Refocusing Math Questions
When I first started thinking about math inquiry, my concerns always circled around: Where can I find inquiry questions? or What resources can I use to help me create inquiry questions? or How do I make a math inquiry question?
Recently my thoughts about inquiry questions have been focused on: how can I refocus or remix this problematic context into an inquiry question that will help push my students’ thinking?
Take for instance the “Muffles’ Truffles” problem from the Fosnot kit, which is designed for grades 3 to 5 students to investigate multiplication and division with the array. I was able to ‘refocus’ this question so students could investigate the following academic goals:
Grade 7: determine, through investigation, the relationships among fractions, decimals, percents, and ratios;
Grade 8: translate between equivalent forms of a number (i.e., decimals, fractions, percents) (e.g., 3/4 = 0.75).
This is a description of our two day math investigation.
Sample of Student Posters
The grade 7s and 8s worked in pairs to create math posters. These were the 2 posters that I selected to bring to congress.
The poster on the left was selected to share first because it provided a solution to the Muffles’ Truffles problem using a chart, division, and open number lines. The poster also focused on providing a ‘reasoning’ for how to deal with the remaining truffles. The poster on the right was selected because it used a decimal representation to explain how many truffles were ‘left over’. It also included a ‘different way’ (for our class) of using the open array to show division where you would yield a decimal value.
Refocusing the Question
The way I refocused this problem to suit my grade 7s and 8s learning needs was to extend the problematic context by asking:
Are you saying that when you divided the number of truffles 218 by the number of spaces in the chocolate box 10, you get 21.8 which means you can create 21 full boxes of 10 truffles, with a remainder of 8?
Well, how many boxes would you need if you had 9 truffles and a box with just 2 compartments? Based on the strategy shared, the answer would be 4.5 boxes. Or, in other words, 4 full boxes of 2 truffles and 5 truffles left over. Do you agree? Talk with an elbow partner.
How about if we had 93 truffles and a box with 4 compartments? What would we end up with?
23.25 So this means that we end up with 23 full boxes of truffles and 25 left over. Right?
Discuss in a group of 3.
At this point the class intuitively knows that something does not make sense but is unsure of why looking to the decimal value when dividing by 10 ‘works’ but that the same checking strategy does not intuitively work when dividing by 2 or 4.
Observations of Student Interaction with the Refocused Question
While I am walking around the class I notice that the conversation is louder than our previous discussion, students are making eye contact with one another, and they are gesturing to the whiteboard with their hands. I also notice triads of students reaching for paper and drawing out 2×2 arrays and 1×2 arrays. I similarly see 1 group of students stand up and make their way to the whiteboard, talking to each other all the while. Another group of students reach for a Smartphone and iPad and begin to use the calculator on one and a sketching app on the other.
After a time, the class reconvenes and we discuss our findings.
In an upcoming blog posting I will include an ‘audio’ clip of this discussion.
Remixing Math Questions
The following day I wanted to check-in with the class to see where their understanding was with respect to the relationship that exists between equivalent forms of fractions, decimals, and percents. Here is the question I posed, which is essentially a “remixed” form of the Muffle’s Truffles problem.
Tyler and Mahaksh own a burger joint.
They have 3 “mini burgers” on their menu.
The “Big Mahak”, the “Tiny Ty” and the “Tiny Ty with cheese”.
Every day they make:
• 102 Big Mahaks
• 93 Tiny Tys
• 127 Tiny Tys with cheese.
The “mini burgers” are packaged in
boxes that look like this:
(one mini burger per compartment).
How many boxes will they need for
each burger flavour?
How many assortment boxes
can they make?
I developed this question because I wanted to use a landmark fraction – 1/4 – to check student thinking as it related to the curriculum expectations I chose to explore with students.
Sample of Student Work
Here is math thinking offered by a grade 8 student for the ‘burger question’.
Here is math thinking (3 pages) offered by another grade 8 student.
An obvious benefit of teaching this concept through an inquiry approach is that the students did not have to rely on me to teach them the basics of decimals, fractions, percents and ratio first. Instead the students grew to rely on each other and ultimately, their own intuitive math sense in order to explore and build their understanding of the relationship between fractions, decimals and percents.
Notes and Question…
An inquiry question designed for one grade can be adapted to suit the needs of a different grade. The trick is to know where you as the teacher are trying to extend student thinking and how you could ‘refocus/remix/or extend’ a given context to get there. This is where familiarity with your mathematics curriculum is key. For me, I need to be familiar with the grade 7/8 Ontario Revised Math Curriculum.
What learning can you share about how you create, develop or remix inquiry (math) questions?