“Suspended sense making” – that was the phrase that stood out to me recently at the BCAMT New Teachers’ Conference that was held in Richmond, BC. Ray Appel, the keynote speaker, spoke how students plow through math questions focused on trying to do the right operation, to get the right answer, but in their quest they often take leave their sense making.
Research from the last decades points to this particular problem (e.g., Radatz, Freudenthal, Schoenfeld, Verschaffel, and DeCorte). Radatz, in a study with 300 children in grades K -5, gave them some unsolvable problems such as this one: “Katja invites 8 children to come to her birthday party, which takes place in 4 days. How old will Katja be on her birthday?”
While only about 10% of the K and grade 1 students tried to solve the problems, the percentages of grade 2 students (about 30%) and grade 3 and 4 students (about 60%) were much higher. The percentage for grade 5 students went down some (45%) but still reflected a poor showing.
Similar results have shown up in other research. In another study other unsolvable problems were given individually to 7-9 year olds and 9-11 year olds. They were asked things like this: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?” Each problem was followed by the question, “What do you think of the problem?” Only 12% children aged 7-9 and 62% of the children aged 9-11 reacted by saying they could not respond properly to the questions. All of the others performed some kind of arithmetic operations on the numbers in the questions without expressing any doubts. One pupil’s response went something like this:
“125 + 5 = 130 …this is too big, and 125 – 5 = 120 is still too big… but 125 ÷ 5 = 25 … that works… I think the shepherd is 25 years old.” While the student did use some sense-making in thinking about how old the shepherd was likely to be, he did not make sense of the big picture: namely, that knowing how many sheep and dogs were in the field could in any way indicate how old the shepherd was.
Lest we dismiss this as being absurd, I experienced a bit of this “suspended sense making” just last week. I had given a class of grade 3 students this question:
The principal at Charlie Lake School was dividing up all the students into 4 house teams. If there were 157 students in the school, how many students would be on each house team?
My students went to work splitting up the students in the question into the 4 required groups using a variety of strategies. (I was impressed with the overall thinking about splitting up 157 and shall have to write a separate blog about the strategies they used!) Partway through the working time one particular boy had arrived at a solution, and before beginning to work on another strategy he asked me, “Is the answer 39 remainder 1?”
His class had been working on basic facts in division and other division questions “close” to those (e.g., 25 ÷ 5 = 5 and 27 ÷ 5 = 5 r 2) so his answer of 39 remainder 1 would have been reasonable had he been given just the numerical problem 157 ÷ 4. But the numbers were in context, and the ‘157’ in the problem was not a vague, meaningless number, but the number of children in the school. I asked the boy what the remainder of 1 represented, and he responded, “That’s the leftovers”. “What was actually left over?” I asked. It took several moments for him to realize a child was left over. I asked what the principal would do with a leftover child, and the boy suggested that one house team would have to have an extra person because the principal would not just leave one student without a team.
Allowing children to do pages of division problems where the remainders are popped into the answer unquestioned as to their meaning seems, in my opinion, to reinforce the practice of “suspended sense making”. Similarly, when students do a page of any algorithmically-based set of problems with no understanding as to the mathematics behind the rules, they are suspending their sense-making. Chants such as, “Ours is not to reason why, just invert and multiply,” come to mind here — a chant that was just said to me this very week!
What do you think? What other kinds of things might we traditionally do that invite children to set sense making aside?
I believe we have to continually provide opportunities for students to make meaning in mathematics, and we have to expect them to do some explaining, some representing of the ideas in question. We want them to be critical thinkers who realize when things just don’t make sense, and not be lulled into only looking at ‘available’ numbers and randomly trying a variety of operations on them. Students should be asking themselves all the time, “Does this make sense?”
The bottom line: math should make sense for every student, every day!!