The other day, I had the opportunity to listen in on the conversation taking place between my neighbor’s children (ages 9 and 11) waiting on line at the ice cream truck. It went something like this (names changed to protect identities):
John: Mommy gave us $5, and she said to make sure to bring back the change.
Mary: Goody! We can both get Bubble Blast Pops!
John: No, we can’t. That costs exactly $5. It’s too much. Mommy wants change.
Mary: Maybe Mommy means that if there is change, we shouldn’t forget to take it from the ice cream man, like we did last time.
John: But maybe Mommy wants change, so we should choose something cheaper.
Mary: But I want a Bubble Blast Pop. You get something else. Maybe get the Firecracker Pop, which is only $2.00. Then there will be change.
John: No, you get something else. I want the Bubble Blast Pop!!
At which point, the conversation morphed into some serious temper tantrums, and I lost interest, glad to know that my own children were past this phase of life.
But as I resumed my walk down the street, I continued to think about the exchange I overheard. These children had a real-life, mathematical problem- the type that we, as teachers, are always trying to bring into our classroom lessons.
But how to we teach our students to solve these types of problems? Do we, in fact, teach children how to really solve problems?
The first of the Standards of Mathematical Practice- Make sense of problems and persevere in solving them - addresses this very scenario. Here were 2 children, struggling to make sense of a problem, and hitting their limit of perseverance rather quickly, as evidenced by the rapid appearance of tears and yelling.
Yet, this is not how a problem solving scenario usually occurs in the grade 3 to 5 classroom. More often, John or Mary is instructed to read problem number 1 on page 53, which says “John buys a Bubble Blast Pop for $2.50. Mary buys a Firecracker Pop for $2.00. How much did they spend together?” or “John and Mary both want to buy a Bubble Blast Pop, which costs $2.50. If they have a $5 bill, do they have enough money?” Do any of these sound like what John and Mary were dealing with?
In the earlier grades, children are focused on “learning to read.” As they get older, this begins to shift to “reading to learn.” In grades 3 through 5, both are often occurring simultaneously. Children can read the words and extract information, but how is their comprehension? In the second “word problem” above, think about the word “both”. What does it mean? Does it mean that they both are going to buy a pop, or does it mean that they both want to buy one, but only one gets to actually make a purchase? I have observed some excellent teachers getting their students involved in some heated debates about what words mean and imply, long before attempting to answer the question. Lucky are these students, for they are developing the ability to make sense of a problem, rather than just learning the proper response to a particular string of words.
As we all know, most real world problems come without written words. Do our children learn how to “comprehend”, let alone solve, these types of problems?
Why is the development of good problem solving skills so important? Simply put, it is because we spend most of our lives doing nothing but solving problems. As adults, we all know that once we solve one problem, the next will not be long in coming.
I was recently directed to an excellent article on problem solving, entitled “Toward a Meta-Theory of Problem Solving” by Dave Jonassen. I have not completed it yet, but am quite intrigued by what he has to say. I would like to quote from an opening paragraph:
Most educators, like Gagne, regard problem solving as the most important learning outcome from life. Why? Because most people, especially professionals and tradespeople, are rewarded in their careers for their abilities to solve problems. No one is paid for memorizing information and completing examinations. Unfortunately, very little education and training requires learners to solve problems, and virtually none engages the kinds of problem solving encountered in the real world. At best, education and training efforts engage learners in well-structured (textbook) problems, while real world problems are nearly always ill-structured.
You can read the entire article here!
For more about Dave Jonassen’s work, click here.
For more on how problem solving should look in grades 3 to 5, visit Teaching Math
Which brings me back to John and Mary at the ice cream truck. They are definitely engaged in an extremely ill-structured real world problem, don’t you think?
As educators of children, we are all committed to doing the best we can to give our students all they will need to become successful adults. Problem solving will be on my mind a lot this summer.
Race you to the ice cream truck!
photo © iStockphoto
As the current school year draws to a close, planning for the upcoming 2012-2013 school year has already begun. New texts have been received, new curriculum maps have been designed, and the full implementation of the Common Core standards is on the horizon. One thing I am noticing is the focus on content (isn’t this always the case?) rather than on practice. The Standards for Mathematical Practice are so important to understand. They supply the answer to “how”, rather than “what”. So, over the next several weeks, I’d like to explore what each of these practice standards really mean.
Make sense of problems and persevere in solving them.
The first of the practice standards is the reason why we learn mathematics in the first place. It is a tool used to solve problems. And not all problems are so easy to solve, which is why perseverance is so important. How many times have your students said things like “I looked at the problem (for 5 seconds, maybe) and don’t know how to do it”? These students lack the ability to persevere in solving problems. It’s usually not their fault. They are probably accustomed to being “taught” how to solve a few standard types of problems in a rote manner. Anything that is slightly different, they can’t do because they haven’t been taught.
Bill McCallum, one of the lead authors of the Math Standards, has created a wonderful diagram (which can be seen at Common Core Tools and is reproduced below) relating the 8 standards and putting them into context.
As you can see, making sense of problems is an over-arching theme (along with precision), providing a reason to develop the other practices and habits of mind.
Children are born with the inherent ability to solve problems through perseverance. All babies have the problem of hunger. They will persevere in their quest to solve the problem through prolonged crying, not stopping until their problem is solved. Older children will be trying to solve the walking problem. Over and over, they will struggle until they get it right. A desirable object placed out of reach will have a child devising all sorts of methods to get what they want.
As educators, it is our responsibility to nurture this innate ability, helping children to learn to use it in more sophisticated ways. Very often, instructions in a traditional classroom of “sit still”, “be quiet”, and “don’t touch” end of destroying the very abilities we are trying to engage. Directing and enhancing those abilities, so that children can use them in more mature and effective ways, is part of our role as teachers.
Many teachers take to opportunity during the summer to invest some time in professional development. A particularly wonderful resource is the Annenberg Learner website, found at www.learner.org. If you are not familiar with this site, I highly recommend that you spend some time exploring all that it has to offer. Of specific interest here is an exploration of what good problem solving looks and sounds like in the K-2 classroom.
An excellent video by Phil Daro about problem solving in the classroom you can watch below!
Phil Daro - Against "Answer-Getting" from SERP Institute on Vimeo.
Finally, anything by NCTM is bound to be amazing. Below are links to some of my favorite student activities related to problem solving, but feel free to investigate all of them:
NCTM Standards Content 1
NCTM Standards Content 2
Until next time-