Building Enduring Mathematical Understanding, one lesson at a time.

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Eradicating the Roots

No, not THOSE roots . . . but bear with me, there’s math in here somewhere. . .

As my youngest enters her senior year, we find ourselves experiencing a series of “lasts.” Some are met with melancholy, and others  with enthusiasm. A few weeks ago we had our “last XC trail-clearing,” which definitely fell on the “so glad this is the last one” side. Our kids are fortunate to run with one of the best programs in the state (not that I’m biased) and lucky to have their own course on school grounds. Each summer (somehow, usually on the hottest Saturday – thanks Coach) parents and athletes gather for “Trail Maintenance,” working to make sure the path is clear for training and races to come, in the fall. One especially “high maintenance” portion runs up a hill alongside a bank of . . . blackberries. (One of the “favorite” practices each year involves “picking” handfuls of berries to smash on fellow teammates, but I digress.) As blackberries are prone to do, each year they begin to encroach upon the running path, high, low, and everywhere in between, lying in wait to “attack” an unsuspecting victim. This summer, one of the Alumni Parents (uh oh, maybe I’m not done yet) used a brush cutter a few weeks before our “trail day” and the blackberries were cut back farther than they have ever been. Hurray! However, if you know anything about blackberries, it’s that they will sprout from the root. As I looked up, I saw new growth at various points along the bank. I decided to try my best to give a gift to future generations of XC runners (and parents) by working on digging out the roots rather than just clipping off the vines. It was less than fun, but also rewarding. Some came out as massive chunks, and others like a long snake. Sadly, some broke off and I couldn’t dig down deep enough to really get at the remaining root. 

Which brings me to math . . . 

How often do we, as teachers, just trim away at the surface of our students’ misconceptions? We “fix” the problem for now, but, leaving the root behind, it will sprout again. Worse yet, the root of the misconception will continue to grow and deepen, even sending out runners to other related concepts. How many areas can be affected by a misconception about place value? It will continue to fester, at least up through scientific notation. A limited “take away” perception of subtraction, with no understanding of “finding the difference” will follow students forever (slope, anyone?) A few weeks ago, @bowenkerins (how do I make that a link?) tweeted about issues regarding the equals sign as a “do this” symbol rather than an indication of “sameness.” Yet another example of a misconception that hinders students from embracing a deeper understanding of, well, properties, equivalence, equations and functions, just to start.

As I continue to make my “final preparations” (as if one is ever truly prepared) for the school year, I find myself pondering how I can get at that root.

I value being “less helpful” and “asking without telling,” but too often I give more worth to the responses I “hope” to hear. Clip, clip, clip, there went the vines for all of the students who just heard the “right answer” or “best strategy” from one of their own, and don’t need to bother getting to the root. Their misconceptions can continue to grow and fester. 

With an individual student, I may ask deeper and more probing questions, attempting to get them to understand the heart of the matter, “Oh, I get it!” Snap! The root breaks off somewhere below the surface, as their false confidence permits them to just “move on,” and I am left wondering how I can get them to go deeper without endangering their enthusiasm. 

I long for the lessons, tasks, and activities during which light bulbs really do come on, and big chunks of misconceptions get tossed aside, never to take root again!

In the meantime, I’ll continue to dig away. . .


Lurking, Learning, and Leaping

About three years ago, after a LONG career as a lurker, I dipped my toe into the waters of the #mtbos. (To be honest, I’m pretty sure that hashtag didn’t yet exist!) While my blogging and tweeting soon fell by the wayside, the lurking persisted, and here I am again, standing on the dock, ready to make the leap. 

For the past three years I have taught Algebra 8 and Math 8, but this fall I will be returning to where I started when I transitioned from high school to middle school, and teach two sections of Math 6 along with some Math 8! I am very excited about this change, for a variety of reasons: Our Math 6 classes are scheduled as a two-period block (yes – be jealous!) I am thrilled to work with our other sixth grade teachers, I feel that this age has the most potential to really change their perspective about what it means to learn mathematics, and I still get to work with my awesome PLC partner from Math 8 ( and support her as she takes on the Algebra.)

At the beginning of the summer I began to contemplate some of the changes I wanted to make for the this year: 

•Weekly homework assignments that include concepts from the previous week (not the current one!) and spiral back through earlier ideas, especially when related to upcoming concepts. 

•Modifying my SBG to include weekly assessments on four different standards, and like the homework, spiraling back through previous content.

•Embracing a problem-solving, conceptual, numeracy-based approach, making full use of groups of 3-4 students on my “monster whiteboards.”

BAM! On my Twitter feed I start reading about:

Make It Stick, and the ideas of “interleaving” and frequent small stakes assessments thanks to: @pamjwilson, @druinok, @mathymeg07, and others, as well as “the” @hpicciotto.

Resources to share (with teachers AND parents) regarding the difference between procedural and conceptual learning in mathematics:, and , thanks to a request by the ever-helpful @jreulbach, and replies by @MathMinds, @fawnpnguyen, @MathEdnet (and others. . .)

VNPS and VRG, thanks to: @wheeler_laura and others, referencing the work of Peter Liljedahl:

Jumping from link to link, I ran across a post from @AlexOverwijk after his session at last year’s TMC: He ended the presentation by putting something like this on the board:  


That’s the REAL leap I want to make this year!  

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