Author: mathmaddicts

I love finding problems that have multiple ways to find the answer and that encourage students to explain how they got their answer.

2015 CMC South Conference

I recently had the privilege of presenting at the CMC (California Math Council) South Conference in Palm Springs this past weekend.  As promised, here are the slides I used.  Thank you for everyone who came and participated in the session, you made it wonderful!

Get your students talking CMC 2015

If you have any feedback, please let me know in the comments.  I am always looking to improve.

Thanks!

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Which Category Are You In?

Given any topic, you (or students) basically fit into one of three categories.

  1. You really understand the idea (and can probably explain it so well to someone else that he or she now understands).
  2. You don’t understand the idea at all…because you don’t care to know (I don’t care about understanding how to hot-wire a car) or you have tried and just can’t quite get it (for me, this would be how to make really good ice cream, I don’t know how they do it, and I’ve tried).
  3. The last category is the most dangerous to be in…You think you know, but, in fact, you don’t know at all.

The third category is dangerous because you won’t actually know that you are in that category until you are tested in some way.  For our students, this is probably too late.  Because for the most part, the next opportunity (if they have another opportunity) for them to prove that they understand, is a final.

This is also a scary place to be in as a person in general.  For example, what if I thought I really did know how to pilot a plane, or swim, or something else that may cause me to get hurt.

But, when I say dangerous, I don’t mean that students are in danger, I more mean it is a bad place to be because of the consequences of thinking they know, when in fact they really don’t.  They will not  ask a question…why would they?  They already understand.  Or learn what they are doing wrong, so then they can do it right…again, because they are already doing it right, they wouldn’t look to learn what they are doing wrong…deceptive, isn’t it?

It is more likely for someone to move from the second category (where they know they don’t know) to the first category (where they really know) than it is for someone to move out of the “dangerous” category.  If you didn’t know how to do something, like make a chocolate cake, you could learn how by looking up a recipe, watching a cooking show, watch someone else,…there are so many options.  Once you learned, you move from the second category up to the first.

One of my greatest fears is that I am (for some areas and topics) living in the “dangerous” category without even knowing.  Maybe my pride, or something.  I guess one of the ways I try to do a self check is by learning more (about whatever), and admitting when I really don’t know (again, dumb pride).

What are you doing to make sure you are not in the dangerous category?  In terms of teaching?  Or anything for that matter?

 

 

 

 

Fixed vs. Growth Mindsets and Math vs. English

Lately, I’ve been really contemplating why I had a fixed mindset, currently trying to cultivate a growth mindset in every area.

Like most math teachers, I am really good at math.  Math is easy.  I figured out how to solve complex math problems.  Honestly, I think I can only remember being stumped in high school once.

So here is how I got to the place of wanting to change my mindset.  I first read Mindset, by Carol Dweck, I also went to a presentation by Jo Boaler (at CMC South) and I read Ed Burger’s The 5 Elements of Effective Thinking.  I am currently reading John Maxwell’s The 15 Invaluable Laws of Growth.  You can say that I saturated myself in this type of thinking.  These are all basically explaining how to continue to grow and not be concerned about failure, because you get to learn from failure (more than you would from success) anyway.

A colleague, who is an English teacher (the kind you wish you had, who by the way has a growth mindset), and I were having a conversation about mindsets.  I had an amazing revelation-type thought.  When you learn math, you try to avoid mistakes or “failures”.  As opposed to when you learn how to write papers, every iteration of a paper is a “failure” until it is revised, which might be another “failure”, and so on.  When you write a paper, you want people to find your mistakes, so you can  improve your paper.  A “failure”, when writing papers, is a momentary location on the road of improvement.  As opposed to in math, mistakes or “failures” are like destinations, in which you now have to start the journey all over.  If we can somehow use these mistakes as temporary locations instead of final defining destinations, we can inherently change students’ mindsets.

I have to say, in my own very humble opinion, the more students (and teachers for that matter) are good at math, the more they are prone to have a fixed mindset.  We try desperately to avoid mistakes, instead of use them to strategically find the solution of the math problem.

What do you think?  Do you have a fixed or growth mindset?  What do you think help you grow that mindset?

My First Attempt at Open Middle Math Problems

Don’t you love when you learn something that makes your life better? Especially if that something is something you wanted to learn, but thought would be impossible to learn.

This happened to me in Palm Springs, at the CMC-South Conference.  Dan Meyer was talking about math problems that had “open middles”.  Open Middle problems had a somewhat traditional problem background, without the truly traditionally predictable solution path.  Traditional problems (i.e. factor the expression x^2 + 6x + 8) had closed beginnings, middles and ends because the way in which a student would arrive at the solution was clearly paved by the teacher…so all a student would have to do is mimic the teacher’s steps.  Open middle problems allowed students to discover the best path (for them) that would help them get to the answer.

This may seem a little unclear so here is the first example I came up with.

Factoring-open-middle

So, here’s what I like.

1.  There are many paths to the answers.  I presented this to a group of Algebra 1 teachers, with hopes of helping them find problems that would help transition to the CCSS and incorporate most of the SMP.  They used guess and check, “the box” (some call this the array or area model) and algebra tiles.

2.  With only a couple of problems, they were able to “prove” their proficiency at factoring.  Ideally, students would find a number and check to see if the quadratic expression was factorable.  In a sense, they are still factoring the traditional type problem.

3.  Math, in general, tends to teach a concept, then, teaches the “backwards” of that concept.  These types of problems do not go forwards, nor backwards, but almost inside-out because a student has to truly understand factoring in order to find the answer.

4.  There are multiple entry points for these problems for students who only have a superficial understanding (they can guess and check to see if the quadratic was factorable) to students who are proficient at factoring and can use a variety of ways to find the missing numbers.

5.  It encouraged collaboration and mathematical discourse.  Because the solution path was not clearly laid out for the students, they had to rely on each other to think things through. I was simply amazed at how these problems naturally promoted collaboration.

6. There are a couple of answers for number 1 (4, -4, 5, -5).  Most found the positive answers. After they read the directions carefully (“integers”), they were able to find more of the answers.  Click here to see the rest of the answers.

So, to make a long story short.  These open middle problems are my new quest.  I want to find more problems like these.  Robert Kaplinsky and I are collecting/making problems like this at OpenMiddle.