Mathematical Habits of Mind, Pt. 1

We are well into our teapots project at the moment.  In one of our brainstorming sessions we discovered  that one of the aspects of design we will need to be well aware of is the concept of surface area and volume.  Without knowing that, we cannot design a box that will be sturdy enough.

This year in Math class, I am attempting to focus on Mathematical habits of mind.  There are countless taxonomies of mathematical thinking out there (it seems every researcher in the field has one of their own, slightly different from the rest), but I have chosen to work with one developed by Al Cuoco, Paul Goldenberg, and June Mark.  I have chosen this one not because it is superior, but just because it is easy to follow and covers a broad spectrum of thinking skills (and it contains the word Tinkerers, which I LOVE).  This will be a recurring series of reflections on how this is shaping our math environment.  It is my hope that this changes their conception of what math is, and how we do math.

[caption id="attachment_1218" align="aligncenter" width="614"] Class Poster[/caption]

Part of how I use these habits is to orient attention to when they are being used.  At the moment, since the vocabulary is new and the kids are not used to them yet (beginning of the year and all) I am the one pointing them out, but as the year progresses on, I hope it is the kids who are bringing them up and realizing them on their own.  Also, I use them in my lesson design.  After only a couple of weeks, I find it is helping me bring different styles of mathematical thinking into our class.  Here is what we did today:

The Task

When the kids came into class in the morning, there were seven containers waiting for them with letters under them.


Each child individually inspected the items and ranked them from most to least (I didn't specify a unit, but most figured out I was talking about Volume).  Next, they described their reasons for choosing the order they did to a partner, and then another partner.  I kept interrupting and orienting attention to use of mathematical terms like volume, capacity, and surface area.  Finally, we had a class discussion about the choices, in which a mini-debate arose about the difference between the orange pan and the measuring cup with the handle (flat and wide versus narrow and tall).


After the discussion, we plotted all of our answers on the board and found the most popular choice, in fraction and percentage.

I then asked them to write sentences about this data.  What is it telling you?  What statements can you make about this?  They came up with things like; most people are sure about the smallest containers (81%), the biggest containers are guesses between two choices, and the middle three are very unsure.  This led to a fruitful discussion about the probability and likelihood of percentages.  We deconstructed, in language terms, what is the difference between 100%, 75%, 50%, 25%, and 0%.  Describing numbers using mathematical language was the habit of mind we were working on.


Next, I asked them to design an experiment to test and prove which one is the biggest, and then to present their experiment to the class in our math congress.  We had three different strategies presented:

  • Fill each container with water, then pour it into a large measuring cup and record the amount in ML.

  • Fill one container with water and pour it into another container; it is overflows it is bigger, if it doesn't fill it is smaller.  Continue this until you know all the sizes.

  • Put marbles inside each container and count the marbles

I originally intended to have them vote on one strategy that we would all do, but I changed my mind and let them each do their own experiment.

To finish it off, we all shared our data and compared to each other.  Each group got the same order, so we concluded that our methods were all sound.  Next, we compared our original guess to the actual answer we found.  We found that the group average was NOT the correct answer.  A few kids had guessed the order correctly, but most had gotten 2-3 wrong.  Finally, we had a discussion and each group made a T-Chart about what was effective about their experiment, and what was ineffective.  We talked about accuracy and how important it is be as accurate as possible.  I hope this type of hands on inquiry and reflection translates into a deeper understanding of Volume when we move to the calculation and mathy side of this concept.  At least we will have something to ground our calculations in, moving from an Enactive experience such as this, to a more abstract and Iconic situation of formal school mathematics (I have started calling it school mathematics since I read Street Mathematics and School Mathematics by Terezinha Nunes, which I HIGHLY recommend). 

Reflecting on the Habits

Finally, we reflected on the differences between the three habits of mind.  I asked them to write a definition of each one in their Evernote reflection journals.  Here are a couple of the reflections:

  • It is important to be a guesser because  if we use a do it, it is faster but it is not accurate

  • the more we practice guessing the more accurate we might be

  • I think it's important to experiment things all the time because if you learn a mathematical thing, you should always want to check it. It's always better to look at the thing, than just listening to details.  Do it yourself to find out.

  • It is important to describe because some one cannot understand you.

I am very interested to see how these habits affect the course of the year.  They struggled with the reflection process, and I will need to provide more scaffolding to help them to be become better reflectors in math.  Orally, we have some great discussions and they provide me with a lot of details about what they are learning, so I am hopeful that these habits will become part of the classroom culture, not only in math, but in all subjects.


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