All Roads Lead to Rome

     I don’t know about you, but when I was growing up in the school
system, math class was the one place in which I could depend on there being
a degree of certitude. Whereas many of my other courses were highly
subjective, mathematics was the class in which there was a definite correct
answer and, typically, one correct method used to find that answer. In a
sense, this was a great relief: it was black or white, right or wrong, yes or no.

     However, as a classroom teacher and mathematics specialist, I have
come to celebrate the trend in mathematics towards open-ended questions
that may or may not have one correct answer. Have I gone soft? Have I lost
my edge? It’s possible, but I don’t think so. Hear me out…

     The inherent issue in mathematics problems in which there is only one
right answer and only one “correct” approach in finding that answer is that
there is little room for creative, higher-level cognition. Isaacs and Carroll
(1999) put it best: “The rote approach encourages students to believe that
mathematics is more memorizing than thinking.”

     This is also a question of equity and access to curriculum. We
understand instinctually that there are myriad different types of thinkers, and
we are helped along in this process by folks like Dr. Howard Gardner of
Harvard University, who introduced us to the Theory of Multiple Intelligences.
We understand as educators that it is both our right and our responsibility to
reach all learners, and this calls for creative instruction on our parts. Perhaps
one of the best ways to creatively teach and reach is to differentiate our
instruction to the needs of all students in our classrooms, keeping in mind
how differently their brains operate. We approach teaching today with
multiple intelligences and multiple modalities in mind and, in doing so, even
out the playing field. We recognize that some students are better able to
demonstrate their learning and understanding using concrete materials,
others are more comfortable drawing pictorial representations of their
mathematics, while others prefer to use an algorithm to solve problems. The
research is very clear that none of these approaches trumps the other.

     So, what can you do as the parent of a math student when working
with your children at home? I highly recommend sharing with them what you
learned in your own education, but keeping an open mind to the new and
creative ways your children are approaching the same problem. You will both
learn something! For example, you are probably most familiar with the
standard algorithm for vertical multiplication. Rest assured your students willbe
exposed to this, but they will also see Base 10 multiplication using physical
 manipulatives, array models, linear models, set models, lattice multiplication,
partial products, etc. Exposing your children to these varied techniques assures
us that they will be able to access the information on their own terms, thereby
taking ownership of, and pride in, their learning.

     All roads lead to Rome, and all approaches being taught to your children lead
them to feeling successful and fulfilled in mathematics.

Mathematics and the Growth Mindset

Every few years, new "buzzwords" enter the field of education and threaten to overtake the nomenclature. Sometimes, things go a little bit overboard; these terms are so overused and abused that they become laughable. Before we know it, Jimmy Fallon and David Letterman are cracking jokes and using our educational terminology in a rather tongue-in-cheek manner. I admit that I crack up right along with them. (Hey! I'm only human, right?)

That having been said, there is a term in education that has hit the scene relatively recently at which I will NEVER laugh. The expression? "Growth Mindset."

What is a "growth mindset" in terms of education and, in particular, in terms of mathematics? I could spend hours giving you my perspective, but wouldn't you rather hear from noted Stanford University psychology professor and author, Carol S. Dweck? I thought so.

Dr. Dweck has dedicated her entire professional life and career to researching achievement and success and to translating these into motivation and productivity. She dares challenge the status quo of teaching, coaching, and parenting, taking an in-depth look at the impact of praise, the philosophy of talent, and the general approach we take to shaping our next generation. Dr. Dweck illuminates the ways in which our "fixed mindsets" unintentionally (but VERY successfully) undermine, subvert, and limit the ones we love the most.

In 2008, Dr. Dweck published a ground-breaking paper entitled, "Mindsets and Math/Science Achievement." The crux of the paper is that:

          There is a growing body of evidence that students’ mindsets play a key role in their math and science             achievement. Students who believe that intelligence or math and science ability is simply a fixed trait               (a fixed mindset) are at a significant disadvantage compared to students who believe that their abilities           can be developed (a growth mindset). Moreover, research is showing that these mindsets can play                 an important role in the relative underachievement of women and minorities in math and science.

She goes on to provide compelling proof of the following:

a) mindsets can predict math/science achievement over time;
b) mindsets can contribute to math/science achievement discrepancies for women and minorities;
c) interventions that change mindsets can boost achievement and reduce achievement discrepancies; and
d) educators play a key role in shaping students’ mindsets.

Even my personal hero and favorite non-fiction author, Malcolm Gladwell, is getting in on the action. Take a look at his article for the New Yorker, "The Talent Myth," in which Gladwell calls out smart people for being overrated.

http://www.newyorker.com/archive/2002/07/22/020722fa_fact?currentPage=all

If you'd like to read Dr. Dweck's research for yourself, it can be found here:

http://growthmindsetmath.files.wordpress.com/2012/08/dweck-mindsets-and-math-achievement-2008.pdf

Are you curious about your own mindset? Take an online quiz! C'mon! It might be FUN!

http://mindsetonline.com/testyourmindset/step1.php

We here at Brassfield are committed to the ideal that our students do NOT have fixed abilities. We are committed to facilitating the discovery on the part of our children that the world is their proverbial oyster and that all things are possible through hard work and commitment.

 

Going for Gold: The "4"

I have had several productive conversations recently about what it means to earn a "4" in mathematics. Many debate the idea of students earning a "3" even in cases in which every answer given is correct. So what more do we want of students if getting all of the answers correct is not enough to earn a "4?"

According to many educational likert scales, a "1" indicates a beginning understanding, a "2" means that a student is working with a more intermediate understanding, a "3" describes a proficient understanding, and a "4" is considered advanced proficiency in understanding and application.

Let's take counting coins as an example. Here is what a scale might look like for counting coins:

1: a student is familiar with the values of basic coins and can recognize and name those coins
2: a student can count a collection of basic coins within 100 and can correctly write the total, including using the symbols for dollars and cents
3: a student can name and count coins within 200, can correctly write the total using numbers and symbols, and can apply their understanding of coin counting to real-world scenarios
4: a student can do all of the above AND can think of and create real-world scenarios in which counting and collecting coins would be relevant; a student can also demonstrate multiple strategies for counting coins (using dimes to count by 10s, for example, then using half dollars to count by 50s); a student can compare and contrast these strategies for counting coins

Here are some great questions to ask students when encouraging them to "GO FOR THE GOLD" and reach that higher-level thinking required to earn a "4":

- can you represent and justify this differently using words, pictures, and/or numbers?
- can you make connections between what you are learning and something else? (math to math/ math to self/ math to world)
- can you create a problem or context using what you have learned?
- can you think of examples AND non-examples in the real world of this particular skill?
- can you explain your thinking clearly to others?

As a general rule, the higher up the ladder of Bloom's Taxonomy we ask students to go, the more likely they are to attain the "4." (See below)

Pi Day: Let's Celebrate!

I don't know about you, but I sure love Pi! In fact, it is my very most favorite irrational number. Most everyone knows that it's 3.14, but do they realize that it's all about the constant relationship that exists between circumference and diameter?

Want to explore that relationship further? Play with a cute puppy dog as he walks around in circles on his leash: http://illuminations.nctm.org/Activity.aspx?id=3547

If you are looking for ways to celebrate Pi day, look no further!

On March 14 (3.14) you may want to take a gander at one or more of these exciting and interactive websites. You'll score extra points if you do so right at 1:59 PM (since Pi is 3.14159).

Here is a Pi day webquest you might want to check out: http://www.mathgoodies.com/webquests/pi_day/

Here is an animated sequence that "unrolls" Pi: http://commons.wikimedia.org/wiki/File:Pi-unrolled_slow.gif LOVE IT!

Want a brief history of Pi? http://www.exploratorium.edu/pi/history_of_pi/index.html

Have you ever seen the first 1 million digits of pi? http://www.piday.org/million/

Did you know you can explore Pi with music? http://avoision.com/experiments/pi10k THIS IS SO COOL! CHECK IT OUT, EMILY NIXON!

Wanna search Pi for number combinations? (Birthdays, Jersey numbers, etc)? http://www.angio.net/pi/bigpi.cgi

Chances of Finding Your Number in Pi

Why can/can't I find my number in Pi? If we view Pi as a big, random string of numbers (which is close enough for our purposes), then we can figure out the odds of finding any string in the first 100 million digits of Pi:

Number Length	Chance of Finding
1-5	100%
6	Nearly 100%
7	99.995%
8	63%
9	9.5%
10	0.995%%
11	0.09995%

Happily, if you include the zeros, birthdays are 8 digits long -- so you have a 63% chance of finding your birthday in the first 100 million digits of pi. Now that we're to 200 million, the odds are up to 86%, so it'll be a while before everyone can find their birthday in Pi. 

PS: I also really love PIE. So...if the mood strikes, feel free to drop off a slice of rhubarb or pecan pie. Seriously.   :)

LEGO MATH! (Yep, you heard that right).

Speaking of fractions, I've been chatting with the 3rd grade team here at Brassfield about using Legos to explore ordering, comparing, adding, and subtracting fractions.

My son, Anthony, who is a 4th grader at Brassfield, was kind enough to let me raid his Lego collection (don't worry...he literally has BINS of them at home) to make these lessons come to life. He even helped me sort them. What a guy. Thanks, Anthony!

Here's an article you might find interesting that shows how all you moms and dads at home can take playing with Legos to a whole new level. Enjoy!

http://faculty.tamucc.edu/sives/1350/tcm2011-04-498a.pdf

Number Sense: It Makes SENSE!

If someone were to force me to identify the one mathematical gift I would choose to bestow upon students, I would whip out my magic wand and shout, "NUMBER SENSE! NUMBER SENSE! NUMBER SENSE!" (Come to think of it, that might just work! I'll have to try it later).

Why number sense?

Number sense is actually quite intuitive. So why teach it? Why reinforce something that comes naturally to students? That's a good question. And I have a good answer. The traditional teaching methods employed in mathematics have actually turned students AWAY from their inherent sense of number. Memorizing "steps" and focusing on traditional algorithms and/or formulas is about as far away from the development of number sense as are the temperatures in Death Valley and Antarctica.

In order to develop a sound sense of number, students must focus on place value, composing and decomposing numbers, understanding the relationships between and among operations, acquiring automaticity and fluency with facts and operations, observing the magic of mathematical properties, comparing and contrasting whole and rational numbers, and more. Students with a strong sense of number are able to estimate well and determine the reasonability of any given answer with ease.They can communicate clearly and effectively about their thinking and can use words, pictures, and numbers to defend their answers. They are extremely FLEXIBLE thinkers and learners and are equally at ease using mental math as recording their math in written form.

The research and data is definitive that number sense with regards to rational numbers (fractions and decimals) is where our students are falling the farthest behind. This is not just a Brassfield problem, nor is it even a problem exclusive to Wake County or the state of North Carolina. Students across the country struggle more with their conceptual understanding of rational numbers than of any other single concept in math.

Please consider taking a gander at this fantastic 2010 article by the National Counsel for Teachers of Mathematics entitled, "Using Number Sense to Compare Fractions."

http://www.ileohio.org/materials/Documents/Using%20Number%20Sense%20to%20Compare%20Fractions.pdf

Teaching Students the Magic of Math

I recently came across this article and video that really encapsulates the spirit of what we are trying to do here at Brassfield in terms of collaborating and crossing curriculum to engage students.

Edutopia/NPR article

Magic of Math