First, a disclaimer: I am not and never have been a Math teacher. After teaching Modern World History for 7 years, I went to the "Dark Side" and became an administrator, so I probably don't know anything about teaching Math.

However, I do know a few things about teaching in general. Furthermore, I have a pretty good grasp of the philosophy of AFL and how applying it in the classroom can increase student learning. So I'm going to give this a shot.

I've noticed that one of the problems that some Math students have is that they don't practice at home. I will in no way advocate stopping the assignment of practice to be done at home. Practice at home is a worthy topic unto itself, but please do not read into what I am about to write that I am recommending not having students practice at home. In fact, I would recommend assigning practice to be done at home every single night. But if:

- Many of our students don't practice at home, and
- We realize that we cannot control what one does at home, and
- We believe practice is required to learn the content, and
- We care MOST about whether or not students learn as opposed to whether or not they make responsible decisions outside of class, then
**It makes sense to provide as much practice time as possible during class since that is the only time in a student's day we can control.**

Of course this idea of giving time to practice in class fits very nicely with the philosophy of AFL. The AFL teacher would give practice opportunities in class that provide useful feedback for the teacher and the student. Frankly, I've never known a Math teacher who doesn't give students chances to practice in class. Usually this comes in the form of a practice problem or getting started on the night's homework. Typically the teacher will move around the room to see how students are doing and to answer questions the students might have.

There is absolutely nothing wrong with this sort of practice activity, but like everything, it does have limitations. For example, if a student chooses to just "go through the motions" of doing the practice, then very little feedback will be received. Also, the student who, for whatever reason, doesn't ask questions will quite possibly not learn as well as the student who does ask questions. Finally, the teacher is only one person and almost always outnumbered greatly by students. It can be difficult to give each student the specific feedback they need during such an activity.

One more background observation before I share my idea. I have noticed that students often take test-like or quiz-like situations more seriously than they do other activities. In other words, kids who will goof around and disrupt classroom practice tend - in a well-led classroom - to sit quietly and do as they're told during a test or quiz situation.

That's a lot of build up and background to an idea that's not all that earth-shattering. In fact, I'm sure the Math teachers out there will respond by saying, "Been there, done that!" But I still figured I'd share a potential practical application of the philosophy of AFL to the Math classroom.

In a nutshell, the idea is to break up the Math process into steps and then give students a daily quiz on each step as they learn the process. It would look something like this:

- The Math process being taught is broken down into steps. For this discussion let's assume we're learning Math Process P which is divided into 3 steps.
- The teacher teaches Step 1 and then gives students a quiz on Step 1. The quiz will ONLY be on Step 1 and it will be worth X points.
- The teacher teaches Step 2 and then gives the students a quiz on Steps 2 AND 1. This quiz will be worth 2X points. The student or the teacher might even choose to erase the first quiz from the grade book or set it to not factor into the grade.
- The teacher teaches Step 3 and then gives the students a quiz on Steps 3 AND 2 AND 1. This quiz will be worth 4X points. The student or the teacher might even choose to erase the first two quizzes from the grade book or set them to not factor into the grade.
- The teacher reviews the quiz on Steps 3, 2, and 1 and then gives a unit test on all aspects of Process P. This unit test is worth 10X points. The student or the teacher might even choose to erase the quizzes from the grade book or set them to not factor into the grade.

Here are some more details:

- A quiz might be given the same day as the respective step was taught. On the other hand, a step might take more than one day to teach. If a step takes a few minutes to teach, then the teacher will quiz on it after giving the students a chance to practice it. If it takes the entire class period to teach the step, then the quiz will open class the next day.
- If any step takes more than one day to teach, then the students will take a quiz on that step on consecutive days.
- There will be a quiz given every day.

To me this seems like a way to make sure students are practicing in class. For example, even if the student did no homework, he would still practice Step 1 three times before the unit test, Step 2 two times, and Step 3 one time. Beyond just practicing the step, the student would be receiving more feedback and more direct feedback than is typically received when the class goes over practice or homework problems. Finally, the teacher would get valuable feedback as he or she would know how each student - as opposed to just the question askers - was doing on each step. Plus the teacher would have the specific feedback necessary to tightly focus remediation efforts, determine what might need to be retaught, and create differentiation efforts.

So, Math Teachers, what do you think? Could this work? What have I overlooked? Would this type of practice - this use of assessment for the purpose of learning - increase the likelihood of students learning?

## Comments

Thanks for the thoughtful feedback, Alisa!

And, yes, that's why you implied that this process would be used occasionally, and only for certain topics :)

This was reasoned out as if thought through by a "math person" with lots of if/then statements, steps to follow, etc. Maybe that's why I agree with you so frequently! This year, unfortunately, my students who are the worst about homework completion are the same students who are absent A LOT, so that complicates the quizzing process a bit. While not all math processes can be broken into steps as mentioned here (in fact, many cannot), your idea is basically one of frequent assessment of "parts" leading up to a "whole." Students definitely benefit from this approach, and teachers obtain valuable feedback. For me, the single biggest downfall to this (as silly as it may sound) is the TIME that it takes to create and grade/mark-up (with valuable feedback) these small assessments. They are so important and so valuable, but they really do add up to more time for the teacher than one might imagine ... especially if done well.

Thanks, Mariann, for the feedback. I like your ideas. Sorry if mine was a little convoluted - some things are easier to say than to type! :)

I thought that was a little difficult to follow and maybe manage. If you would like more practice time in math class (and I agree with you, that while we don't want to kill them with homework, there is value in assigning work outside of class)....one thing I have done is "flip" the order of my class from time to time. The typical order of a math classroom is to start with last night's hw, then teach new lesson, then guided practice, etc. I would intentionally flip that order for days, by teaching a new lesson at the end of class, and then going straight into practice the next morning....you wrap up that lesson, then go to the new lesson again at the end of class. Also there's great value in teaching "units". For example, if you are teaching quadratic equations....try this. Start with a hook (something real-world) showing students how to solve quadratic equations all ways (students should always be taught to think numerically, algebraically, and graphically), then for days teach every way to solve rather than factoring one or two days, graphing one or two days, quadratic formula one or two, etc. Teaching the methods as just that....the methods....helps them to tie it all together better. And my final thought....you can teach an entire unit with just taking homework (completion grades) and then when the material is mastered, then go back and quiz/test. For example: Let the kids tell you when they have it all, and then say okay....tomorrow, we're going to quiz on just factoring, the next day quadratic formula....etc. or give them a test where they work the same problem 4 ways, the next problem 4 ways. That's a great way to test and I suggest doing both. Giving the students input into the testing times works great. (With you guiding this of course!) You can do this in all courses.....Algebra to AP Calc.....master a concept then go backwards and pick up the grades.....and as far as "time" goes....it all comes out the same....promise! Hope this makes sense...kind of hard to explain in a post.

Beth - thanks for the feedback. I hope I didn't make it sound like every single lesson in Math would be taught this one way and this one way only. Definitely didn't mean to imply that. Also - I don't think time would have to be an issue. In theory, all 3 steps in the example I created could take place in one day. In other words, it wouldn't have to be "a step a day". On top of that, the quizzes might be very small and very quick - whatever made the most sense.

I definitely like the idea of making good use of the Do Now time - especially if you used it in a way to provide feedback to kids and to get feedback for yourself. Not sure if I'd advocate grading the Do Now as it could inflate or deflate the grade and get in the way of the grade representing mastery. However, I could see future tests or assignments canceling out previous Do Nows.

In theory it sounds great but in reality probably not so much. First, not everything can be taught in steps in math - a lot of things in Geometry come to my mind. Many times the overall concept needs to be taught - kids may need to see the big picture first. Second, time would be a huge factor. I think this process would take much longer than the average math classroom time-frame permits. Perhaps in an ideal world where math class is an extended period would something like this work. Thirdly, it doesn't take into account different student learning styles - if a student learns by doing the same process over and over this would be good but what about the student that needs it another way?

While I agree students not practicing is a huge issue in today's education setting, one way I deal with it is by doing spiral Do Nows (or Warm ups) at the beginning of class. This way students do see the material multiply times. I usually take a grade on these at the end of the week.