For this module we are investigating ISTE Student Standard 1: Empowered Learner – “students leverage technology to take an active role in choosing, achieving and demonstrating competency in their learning goals, informed by the learning sciences.” In response to Empowered Learner Indicator 1c, “students use technology to seek feedback that informs and improves their practice and to demonstrate their learning in a variety of ways,” I asked the investigation questions:
What are some methods for sanity checking in mathematics, examples of a time when it was needed, and how do you teach this skill? How can students demonstrate sanity checking?
My own working definition of sanity checking is: the act of using tools, techniques, and information to answer the question “Does this even make sense?” This definition is very general and not math specific, though I will focus on its application in math. Sanity checking can be used while solving problems, or problem solving, and to check a final answer.
I felt sanity checking was related to ISTE 1: Empowered Learner because when a student spontaneously sanity checks their own work, they are taking an active role in their learning and they are seeking some type of feedback to inform what they are doing. Tools such as calculators, mathematics software, and Wolfram Alpha can be creatively utilized to get feedback/aid in sanity checking, and of course there’s always Google. I figure that while they may have not articulated their own learning goals, they are implicitly demonstrating an achievement-related goal, whether it’s to achieve sense-making or achieve the right answer.
In my experience, sanity checking was not something that was ever explicitly taught, in spite of its value as a skill and way of thinking. This led me to wonder what kinds of educational tools and research exist related to teaching sanity checking. To my surprise, I found exactly zero published journal articles related to the combined key words “sanity checking” and “mathematics education” (and related searches, like “sanity testing”). This makes me wonder if sanity checking goes by different name in research. Turning to Google, I found a few websites and books with some relevant information.
Q: What are some methods for sanity checking in mathematics?
Finding a list of methods was more challenging than I imagined. But from a few resources, I have put together the start of a list:
- Estimation: For example, you can use estimation to check that an answer is reasonable (Petrilli, 2014).
- Plug in numbers: If two things should be equal, are they in fact equal when you plug in a random number (Wood, 2015)?
- Comparing against external information: Suppose you know that a penny weighs about 3 grams, and you calculate that an adult weighs 6 grams (i.e., two pennies). Compared to the external information, this answer does not seems reasonable. Wood (2015) and Yaqoob (2011, p. 33) mention related things.
- Do the units make sense? (Wood, 2015) or Dimensional analysis: If you’re calculating a velocity (meters/second), but end up with kilograms*meters/second, something went wrong. More generally, you can often use the “fundamental dimension” like length, time, and mass instead of meters, seconds, and kilograms to sanity check.
- Definition/rule/fact based: For example, checking that the hypotenuse is the longest side of a right triangle, since it always is. Fenner (2013) gives many examples of this kind of sanity checking.
Q: What are some examples of a time when it was needed?
We have to show students that sanity checking can be a meaningful activity but I did not find any information on how to do this. So I would love to know some real examples of when sanity checking was needed or valuable. More specifically, I would love to know about the experiences students have had where they found value in sanity checking. Perhaps this could be an interesting assignment – have them turn in a reflection at some point throughout the quarter explaining a time when sanity checking helped them while working on the coursework.
Qs: How do you teach this skill? How can students demonstrate sanity checking?
I did not find any information on these questions either. Of course, demonstrating real-time sanity checking while working with students can help teach the skill by exposure. Additionally, let students see your “backstage performance” (Olitsky, 2007) to show them the struggles you have and how sanity checking is helping you.
One of the difficulties with demonstrating sanity checking is that sometimes it leads you along a messy path. You might erase little things here and there. You might scrap the whole page and start over. I’m not sure that turning in a homework set including all those changes would be easy or very valuable (maybe it would), but this is one reason I like the idea of a reflection assignment that has them explain one meaningful sanity checking experience from the quarter. I will also keep thinking about ways that students could utilize technology to share their sanity checking stories or resources with each other.
Sanity checking is a valuable skill in mathematics, and teaching sanity checking can help students become self-directed learners, which Kivunja (2014) considers to be an important 21st century skill. Considering the value of sanity checking, I would like to find some more resources on the topic, for example best practices to teach it or how students utilize sanity checking. I will keep my eyes out for synonyms that might lead me to the vein of research on this topic, and if none exists, then perhaps this is something I would like to research in the future.
Fenner, S. A. (2013). Basic mathematics for engineers (8th Ed.). Lulu Press, Inc. (link)
ISTE: International Society for Technology in Education. (2016). 1: Empowered learner. ISTE standards for students. Retrieved from https://www.iste.org/standards/standards/for-students-2016
Kivunja, C. (2014). Teaching students to learn and to work well with 21st century skills: Unpacking the career and life skills domain of the new learning paradigm. International Journal of Higher Education, 4(1), p1. Retrieved from http://files.eric.ed.gov/fulltext/EJ1060566.pdf
Olitsky, S. (2007). Facilitating identity formation, group membership, and learning in science classrooms: What can be learned from out‐of‐field teaching in an urban school? Science Education, 91(2), 201-221.
Petrilli, M. J. (2014). The Common Core sanity check of the day: Estimation is not a fuzzy math skill. Retrieved from https://edexcellence.net/commentary/education-gadfly-daily/flypaper/the-common-core-sanity-check-of-the-day-estimation-is-not
Wood, B. (2015). Sanity checking. Retrieved from http://mathmisery.com/wp/2015/04/06/sanity-checking/
Yaqoob, T. (2011). What can I do to help my child with math when I don’t know any myself? Baltimore, MD: New Earth Labs. (link)