Engaging in a Math Debate

"One train may hide another."

Recently, Lisa, a sixth grade teacher sent me a link to this Amazing 6th Grade Math Work! blog post that she had written up to document a math debate that she released into her classroom around the difference between ratios and rates.  It's definitely worth reading her post, but if you don't have time, here's a synopsis...

Lisa began by admitting that, although she knew the official definitions of "ratio" and "rate," she wasn't sure what the practical difference was.  So, she put that out there and had her students research the terms, debate the differences and come up with a working definition that they summarized in a table.

After she sent this to me, I had some thoughts, too, and then the two of us debated back and forth over email a bit (for those of you who are riveted by such things, I'll paste these in at the end of this post).  Then, Lisa brought these ideas back to her students.

Is there room for debate in math class?

Typically, we might think of debate as an engaging instructional tool for a social studies class.  We might think of math concepts as being more "cut and dry" and less... well, debatable.   However, Common Core Math Practice #3 states that students should be able to construct viable arguments and critique the reasoning of others.   Unpacking this a little, this means that:

• Students understand and use stated assumptions, definitions and previously established results to construct arguments.
• Students make conjectures (a guess about an idea or pattern that they think is true) and logically explore the truth of their conjectures through examples and counterexamples.
• Students listen to the arguments of others, decide if they make sense and ask useful questions to clarify and improve the arguments.

At the heart of this practice is inquiry - using questioning, definitions and observations systematically to seek truth, and we accomplish this through mathematical discourse.
Mathematical discourse can happen lots of different ways.  Why not make it playful and engaging?  Why not release a debate in the room?

The power of misconceptions and how to correct them...

There is research demonstrating that students' misconceptions are enormously powerful in blocking learning.  For example, in science many students believe that the Sun revolves around the Earth.  Language like "the sun rises and sets" coupled with misleading observations gives student deep misconceptions.  These misconceptions stubbornly resist our best efforts to dispel them because students have to delete a mental image that makes sense to them in favor of one that intuitively does not.  

At one point, I encountered a professor specializing in cognition who described this kind of blocking using the phrase, "one train may hide another..." He pulled it from a warning sign that he saw at a railroad crossing.  Those crossing needed to take care because a train going one way would block the view of the train going the other way. 

How do we break down these misconceptions?  Simply teaching students the concept the "right" way is sadly ineffective.  In order to remove the blocks, we must:

1. Identify students' misconceptions.
2. Provide a forum for students to confront their misconceptions.
3. Help students to reconstruct and internalize new knowledge based on correct models.  

I'm thinking that debate in the math classroom is an excellent, lively way to do this.  What do you think?

Here's a nice article about how to promote math debates.

If you're interested in other ways to confront misconceptions check out:

• This blog post about facilitating mathematical discourse.   
• My Favorite No.  I've posted this before, but it's worth repeating (and good for all grade levels, don't let the algebra fool you).

So, what's the difference between ratio and rate?

Here's what Lisa and I had to say about it...

Ellen:

 ... a rate is a special kind of ratio that compares two things in different units (miles walked to feet walked).  So "rates" are a subset of "ratio" and a unit rate is a subset of "rate" (picture a bulls eye Venn diagram) where one of the units being compared is 1.
So to answer your question about when a ratio is NOT a rate... when it compares like things in like units (# of boy to # of girls, eggs broken to egg intact, miles driven to miles walked).  
With a ratio you don't need to include units, you can say "the ratio of boy to girls is 3 to 1."  It is understood that you're talking about comparing numbers of children.  Whereas with a rate you need to label units. For example, in 30 minutes the barber can do 3 haircuts.  The ratio (which is also a rate) is 30 minutes:3 haircuts.   The unit rate (which is both a ratio and a rate) is 10 minutes per haircut or 10 cuts:1 minute.
Here is a notebook entry that I've use with my students: https://docs.google.com/document/d/1lAtjtxorZGaWMQQW0QP1T2gXaC8AlO5cZYOrNY9TNcQ/edit

Ellen:

Thinking about this more.  It's important to see what this is building towards... Ratio leads to rate which leads to unit rate. These concepts escalate in complexity, right?   Then we move from unit rate to rate of change and slope. Unit rates are constant for proportional relationships and proportional relationship are always linear, but unit rates can vary for some relationships even if they are linear. 

Think of an all you can eat pizza deal that doesn't include drinks. Say it's $5 and $1 per drink. At one drink, the unit rate is a $6 per meal. At two drinks it's $5+2 or $7 per meal. The overall unit rate changes even though the rate of change is constant (at $1). From here student move to looking at how slope represents a ratio, a rate, a unit rate, a rate of change that can be either constant or changing. This helps them to compare and define functions.

Ratio is the trunk of the tree. Fractions are the roots.

Lisa:

Aren't boys and girls different units? Broken eggs/whole eggs?

And then, what we found is a lot of rate definitions say that a rate "Usually" includes a comparison with measurement.  What would be an example of rate that is not measurement based?

Ellen:

It is relative, for sure. Which is good. It's important how you define a unit. And it is admittedly a little ambiguous. But rate allows for comparing units in a way that ratio is too simple to accommodate. This builds to looking at linear and nonlinear functions where relationships between quantities (regardless of units) can be compared.

Regarding boys and girls or intact or broken eggs, you are interested in comparing categories of items using the same unit (kids, eggs, etc).

Want to weigh in?  What do you think?