Collaborative Group Work

To model collaborative group work, our THSP math conference presenter divided us into groups of three or four and gave us the following math problem:

"Farmer Brown sent her children to the field to count the number of horses and ducks. Her son returned with the count of 30 heads while her daughter came back with the count of 94 legs. How many horses and how many ducks does Farmer Brown have?"

The objective was for each group to, not just solve the problem but, determine multiple solutions and to make sure that each member will be able to explain correctly all solutions that the group finds. Our presenter, Robert Knittle (University Park Campus, Worcester, Massachusetts), had indicated earlier that research supports activities such as this that require collaborative group work, one of six components of THSP's common instructional framework.

We all thought we knew the problem well, so we proceeded with less interest.

Tonya Milburn of Early College High School, Galveston, Texas suggested that we solve the problem using a system of equations and solve it using three different methods: by graphing and by using both processes of elimination and substitution.

I volunteered to do the latter. Angelica Vega, the dean of instruction of East Early College High School in Houston, Texas said she will use guess-and-check in tabular form.

While we were all preoccupied with our individual tasks, our fourth member, Canary Branch Bui of Empowerment Early College, also in Houston, remembered something from her younger days back in Vietnam.

"I have solved a similar problem when I was in the elementary," she announced.

Well, I wasn't the only one who got intrigued. Everyone listened intently as Canary explained her unconventional solution.

"Assume that all thirty heads are horses. Okay?" The three of us nodded.

"Since each horse has four legs, then we have 120 legs altogether. But there are only 94 legs according to the farmer's daughter."

Canary proceeded to subtract 94 from 120 to get 26. Then she announced that these 26 legs mean that there are 13 ducks, and there are 17 horses.

Initially, I didn't understand her explanation why the 26 legs mean that there are 13 ducks, but I thought that out of those 30 four-legged animals, there will be 13 of them that should lose two legs each. These 13 must be the number of ducks.

We were all excited to have found a "new" solution to an extremely familiar problem situation.

Angie, whom I met at the Park City Mathematics Institute in Park City, Utah about five years ago, promptly raised an intriguing question. "Will it work if all 30 heads are assumed to be ducks, instead of horses?"

Here's how I convinced my group mates that said method works:

Thirty ducks give us a minimum number of 60 legs. Since all 30 animals have a combined total of 94 legs, then there are 34 legs that are unaccounted for. This means that there are  17 (34/2) heads that require two more legs each. Note that only horses have four legs, therefore there must be 17 of them and the rest are ducks as our other conventional solutions prove.

Canary was requested to show her crowd-pleasing method to everyone in the room, and I volunteered to show my sweet "corollary" to her "theorem."

A problem-solving exercise meant for our classroom use with our students turned out to be a tremendous learning experience even for us!

Without a doubt, subjecting our students on a regular basis to collaborative group work activities such as the one given above is not such a bad idea after all.

Cursing Students

Today, the first day of the two-day Texas High School Project (THSP) mathematics conference here in Dallas, Texas, our presenter, Robert Knittle (University Park Campus School, Worcester, Massachusetts) read to us the children's illustrated book "Math Curse" (1995, Viking Press)  written by Jon Scieska and illustrated by Lane Smith.

Here's the book's plot according to Wikipedia: "The book, which is told by a nameless female student, begins with a seemingly innocent statement by her math teacher- "you know, almost everything in life can be considered a math problem." The next morning, the heroine finds herself thinking of the time she needs to get up along the lines of algebra. Next comes the mathematical school of probability, followed by charts and statistics. As the narrator slowly turns into a "math zombie", everything in her life is transformed into a problem. A class treat of cupcakes becomes a study in fractions, while a trip to the store turns into a problem of money. Finally, she is left painstakingly calculating how many minutes of "math madness" will be in her life now that she is a "mathematical lunatic." Her sister asks her what her problem is, and she responds, "365 days x 24 hours x 60 minutes." Finally, she collapses on her bed, and dreams that she is trapped in a blackboard-room covered in math problems. Armed with only a piece of chalk, she must escape-and she manages to do just that by breaking the chalk in half, because "two halves make a whole." She escapes through this "whole", and awakens the next morning with the ability to solve any problem. Her curse is broken...until the next day, when her science teacher mentions that in life, everything can be viewed as a science experiment."

After the book reading, we were asked to write how we "curse" our students.

In my table, there were two gentlemen whose responses caught my interest.

Gustavo Alvarado (Cotton Valley Early College High School in El Paso, TX) wrote: "Everything in your daily lives is a function."

Andrew Waxman (Mission Early College High School, El Paso, TX) wrote: "Life is (full of) word problems."

I didn't understand the question then, so my response was totally off. But now that I do, I say that I "curse" my students by telling them, "Finding something difficult to do means that you are about to learn something new."

How do you "curse" your students?