by Kate Fanelli
Research is clear on how everyone can develop accurate and deep conceptual understanding of mathematics: start by understanding the quantities involved (often done through the use of manipulatives, stories, or drawings) and move toward symbolic or procedural representations. Even in high school, when teachers might expect that students are internalizing quantity, the lack of this understanding can be a major barrier to higher level learning.
Having a sense of quantity means that when I say “3x + 2y = 7” the student understands that there are 3 of something, and how many 3 is, and 2 of something different, and how many 2 is, and then 7 things. In fact, equations like this, that lack labels or stories, can further confuse the issue of quantity as students are looking only at symbols, and not necessarily thinking about amounts.
The student who must count out everything before calculating is still developing that sense of quantity, and if she did not count out, represent, or place individual physical objects out, the numbers on the page would lack meaning. I had a student who would do the problem 19-8 by drawing 19 sticks, going back to the beginning and counting 8 sticks, circling them and drawing an X through the circle, and then counting the 11 remaining sticks. This student, although a high functioning teenager, needed to go back to quantity to work with subtraction.
For students who already have a firm grasp of quantity, a refresher on a procedure may be just what is needed to efficiently apply mathematics with meaning. For those who never gained the conceptual understanding, procedural instruction on its own can be insufficient for true learning to take place.
Three websites present engaging ways to introduce mathematical concepts in a contextual manner. They are problem-based, which is to say students are presented with a situation that is open-ended, builds curiosity, and begs for some work to be done to bring resolution (i.e. problem solving).
Estimation180.com is a collection of 210 (and counting) short estimation activities created by Andrew Stadel, subtitled “Building Number Sense One Day at Time.” Students are presented with a photograph and asked to estimate amounts, distances, values, etc. Students enter values for “too low,” “too high,” “estimate,” “reasoning,” and “name,” and submit their answers to a Google spreadsheet, where they can see hundreds of other entries.
Using data provided in the Google spreadsheets, students could use regressions, averages, and other statistical analyses to identify patterns and trends and improve their own estimations.
To get an idea of what this looks and sounds like with students, watch Stadel describe the power of estimating in his Ignite Talk.
Three-Act Math Tasks from Dan Meyer engages students in problem solving through carefully constructed mathematical stories that tap into the natural curiosity of students. Meyer describes the structure in his blog and offers a spreadsheet with his videos, alignment to standards, and suggested questions.
The What If? archive is a collection of questions people have wondered about that can be answered using math. What if everyone on earth got as close to each other as possible and jumped at the same time? What if a rainstorm dropped all of its water in a single giant drop?
Kate Fanelli is the math accessibility specialist for Michigan’s Integrated Mathematics Initiative (Mi)2, a state of Michigan initiative that promotes and supports high quality mathematics education for ALL students. Follow (Mi)2 on Facebook (www.facebook.com/mi2.page) or on Twitter (@MI2_Math). Contact Kate at firstname.lastname@example.org.