It was fascinating to read the book and everyone's thoughts and examples. I especially appreciated the example of the young students using socks, shoes, and place settings to learn about one-to-one correspondence (and perhaps addition) as Steven's examples were all for older students, as is my experience.

These real-life examples (or even unreal problems) engage students, involve students in true mathematical practices, and check student understanding. One concern that I read and have often heard was from Heather and Kim:

We learned a ton about who really knew how to compare fractions with unlike denominators, who could demonstrate it with a diagram/drawing, and the numerous misconceptions that abounded! But, we can’t help but wonder how to help those students who simply don’t get it.

I would say that, as they found, this type of problem really asses if students understand the math as opposed to memorizing a procedure. In addition, students who think the mathematics are more likely to be able to solve the problem in the future, something that is very important for the CC in which each years math builds upon previous understandings. Lastly, in my experience (and I'm an "old man") some of the students who we think "get it" because they can solve the traditional problem are least able to apply the math in different types of problems. They are so good at memorizing and applying algorithms that they don't develop understanding.

I was disappointed to not see samples of student work, and have none myself, but I have been inspired to save examples so I can share them on future prompts (if I can figure out how - maybe copy and paste or link to a site such as photobucket.

Now, my own examples/thoughts. I try to give problems related to real life regularly, but they are seldom connected to the world as artfully as the burger king or car speed examples. I think I need to do a better job of keeping my eyes open (and using my students' eyes once they have seen some of these problems. Any problems that connect to real life are excellent formative assessments - they show a lot about student thinking. I have had much support for this, especially from our math interventionist, Tracey Harnett.

I believe it is difficult to ensure that all content targets are included using Problem Based Learning. We have an alternative, hands-on program at our middle school. The program integrates literacy, science and engineering, and math into learning projects such as building a racing car with support of a local racing team, developing and maintaining a community garden, and learning about ecology. Despite this, the one class the kids must participate in separately from this program is math.

I have done Service Learning with my classes the last several years and have not done a particularly good job at integrating math. This year, my class will be divided into groups and take on three projects at Robert's Farm, a local community garden and natural area. One will involve building a chicken coop. Proportions and fractions can fit nicely into this. Another group will work on a nature trail, a bit more difficult to incorporate math unless we do some statistical analysis of what grows where or mapping. Our third project will be growing in a hydroponic system. I'm not sure yet how math can work into this; the nutrients are supplied by a "pill" put into each plant container. Perhaps some geometry looking at volume, measurements, and planting density.

I hope I have success incorporating math into these projects!