Now I know that most all of us are familiar with clue words to solving word problems. We were either taught them, or we have taught them ourselves, or we have had children who came home with them in their math notebooks.
For example, the following words supposedly indicate addition: in all, together, altogether, total, both, etc.
But consider the following problem.
Max likes to collect marbles. For his birthday his grandparents gave him 100 more marbles. Now he has 567 marbles altogether. How many marbles did he have before his birthday?
According to the clue words a student should add the numbers in order to solve the problem. However, adding the numbers would not yield the correct answer, nor would it even make sense.
I am NOT saying that students should not understand the meanings of words. But the meanings of words can be taught and learned without being taught as a ‘clue’ for what operation someone should use to solve a word problem.
I have a personal experience to share. Several years ago I went to my niece’s and nephew’s school to facilitate a mathematics lesson in each of their classrooms. My nephew’s group of four were solving “Max has 39 cupcakes. 17 are chocolate and the rest are strawberry. How many strawberry cupcakes does Max have?” He was so excited as he called me over to his group and told me that there were 22 strawberry cupcakes. I asked him how they knew that. One of the students said “Well, we subtracted to find out how many were strawberry.” All of the sudden my nephew said, “Oh no, we have it wrong. We were supposed to add”. Not showing any emotion I asked why he thought they should have added. He said, “Well, because it says ‘in all’ and it says on that poster over there (pointing to the wall) that ‘in all’ means to add.” I thought I would cry. When the students were thinking about the actual problem and making sense of the actual content they reasoned their way to a solution – the correct solution. Yet, when an operation was mentioned they immediately gave up reasoning and (in my translation) thought “Uh oh, we’re not supposed to pay attention to all of the content. It will confuse us. We need to use that helpful chart with the clue words that will help us to always get the right answer.”
I have often said that I do NOT believe that teachers walk into a classroom thinking “Oooh, I’m going to mess my kids’ up so badly today…” Teachers in general love children. Most if not all have a nurturing nature. We want to help our students. At times clue words have proven to be helpful. The problem is however, that only the most traditional word problems follow the ‘rule’ of the clue words. If I am standing in front of a classroom and creating a word problem ‘on my feet’ I am more likely to create problems where the end of an action is what is missing. Two amounts are put together and the student is asked to find the total. Or two things are separated and the students are asked to determine what is left. But there are many problem structures that we rarely provide for our students.
I would like to introduce all mathematics teachers to Cognitively Guided Instruction (CGI). http://cognitivelyguidedinstruction.com/index.html and http://www.promisingpractices.net/program.asp?programid=114 are two links where CGI is discussed in detail. To be honest I know nothing about either of these sites but they seem to provide the background information needed to get a good idea of CGI. The first site provides 4 ‘GOTTA HAVE’ books for all teachers of mathematics.
So, if you are a teacher of mathematics and you have a poster of clue words on your walls, please remove it immediately. Order and read the Children’s Mathematics: CGI book and then provide for your students opportunities to work with a variety of problem structures without ‘telling’ them how to solve the problems.
I realize that many of my posts may cause discomfort for some, clarity for some, confusion for some, and maybe even anger for others. Please feel free to contact me if you have any question regarding any of my posts. firstname.lastname@example.org