What is it?

The “math is only computation” belief is the belief that all math is formulas and just working through numbers.  In this belief, it’s safe to say you also believe calculators can solve nearly every problem.  To be completely honest, I bought into this belief until approximately a year ago.  I didn’t even know “other” math even existed.

What causes it?

Our testing methods lead give this belief life.  Our assessments primarily consist of solving problems that involve our students to work their way through questions.  Our homework is typically very similar.  We have failed to show our students that math is more than this.  How often do we take a step back and talk about the logic and thinking processes involved in math?  Not nearly enough.  To be fair, our traditional curricula hasn’t really allowed much breathing room for such conversations.

There have been many times where I have heard people, other teachers included, say they love math because it’s so “black and white” (I’m sure just as many people hate math, by that same train of thought). You either get it right, or you get it wrong.  All of the math I recall taking in school fit under this category. In a math assessment class I took last year, we discussed that reading skills and pattern recognition are the best predictors of success in mathematics.  I don’t think we help students with either of these nearly enough.

When all of the math our students see is computation based, why would they have any reason to think math is ever any different?

What does it look like?

Students may not see the value in non-computational math.  If you choose to do brainteasers with your students, they may enjoy them, but not see the point in doing them.  The few brain teasers I have done with my students, have ended with: “Now let’s get back to real math”.   Our students don’t see that skills like recognizing patterns and developing game strategy is considered math.  I wonder, if we began to use more this kind of math in our classes, if students would start enjoying math class more.

What can we do about it?

Rita talked about having a game day every now and then in her math classes.  These days would encourage problem solving and building strategies that would give you the best chance at being successful in the games.  She mentioned the game of Nim, which I have never seen before but I am interested in trying to play it with someone as soon as I get the opportunity.

Using sequences or analogies that don’t use numbers in them may allow students to start looking at math as not only computational.  We should encourage our students to look for patterns in absolutely everything.  It could very well be the most valuable thing we teach students in math, but I think it is often overlooked.

It might be valuable to discuss what real mathematicians do.  You’re students might be surprised to find out that they don’t just show up work and get given a sheet of problems to solve for the day.  Students seem to know very little about math beyond high school.

Image from Flickr:  Some rights reserved by ZeRo`SKiLL

Five unhealthy beliefs in math that exist today are:

• The “five minute” belief,
• The “you have to be taught how” belief,  
• The “math is mostly memorizing” belief, 
• The “math is only computation” belief, and
• The “some people just naturally can’t do math” belief. (blog post to come)

What is it?

The “math is mostly memorizing” belief is the belief that you can be good at math by just memorizing a few things.  If you simply memorize a few formulas and the step-by-step process that your teacher uses, you can get through math easily.  Unfortunately, I think this is a belief that many of our students buy into, even though they may not find much success with it.

What causes it?

This belief is, once again, largely caused by our current teaching methods.   When we just emphasize teaching skills, students see this as opportunity to follow a step-by-step checklist to arrive at solutions.   We tell students to “do what we do” and, if they do, they will see success in our classes.

It might also stem for our questioning as teachers.  Typically, when we’re teaching students to replicate what we are doing, we only engage them in low-level thinking questions.  Instead of asking our students to think about what we’re doing in a problem, we’ll ask them to tell us what comes next.  Students can easily get by knowing the step-by-step and not having a clue as to why we progress throughf problems the way we do.

Additionally, teachers may cause this by creating assessments (tests or quizzes) that replicate the questions done in class or completed by students for homework.  If a teacher consistently does this, students could very easily get through math without ever actually understanding the questions they are “solving”.

What does it look like?

One of the most noticeable ways we see students believing this is when we hear them respond to questions with “You said…”.  Right away, this shows us that the student has tried to memorize what you had said at an earlier time.  I have run into this many times and I have usually found that the students haven’t memorized it properly, or have mixed it up with another similar concept.

Students also have difficulty generalizing or applying the concepts to different situations.  They may struggle to make new connections between concepts or use the concepts in real world situations.  If you were to present them with a WCYDWT problem, I would imagine they would have difficulty making the connections between the real world aspects of these problems.

What can we do about it?

One of the best ways to address this memorization is derive any formulas we use in class, instead of just telling students to memorize it.  I like this idea, but I wonder if students that believe math is memorizing will just zone out during the derivation and memorize the final formula.  There has to be a better way to do this.  Maybe some sort of inquiry-based activity is the answer here.

Rita also suggested that once a concept is derived, let your students name it.  This will allow students to take some ownership of the formula and maybe they’ll be more inclined to understand where they got it from.  Over time, you can begin to explain that the math community actually calls it something else, opening the door the “correct” math vocabulary.

Image from Flickr:  Some rights reserved by Pragmagraphr

Five unhealthy beliefs in math that exist today are:

• The “five minute” belief,
• The “you have to be taught how” belief,  
• The “math is mostly memorizing” belief, 
• The “math is only computation” belief,  and
• The “some people just naturally can’t do math” belief. (blog post to come)

What is it?

The “you have to be taught how” belief is the belief that you can’t do anything mathematical until someone shows you how.  When I think about this belief, I wonder if it were true, how math would even exist to the extent it does today?  Clearly, the mathematicians who have established the mathematics we teach our students, weren’t held back by this belief.   That being said, I know that not every student that steps into my classroom will be a mathematician.  Maybe none of them ever will, and that’s okay.  I’m not in the business of generating math Phd’s, but rather, I hope my students can think critically and not be afraid to approach problems they might not yet understand.   And, if students feel that they need to be taught how to do something before trying it, we have a big problem.

What causes it?

Me. You. Teachers everywhere.  When we require students to do things a certain way, we lead our students to believe this, completely unintentionally.  If a student tries to solve a problem, and arrives at the answer in a different way than we’re expecting, we might thank them for approaching the problem this way but encourage them to look at it from a different approach (our approach).  I’m certain this is largely to make our assessment easier.  It’s much easier to assess process when students are supposed to fit a specific template of process.  Why would a student continually try to do things their own way?  We’re just going to “fix” them later anyways.   We like to encourage students into solving our way by asking them to “show their work”.  Sometimes, teachers may refuse to give full grades if work is not shown.

Rita says that when we teach shortcuts, we can build on this belief.  Often in math classes, we teach students an idea a certain way and then a few days later we make that idea obsolete by showing them a quicker, easier way to do the exact same thing (the concepts of polynomial long division and synthetic division come to mind).  I struggle with this idea, because I know why it is valuable to show students the longer, harder way of doing things so they can appreciate and understand the shorter, easier methods we introduce later.  But if we can do something quicker and easier, why wouldn’t we just teach that?

I think that the curriculum we are expected to deliver to our students also contributes to this belief.  We have so many topics and concepts we are expected to get through, it doesn’t leave much time for students to inquire and discover math on their own.  So, to make up time, we take some shortcuts, show some quick tricks.  These quick tricks don’t teach our students how to think about math.  If we are going to show them these tricks and shortcuts, there is no motivation to make connections on their own and find their own shortcuts.

What does it look like?

If students buy into this belief, they likely won’t even attempt to solve a problem until you tell them how to.   They may simply skip problems they haven’t been taught how to solve, with the expectation that you will come back to teach them how to do it later.  This belief becomes a big problem when students begin to miss class time or stop paying attention for any length of time.

What can we do about it?

Once again, Rita suggests brain teasers as a way to counter this belief.  Since the teacher doesn’t just provide a solution, students will need to find a way to solve it themselves, hopefully allowing them to realize that they don’t need to be taught to how to solve a problem.

Another idea is to get your students to solve problems in more than one way.  This will show them that there is no one “right” way to do arrive at a solution.  And, if you have a student who finds a solution in a way you have not seen, or did not expect to see, make a big deal about it and acknowledge that extra thinking and creativity that was used to find it.

We need to find a better to get our students to “show their work”, because we want to into their minds and assess the process of their problem solving.  Instead, maybe we could ask students to “show their thinking”.  It might seem like a very little difference, but it could open the door to allow our students to be creative in explaining themselves.  Some may draw pictures, use diagrams, or even write sentences to describe what they’re doing.  I would be very interested to see how students would choose to do this.  This is a very simple change that I hope to make immediately.

Image from Flickr:  Some rights reserved by Jiuck

Five unhealthy beliefs in math that exist today are:

• The “five minute” belief,
• The “you have to be taught how” belief,  
• The “math is mostly memorizing” belief, 
• The “math is only computation” belief, and
• The “some people just naturally can’t do math” belief. (blog post to come)

What is it?

The “five minute” belief is believing that if you can’t figure out a problem in five minutes or less, you won’t be able to figure it out.  I think it’s safe to say, that many people give it much less than five minutes.  There are times where I wish students would just try it for five minutes.  How often have students tried a problem and gave up after only a moment because it has become challenging?  I see this belief every day I teach.

What causes it?

Our traditional teaching methods, where we tell kids how to do something before allowing them to try it on their own.  By telling our students how to do everything before they even try, we kill the curiosity in our students that would allow them to try to find their own way to solve a problem.   In many cases, they have never had a true opportunity to find, or try to find, solutions on their own.  And, if this is how we are going to teach our students, why would they bother to try to figure it out? We’re just going to tell them the answers anyways, right?

I think this belief might also stem from our students fear of failure and fear to make mistakes.  If every time they make a mistake, we are lowering their grades, why would a student take a risk to figure it out on their own?  When students are only trying to get the highest grade possible (and who can blame them with the education system we find ourselves in today), why would they care about learning if they can find an easier way to get the grades they need (mimicking and copying you, the teacher).  This is another topic in itself that I’ll save for another day.

Rita also suggested that students have come accustomed to seeing problems solved very quickly, through  current TV shows that solve numerous problems in under 30 or 60 minutes.  Maybe it’s not just TV shows, maybe seeing us (the teachers) work so easily and quickly through example problems make things look as if they should go much quicker and smoother.  Maybe, as teachers, we need to make mistakes and display struggles more frequently to let our students know that it is normal and acceptable.  I have a hard time picturing how I would do that, but it’s something I’m thinking about now.

What does it look like?

A student might give up on problems quickly, before you even feel they’ve given it an honest effort.  Sometimes, before beginning a problem, hands will shoot up in the air, asking where to start.  Other times, it might be an “I don’t get it” before even reading or looking at a problem.  And instead of asking valuable questions, they simply expect you start it for them and work through it “with” them.

I would also doubt that students who possess this belief ask a classmate, a friend, a family member at any point.  They probably also don’t look back at any resources they have to help them out.  They didn’t get it in the first five minutes, it’s hopeless.  Why bother trying at all?

What can we do about it?

Rita suggests that teachers can generate a “did you?” list.  A “did you?” list is a checklist that students would need to go through before getting help from you, the teacher.  It would include items like “did you ask a friend?”, “did you check your notes?”, and any other “did you…?” questions that you feel appropriate for your students.  The hope is that over time, students should get used to going through this check and internalize this checklist.

She also suggested that brain teasers can be an effective way to counter this belief.  They can encourage students to think in various ways to try to approach a problem.  The brain teasers may get students in the habit of thinking and trying, instead of giving up.  When you give students a brain teaser problem, you do not give them the answer.  Even if they can’t arrive at a solution in the time you give them.  Rita told a story about one brain teaser that stayed on her board for over a month before a student was able to solve it.  This helps illustrate that the math done by mathematicians (and arguably anyone else on a day-to-day basis), is never complete and answers don’t always come quickly or easily.  Rita suggested that talking about the Four colour theorem, and the history of it, would be a great way to further illustrate this point.

Have you seen this belief in your class?  What have you done to counter it?

Image from Flickr:  Some rights reserved by * hiro008

Five unhealthy beliefs in math that exist today are:

• The “five minute” belief,
• The “you have to be taught how” belief, 
• The “math is mostly memorizing” belief, 
• The “math is only computation” belief,  and
• The “some people just naturally can’t do math” belief. (blog post to come)