What Happened When I Used Numberless Word Problems.

 

The other week, I wrote about numberless word problems.  I spent one week giving my students numberless word problems and a simple question.  Such as the first day, I asked students what they noticed and what they wondered about the problem.  Almost every student noticed I didn’t include any numbers and they promptly provided their own numbers and solved it.

The second day, I selected a different problem, again without any numbers and asked students what operation they needed to solve the problem.  I didn’t ask anything other than what operation is needed to solve the problem and most students answered by substituting numbers into the problem to get a numerical answer.

The third day, I provided a different problem and asked why I chose not to use numbers in the problem and again too many students put their own numbers in the problem to solve it.  The same thing happened on the 4th and 5th days.  

So based on this short trial, I can make some reasonable conclusions:

1.  Students do not actually read the problem.

2. Students have had it drilled into their thinking that all word problems must be solved, even if no numbers are provided.  

3. The solution seems to be more important than even deciding what operation is needed to solve the problem.

So I”m wondering if we spend too much time teaching students how to solve word problems rather than teaching them how problems are structured, how to “read” the actual problem, and how to look at a numberless word problem without the urge to solve it.  

Unfortunately, by high school, students are so into finding numbers and just solving the problems that I don’t think they really take time to read the problems.  I have one class that has a lot of word problems.  When they ask me how to solve each problem, I ask them to read it out loud to me before they even try to write out the problem.

I think it is important to move students past the feeling that they must have a numerical answer from any word problem immediately, rather than taking the time to explore the problem itself. It is almost like chugging a glass of wine rather than taking time to look at the color, checking out the aroma, and enjoying the flavors.  

I don’t know what it would take to get students to savor solving word problems.  Let me know what you think, I”d love to hear.  Have a great day.

Numberless Word Problems

The other day, I ran across the idea of using numberless word problems in class.  Most students when they see a number problem look for the numbers, apply one or more operations to it in the hopes they arrive at the correct answer.  A few understand that the material taught in the section is to be applied to the word problem but most haven't gotten the hang of it.  

Numberless word problems have a every thing a regular word problem has except for the amounts.  We know students automatically look for numbers rather than reading through the whole problem.  An example of a numberless word problem might be "I travelled ______ miles on a tank of gas.  I started with ______ gallons of gas, how many miles per gallon did I get?"

When the numbers are removed, students are forced to look at the meaning and context behind the problem. It helps develop mathematical reasoning.  They are forced to move away from the numbers and finding a solution to understanding exactly the problem itself.  It allows them to focus on context and the underlying structure of word problems.

In the past,   I've given students numbers and asked them to create their own word problem using the values and over half of the incoming freshmen wrote a one sentence problem such as two plus half of twelve is 15.  This indicates they had not yet learned how a word problem is put together instead they'd learned to focus on looking to create an equation to find the solution.

The great thing about giving students numberless word problems, is that you've removed all numbers so students can't "solve" the problem.  They have to slow down, read the problem itself to determine the context of the situation.  It gives them time to figure out how to go about finding a solution or a plan of action. 

Furthermore, by removing the numbers, students have an opportunity to notice that certain problems are similar to each other such as addition problems have them combining things  rather than treating each word problem as something you look for key words to tell them what to do with the numbers written down.  They have to really think about how they would solve it by creating a plan.  

In addition, many students hesitate to solve word problems that have fractions or large numbers or scientific numbers because they are uncomfortable working with them.  In situations like this, many students shut down and refuse to try but by removing the numbers, a limitation is removed and students are more willing to look at word problems because they cannot get a "wrong" answer.  

One way to utilize numberless word problems is to create a set of numberless word problems so students can sort them by operation.  This can easily be done in pairs or small groups where students talk about the phrasing to determine if the problem is asking for addition, multiplication, subtraction or division.  

A second way is to provide one problem to a small group of students and have them do a "What do you know about this problem?", "What do you need to know?" before answering questions such as "What is a possible answer to this problem?"  Then pass out a second problem which is the first problem with one piece of information added so students can answer "What changed in this problem?" "Does this piece of information change the problem?" if so "How?"  Then ask them the "What do you know and what do you need to know "questions before asking for possible solutions.  Finally, provide a third problem with all the information for students to solve independently before they work with a partner to prove their solution.

This way of teaching word problems can be used from Kindergarten to seniors in high school.  Perhaps by using numberless word problems, students will unlearn the idea they have to grab all the numbers, combine them in some manner to get the "right" answer.  Let me know what you think, I'd love to hear.  Have a great day.