# Deconstructed math

Common Core’s advent has inspired math teacher Jessica Hiltabidel to find ways to instigate thinking and encourage “productive struggle” she writes in Educational Leadership.

Pre-Core, Maryland middle-school teachers would guide students through a word problem, such as:

To receive a discount at the Half-a-Dozen Flags Amusement Park, a group of visitors must purchase a minimum of \$600 worth of full-day and half-day tickets. Twenty-six tourists in a group purchased full-day tickets at \$15 each. Write and solve an inequality to calculate how many more tourists in this group would need to each buy a \$9.50 half-day ticket so this group qualifies for the discount.

The “inquiry-based instruction that undergirds the Common Core State Standards” calls for “open-ended questions,” Hiltabidel writes.

One textbook question read, “To go to Disney World costs \$50 for each adult ticket and \$30 for each child’s tickets. X tickets are bought of each. If the total spent is more than \$800, how many tickets were bought?”

She “deconstructed” the problem to read: “You’re taking a trip to Disney World. How many tickets do you buy?”

Students were told to solve the problem any way they wanted as long as they could explain the process.

Some students . . . bought enough tickets for their family and friends. Others bought enough tickets for the Baltimore Ravens football team or all members of their favorite boy band.

After students had presented their ideas and we discussed them as a class, I asked, “How much money would you spend to take all these people to Disney World?”

This time, students were energized and focused on the process of discovery. One group used the Internet to figure out the actual cost of a ticket. . . . By the time we were finished, all my students could identify what key information they would need to solve a word problem like this and what simple operations they could use.

Then she showed them the textbook question. They used “the structure they’d discovered” to “see this complicated question” as a set of steps: Multiply the cost of an adult ticket by how many adults went; multiply the cost of a child ticket by how many children went; add the total, and make sure the sum is larger than \$800.

I don’t understand the textbook question. Were an equal number (X) of adult and child tickets purchased? Or is it just an unknown number of both? Is the question asking for the minimum number of tickets purchased that would total more than \$800? If it’s not, then there are an infinite number of answers.

Her “reconstructed” question doesn’t involve math. And I’m not sure speculating on who you’d like to invite to Disney World leads to an understanding that you need to multiply the number of adults and kids by the cost of their tickets, add the sum and then . . . What if the total is less than \$800?

Am I missing something?

1. Mark Roulo says:

“To go to Disney World costs \$50 for each adult ticket and \$30 for each child’s tickets. X tickets are bought of each. If the total spent is more than \$800, how many tickets were bought?”

Are they looking for “an even number more than 20”? Or is any specific even number (e.g. 1,234,568) an acceptable answer? Or do they want an equation (since we weren’t told how much more than \$800 was spent)?

2. Deirdre Mundy says:

Well, ‘deconstructed’ math will certainly be more time-consuming!

3. Rob says:

>all my students could identify what key information
>they would need to solve a word problem like this
>and what simple operations they could use.

OK, but could they actually SOLVE the problem? I’m betting that many could not. The Disney World problem is problematic: an adult ticket plus a child ticket would be \$80 and that divides neatly into \$800, BUT it says “more than” \$800, so you have to answer 11. Why make it so you have to buy a whole extra ticket to be more than \$800?

The first problem avoids this and I think it’s actually considerably more complex. “Deconstructing” the problem seems mostly to be about avoiding all of that scary algebra in favor of wasting an hour on a five minute problem. At the end of their hour of math, the kids would be ready to go to Disney World, but no closer at all to understanding the application of algebra in daily life.

• cranberry says:

For the textbook question, I think they want the answer, 22. 50X + 30X = (more than) 800. X can’t be 10, because then it must be more than 800. Remember to double X for the answer, as there are X adults and X children.

As they phrased it, though, the answer is 22 or more, on to infinity, or the capacity of the park, whichever comes first.

An occasional extended discussion of how to solve a type of problem could be useful. Every day, though? With a warm-up of shaking limbs 144 times? (each limb 8, 7, 6, 5,…)

4. At least 22 tickets — but that’s a kind of silly question.

(Explanation: Equal numbers of tickets were bought for children and adults. If \$800 had been spent, that would have been 20 tickets, but since more than \$800 was spent, they had to have bought at least 22.)

5. wahoofive says:

Clearly it’s been a while since some of you took math. Obviously the topic of the day is inequalities, whose solutions take the form of x > 10 or whatever. Saying x = 1000000, or 22, would be incorrect.

Joanne’s right that the “textbook” question is ambiguous, although that’s hardly the point (and maybe was quoted incorrectly). If the point is to get kids interested enough to want to solve the problem, who cares what extraneous information they generate? Conversely, can we afford the time to do something like this on every problem?

• SC Math Teacher says:

No, we cannot afford the time. These folks want in depth discussions among the students (to construct their own knowledge!), many of whom will be so clueless in the first place as to make the process interminable.

• Paul says:

I’ve always been amazed at strict constructivist approaches. They want the average 7th grader to recreate the mathematical thinking and creative leaps that took the greatest minds of their generations 3000 years of human history to put together.

6. I am convinced that the goal of teachers is to turn all subjects into English. They are scared of math. And, frankly, that scares me.

• Bill says:

Suzanne,

The only reason that students struggle with math, and this starts in the K-5 grades is that we try to re-invent how math is taught every few years, which leads to students not being able to learn it (thus, I HATE MATH, or I’m NO GOOD at MATH syndrome continues).

Sigh

7. Richard Aubrey says:

An adult, interested in intellectual activities, will find rote work too simple, not engaging his or her interests in teaching and learning and all that stuff. It would seem to be much more interesting to dream this stuff up and go through it with the kids than to rejigger the test on simple equations.
Besides, as somebody said, where do you get the time. ?A school board–in SD, iirc–has dumped the Pledge of Allegiance because, they claim with a straight face, they can’t find the time for it in the school day.
Surprised that many people from San Francisco would move to SD and get on the school board.
But if ten seconds is too dear, nonsense like this math is either the problem, or ought to be dumped.