Regular Class work is so Helpful.

I find that a regular activity is a good thing. The students look at it as a useful digression and happily go about working on it ... then they realize it fits right in with what you've been doing.

@fawnpnguyen: "I was brainstorming with a couple of 6th grade math teachers at another district, and we were listing out a possible warm-up/math talk schedule, something like Monday: number talk; Tuesday: visual pattern; Wednesday: estimation 180; Thursday: fun fact, or WYR, or Keeping Skills Sharp, or SBAC/review question; Friday: personal reflection.

Here are a couple of those ideas and one or two others:

Estimation180 - building number sense by estimating values from images or video. This is valuable because so often we say "Does that answer make sense?" If the students have little to no experience with the subject of the question, and no practice making estimates, then answering our "Sense?" question is an exercise in random answer generation.

• There's a worksheet or develop your own.
• estimation180.com  

One Hundred and One Questions - An image or video is presented and students ask any question that comes to mind. While I personally wish the prompt was "What math question comes to mind here?", it is a good place to help them develop the ability to ask questions of the world round them and to see that math isn't just a classroom activity. Browse beforehand and record the links of the ones that fit your current material or your mood.

• 101qs.com 

Math Arguments 180 - The goal is to have students question their assumptions and bring those assumptions to the front of their minds for conscious consideration instead of letting them hold on to common misconceptions that mess up their thinking.  Still in the development stages. The Math Concepts Challenge is also for teachers, though your students might be charged up for it.

• matharguments180.com
• Math Concepts Challenge

Visual Patterns - Practicing the art of understanding the pattern and setting an equation to it in order to predict the value at step 43. I've been thinking it needs more patterns that aren't straightforward linear functions, but if that's the age you're working with then here is a bunch.

• www.visualpatterns.org

Math Talks - Prompts for discussion with your students.

• mathtalks.fawnnguyen.com

Would You Rather? - students are presented with a choice. They choose and then have to justify their choice. "Would you rather have a bag of nickels that weighs as much as you do or a stack of quarters as high as you are?"

• wyrmath.wordpress.com 

Graphing Stories - a video is shown and the students need to create a graph of some data from it. The video contains a graph blank that shows the independent and dependent variables. Usually, these are time-series graphs of height or altitude, but if you only show the "action" portion, you can have them graph whatever quantities that come to mind.

• graphingstories.com

The UVM Math Contest - Problems from the University of Vermont High School Math Contest. These are given in the spring of Pre-Calculus and are meant for mature students who have had a good algebra II background. These questions are to be solved without a calculator or technology of any kind; figuring out the method is the whole point. Attempting a whole test in the allotted two hours would challenge even the best math teachers. Scores of 15 out of 41 are considered excellent.

• mathcontest.blogspot.com

Drilling basic facts pays off.

"Practice makes perfect."

Okay, we do have to differentiate between mindless drill & kill and what really makes kids better at math - focused practice, error correction, and repetition of correct processes.  Just as hacking around on a soccer field doesn't help the team nearly as well as a directed practice, endless pages of addition problems aren't useful in developing math skills.

Do ten at a time until you get them right. Examine the work, identify errors and don't repeat them .... which sounds like what Common Core asks math teachers to do and what math teachers have been doing for a long time.

This all sounds like what you've been doing, unless you have Curriculum coordinators like mine who denigrate practice as "drill and Kill. We should have the kids do critical thinking", as if critical thinking and error analysis were two different things.

"Healthy children start making that switch between counting to what’s called fact retrieval when they’re 8 to 9 years old, when they’re still working on fundamental addition and subtraction. How well kids make that shift to memory-based problem-solving is known to predict their ultimate math achievement. Those who fall behind “are impairing or slowing down their math learning later on,” Mann Koepke said. 

So, those teachers who insist that the kids should have a calculator and an internet connection so they can look it all up are really doing a job on their kids.

It turns out that adults don’t use their memory-crunching hippocampus in the same way. Instead of using a lot of effort, retrieving six plus four equals 10 from long-term storage was almost automatic, Menon said. In other words, over time the brain became increasingly efficient at retrieving facts. Think of it like a bumpy, grassy field, NIH’s Mann Koepke explained. Walk over the same spot enough and a smooth, grass-free path forms, making it easier to get from start to end. If your brain doesn’t have to work as hard on simple math, it has more working memory free to process the teacher’s brand-new lesson on more complex math. source.

So the next time someone says that elementary students should have a calculator, that "you can look that up" and "all we need is to teach them information retrieval skills", remind them that the brain needs to have some basic facts to work with before it can handle more complex tasks.

Currently, the argument point brought up here is the Japanese education system, which has been moving away from drill and practice and turning to the higher-order thinking that reformers here in the States are always going on about.

Remember, however, that a typical Japanese high school teacher does NOT have to do as much of this preparatory drill and skills practice ... because of the ubiquitous "jukus" - after-school academies that nearly every Japanese student attends to make sure that he/she has a shot at higher education.What do the "jukus" do? Drill. Basic skills. Practice, practice, practice. Because it works.

Here's a description of one of these "cram schools":

"This school broadcasts lectures to its 800 satellite schools all over Japan and supports students via telephone, fax and the Internet. Students can take math training menu and other drills on the Internet through a service called Toshin Home Lesson. They can choose from one series of lectures to a whole package."