Geometric Landscape

I got shifted from my usual (of late) secondary student teacher supervision to elementary preservice teacher prep this fall. (We have an unusually low number of student teachers this fall.) I love this teaching, too, so it will be a treat. Pam Wells, David Coffey and Jon Hasenbank were already coplanning a revision to the course, so it gave me a chance to dive in and collaborate. And gave me my first chance to look in detail at the K-5 Geometry common core. So I thought I'd share what I saw:

First I collated them, then tried to look for a way to organize them more sensibly. They are pretty unevenly written. From vague generalities to hyper-specifics. The best common threads I saw were the action verbs about what the students were supposed to be able to do.

Our assignment was to sort them into a concept map or landscape of learning.  I'm very fond of the landscape of learning model for teachers. I first saw the idea in Fosnot and Dolk.  In addition to those Young Mathematicians at Work books, they are involved in the great Mathematics in the City project and the excellent curriculum Contexts for Learning Mathematics. Here's a sample chapter from the YMAW: Algebra book. This sample chapter from Contexts for Learning has a Multiplication Landscape of Learning (page 16).

A landscape emphasizes the many paths through understanding that students might take, and are loosely organized from bottom to top in terms of students development. (Read also Christopher Danielson on landscapes. Here's a landscape from years ago I developed with novice teachers for teaching money.)

Here's what I came up with. I'd love feedback on ordering from top to bottom, what you would add, and classification into strategies, concepts and models.

(Here it is as a PDF.)

There's things that are quite sophisticated present (hierarchical structuring, Van Hiele level 2 and level 3 reasoning) and very accessible things missing (motions, congruence and similarity). Even though they are not included, of course, you can still teach them; use those ideas to help students access the ideas that are required.

As I develop and revise activities for the course I'll be sure to share them. Again, if you have feedback about the landscape, shout it out!

More Money

My preservice elementary teachers are preparing to do some 2nd grade tutoring, and the teacher asked for money and time.  (Really, who couldn't do with more of both?)  Those are always challenge areas, in my experience, so I thought I'd share our resources.  Here is a collection of some money activities.  It includes Change for the Better and Make It Take It which were described in an old post, which are two of my favorites.   It's been constructive to use games for practice, interesting problems to help with concept development, and explore multiple representations.  One thing that's often lacking is a visual representation to help understand the relative value of coins.

A quick easy game:  Monopoly Money Madness

Materials:  play money, 2 dice.
Math content:  addition, money recognition, unitizing (grouping into new amounts.)
Game play:  Very simple – roll two dice, and take that much money.  If you can group your money into a larger bill (for example, a five and five ones into a ten dollar bill).  First player to $100 wins.
Variation:  have players “shop” a catalog or the web for something they would like.  Play until they have enough to buy the item they would like.


Money as a context offers some nice practice for developing unitizing, the understanding that allows learners to flexibly exchange between a groups and individual members, or the ability to switch what you are considering as a unit.  (I.e. switching from dollars to quarters or cents.)  It is such a key concept for 2nd grade, as students move from single digit arithmetic to multidigit arithmetic.  Lack of understanding in this will follow them for the rest of elementary school.

From Mathematics in the City: Measuring Teacher Change in Facilitating Mathematizing, Catherine Twomey Fosnot, Maarten Dolk, et al. (link goes to a pdf of the article)

Unitizing requires that children use number to count not only objects, but also
groups—and to do them both simultaneously.  The whole is thus seen as a group of a
number of objects. The parts together become the new whole, and the parts (the objects in
the group) and the whole (the group) can be considered simultaneously. For learners,
unitizing is a shift in perspective.  Children have just learned to count ten objects, one by
one. Unitizing these ten things as one thing—one group, requires almost a negating of the
original idea of number.  It is a huge shift in thinking for children, and in fact, was a huge
shift in mathematics, taking centuries to develop. Understanding that a square in a tiled
array can represent a column and a row simultaneously also involves a construction of
part/whole relations (Battista et. al., 1998), as does the relationship between
multiplication and division. There are many more. Because “big ideas” involve
part/whole relations, they require a shift in perspective by learners.

As we look at state content expectations, it's pretty clear they are focusing on skills.  Fosnot and Dolk (in Math in the City, their books on Young Mathematicians at Work, and their curriculum Contexts for Learning) have a nice way of organizing content into skills, ideas, and models, and then representing them on a landscape.  The preservice teachers took their information and tried to do the same for money.  Here's what they got: (as a pdf)