While the balloon video may seem off the wall to a high school math teacher, Dan’s explanation beautifully illustrates the difference between a “word problem” and the real-life modeling intended by the CCSS The balloon problem has a closed beginning but has an open middle (students design how they will analyze the problem) and open end (because the environmental factors will cause answers to vary). The applicable CCSS standards for that activity would depend upon the grade level for which the activity was used. It could easily be a 7th grade problem or an Algebra 2 problem depending upon how students consider the variables. Would such a dog pop balloons faster as she got the hang of it and then slow down when fatigued? That could be modeled by a quadratic function.
The ultimate goal is that, by Algebra 2, students will be able to tackle open-open-open problems using a scientific thought process according to the CCSS Modeling Progressions:
The CCSS has freed up time in HS math curriculum such that we have time to do some of this modeling. So far, though, I have not been able to find any open-open-open problems that would interest my high school students enough to motivate them into pursuing them in such a way that it would be wise use of class time. Most of the problems I have seen so far would bore my students to groans, so I have been using the time normally wasted on those to occasionally test home-made 3-acts and other modeling problems with some degree of openness. Those experiences have been glorious, but I am not yet meeting all the modeling standards.
Math teachers often think they are doing a better job meeting the standards by hammering in rote procedure with an emphasis on concept; but I find the hammering is much easier when the students are motivated to learn the procedures (and so did Schwartz and Martin). My students say things like, “We just learned about that in my environmental science class!” (or computer science, biology…) Next, the modeling standards are what move students from executing procedures to useful problem solving. It appears to me that US math teachers, in general, have a long way to go, and I am genuinely concerned we have gotten stuck. I argue that we have a professional obligation as math teachers to figure this out.