What makes a good math problem?
For the past 7 years I’ve moderated a group initially called “Common Core task Based Math” but we changed the name to “Low Floor, High Ceiling Math Tasks” a few years ago. Great discussion, sharing of math games and tasks usually happens there. Recently there has been some discussion over what kinds of discussion should happen — what started as a facebook message has turned into a blog post. Specifically someone posted about how an SAT question hit certain standards. It caused debate not so much because of the questions themselves I don’t think but because the author was advocating about giving students harder material to younger students. Eg it was a question for algebra II area standards and he asked if it hit middle school standards. But some folks also just recoil when they see a standardized test question, which I don’t think is right either in certain contexts! Let me explain.
I have previously worked for a standardized test company and it’s actually an interesting space. Testing companies are pretty much common core or state standard-agnostic, and instead utilize their own internal taxonomies of what knowledge is essential for success in college, and align their questions then to the specific state standards via careful alignment studies. ACT released their Holistic Framework publicly in 2017 which is actually a huge great step forward- I am also biased because they released it on the platform I was product manager of! But academic standards are fairly arbitrary — they all attempt to capture essential skills in a way that teachers can teach the topics most important to future success. But this is not about the common core standards, it’s about mathematical learning to any standard in any country.
Thus, it seems clear standardized tests are not where you go for innovation in seeing student thinking — process. By the very nature of their large scale, it’s hard to be innovative with question formats and the like because everything has to be calibrated to be free of bias, etc. In other words you have to remove caring about the process because in the end the answer is all that matters.
A so-called Low Floor High Ceiling program (popularized or even coined by Dr. Jo Boaler in Mathematical Mindsets and before) gives entry points to all students and also in the same context gives students an opportunity to go further in their mathematical thinking. A great example of this is the Albert’s Insomnia game.
Another is the entire set of openmiddle.com problem sets which encourage students to think and come up with multiple solutions — the problem shown below has about 700 possible solutions!
For me still the exciting part of what makes a problem worth solving — even if it came from a textbook factory — is this explanation from Dan Meyer’s Ted Talk over 10 years ago about taking out information from the problem until the students come up with both the questions and the answers. True, making the problem worth solving isn’t necessarily ‘low floor high ceiling’ but for me anything a kid can guess at and then refine is indeed worth solving.
So can a multiple choice question offer multiple entry points and extensions? YES… but not in that format. Just as worksheets are often derided for being unoriginal and boring — and likewise we are told “but students need practice solving multiple problems in the same way…” I don’t think there’s ever going to be an easy answer. Modern school wants a number to know how well kids know something. Equitable math pedagogy says “show me more than one way to solve the same problem.” The solution I feel is more of the kinds of responses that technology enables — video or audio based explanation answers, portfolios of work instead of grades, interactive and portable transcripts, and a change in the entire education-work ecosystem. Side note, that’s the problem space I’m working in now — skill based hiring in the workforce. Standardized tests have their place in objective measurement of student results. Yet, anything used in the classroom for me always should be helping make student thinking more visible, accessible and equitable — multiple choice tests don’t do that.
