Well, school is closed…again. It just made me wonder, what if…



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