Friday, November 16, 2018

Ditch C.U.B.E.S. when solving word problems!

No I'm not referring to manipulatives when I say ditch the C.U.B.E.S. when solving word problems. Full Disclosure: In the past, I probably fell into the C.U.B.E.S. supporter bunch. I've learned a lot over the past few years and now focus on other ways of teaching students how to solve word problems.

First of all, some of you may be asking, "What is C.U.B.E.S.?" C.U.B.E.S. is an acronym that you may have seen posted in math classrooms throughout the years. It may take the form of a different acronym but they all get to the same basic idea: when solving a word problem use C.U.B.E.S.
  1. Circle all key numbers.
  2. Underline the question.
  3. Box any key words.
  4. Eliminate unnecessary information.
  5. Solve and check.
Rather than telling you why I no longer use this, let's just jump right in and try an example.  As you look at each stage (before skipping ahead to the next image), ask yourself "What's the answer?"

You see the following word problem that makes no sense to you...

so you faithfully follow the steps...

Step 1: Circle all key numbers

What's your gut instinct for the answer?

Step 2: Underline the question:
Do you feel pretty confident about that gut instinct?

Step 3: Underline the key words:
Still feeling pretty confident?

This is where most students stop when they don't know how to solve a word problem. What answer do you think comes to mind?

Let's look at the full question. Were you right?

What operation is used in this question?

Did you know that most students who aren't sure what to do when solving a word problem will take all of the numbers they see and just add them?

Do you have a math word wall? Have you taught students to identify key words? What have they learned "in total" means?

Let's try another example:

Let's start with that key word: "in total"

Circle all the numbers:
What do you think the answer is?

Are you trying not to add? But if it's not adding, then what is it?

Let's look at the question: 

Woah...did that 5 just get removed from the calculations?  Oops! How many students might get thrown off by that?

Let's look at the full question....

What operation is used in this question?  Were there any other "key words" that students might have identified? Would they have helped or hindered their understanding?

Let's look at another example:

Here are the numbers...

Here's the question...

What are you going to do with the information?

Again, let's look at the full question. Does identifying key words help here?

You know comes another example.

Let's pull those numbers out.

What's the question?

What's the full set of information?

So, what's the commonality in these four questions? They all focus on the key word "in total" which is typically taught as "addition" but every question actually addresses a different operation and some have extraneous numbers.

Here's a word problem for you to try: "There are 125 sheep and 5 dogs in a flock. How old is the shepherd?"  I'll wait while you figure it out.

Did that seem like a silly question to you? Obviously, you can't figure it out. 

Robert Kaplinsky asks 32 students to respond to this question. How many do you think agree with you and how many do you think attempted to solve it? Once you've made your prediction, watch the video.

Were you surprised? Why did so few students make sense of this problem?  How do we help students dig deeper into the word problem rather than relying on following steps like C.U.B.E.S.?

Brian Bushart introduced me to Numberless Word Problems and I fell in love.

As always, let's begin with an example.

There are some mice on the field. Some more mice come.

  1. What do you notice?
  2. What do you know to be true?
  3. What do you think to be true?

Take some time to think about the three questions in relation to the two statements above.

I have used this example in many of my workshops and I'm always fascinated by the discourse that occurs. If the participants/students have never experienced a numberless word problem before, they might experience confusion - there are no numbers! They sometimes start by retelling me the information that's on the board. I have had to prompt them to think about what "numbers" are in the question. What words might imply numbers? They will focus on the word "some". What does some mean? This is where it gets interesting. In my experience, most will say "more than one" but some will disagree and state that it means at least three. Why three? If the writer meant two, then they would have said "couple". We often spend quite a bit of time talking about the differences between "couple", "some", and another examples they might bring up. It's really very interesting. I'll then ask the question, "If -some- means -more than one- what do we know?". Students will have time to talk about that. They'll naturally start to think about the question "How many mice are on the field now?". Invariably, I'll hear the answer 3 but then they'll discuss how more than 1 and more than 1 has to be more than 3. They'll share their reasoning.  I'll follow up this question with "if -some- means -more than two- what do we know?". After we have fully dissected this question, I will replace one word:

There are 7 mice on the field. Some more mice come.

  1. What do you notice?
  2. What do you know to be true?
  3. What do you think to be true?

We'll discuss what's changed and how that changes the rest of the information we talked about. After a full discussion, I adjust the question again.

There are 7 mice on the field. 4 more mice come.

  1. What do you notice?
  2. What do you know to be true?
  3. What do you think the question is going to be?
  4. Solve that question.
At this stage, students solve the question.

Questions like this are easy to create. Just pull out a question that you would give them anyways, replace the numbers with generic words like "some", "many", etc. Check out #numberlesswp on twitter. 

CAUTION! This is great at the beginning of a class. Do NOT spend an entire class talking about addition and then pull out this question. Why? If we do 40 minutes of addition questions and then we have a "word problem", students are just going to assume that it's addition. The thinking stops and they revert to running a procedure rather than delving into the meaning behind the words.

Can you run a numberless word problem in other formats? Absolutely! Brian gave me permission to turn his numberless graph into a presentation I can share with teachers. I've embedded it below. Click through it, thinking carefully about each slide, asking yourself the same 3 starting questions as in the mouse question above. I've inserted a blank slide between each image mainly so I didn't accidentally click too many times in a row when presenting. :) You can download the slideshow from my account and you definitely should check out Brian's "original blog post" on the activity. 

Once students have spent time making sense of a problem without numbers, when they get to a question they are not sure how to solve, they will slow down and think more deeply about the information provided. They won't just take the numbers, throw in an operation and "solve" the question.

Question for Reflection:
How do you think your students would do in the "How old is the shepherd" problem? Try it in your class and see how they do. Are you surprised by their responses?

Activity to Try:
There are many words / phrases like "in total" that have many mathematical meanings depending on the context of the problem. Can you create 4 word problems that each use that word/phrase but utilize a different operation (like my examples above)?

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