Teachers,
Please click here: NAEP
Thanks and have fun!
Larissa
Monday, June 30, 2014
Tuesday, June 17, 2014
Cross Multiplication: A Slippery Slope
As students progress towards their last few years of elementary school, they are often tasked with learning to compare fractions. A very common approach to teaching students to compare the value of two fractions is to use cross multiplication, which is GREAT if all you care about is getting the "right answer." IS all we care about getting the right answer? Absolutely not. In fact, cross multiplication only works as expected in a very narrow range of problems and is not a good foundation for understanding proportional relationships at their core. The most insidious issue about the use and overuse of this technique is that it neither requires nor supports a conceptual understanding of patterns and relationships in numbers.
Cross multiplication was originally designed to determine the proportionality of two ratios or to solve for a missing value in a proportion problem. Historically, there has been little to no instruction for students explaining HOW or WHY cross multiplication works. So, ask yourself: do YOU know how or why it works? If not, how would you go about explaining the approach to a student without it coming off as a "math magic trick?" The perception on behalf of students that math is full of magic is dangerous, to say the least. At some point, the student comes to view the teacher or parent as the magician and gives up trying to understand how the rabbit magically appeared in the hat. In math terms, that's the moment a student hits the proverbial wall and decides that he or she "isn't good at math."
So, HOW does cross multiplication work, and WHY? I'll try not to get too "mathy" here. Cross multiplication is a basic shortcut for finding the lowest common denominator, then comparing numerators. There are two common approaches to cross multiplication:
A) Multiply both sides of the equation by a fractional equivalent of 1 to yield a common denominator OR
B) Multiply both sides of the equation by the product of the denominators.
Cross multiplication was originally designed to determine the proportionality of two ratios or to solve for a missing value in a proportion problem. Historically, there has been little to no instruction for students explaining HOW or WHY cross multiplication works. So, ask yourself: do YOU know how or why it works? If not, how would you go about explaining the approach to a student without it coming off as a "math magic trick?" The perception on behalf of students that math is full of magic is dangerous, to say the least. At some point, the student comes to view the teacher or parent as the magician and gives up trying to understand how the rabbit magically appeared in the hat. In math terms, that's the moment a student hits the proverbial wall and decides that he or she "isn't good at math."
So, HOW does cross multiplication work, and WHY? I'll try not to get too "mathy" here. Cross multiplication is a basic shortcut for finding the lowest common denominator, then comparing numerators. There are two common approaches to cross multiplication:
A) Multiply both sides of the equation by a fractional equivalent of 1 to yield a common denominator OR
B) Multiply both sides of the equation by the product of the denominators.
Remember that if you have two equal quantities and multiply them
by the same amount, the products will again be equal. So if we
multiply the fractions a/b and c/d by b, the results are equal:
a c
--- * b = --- * b
b d
which can be written as
bc
a = ----
d
Now we can multiply both fractions by d:
bc
ad = ---- * d
d
which, of course, means
ad = bc
There are a number of pitfalls of using cross multiplication. One is that it detracts from paying
attention to the relationship between the two values. Are we comparing unit price to unit
amount?
Cross multiplication gained ground and its popularity swelled in a time when rote methods wereapplied without thought or context. The fundamental drawbacks of cross multiplication may not be as obvious in elementary school as they will surely become in middle and high school, when functions, graphs, linear equations, and other key algebraic ideas depend upon a more dynamic understanding of the relationships between and among numbers. It is our responsibility as educators to ensure that we are setting up our students for success and that we are keeping in mind the need for vertical articulation from start to finish. After all, it takes a village.
Tuesday, June 3, 2014
The Role of Rigor In Preparing NC Students for Life (Rene Herrick)
Recently, my friend and colleague wrote a piece that was published in the News and Observer. I would like to share what Rene wrote because I think her words are of great import. The article can be found here: http://www.newsobserver.com/2014/05/26/3883639/the-role-of-rigor-in-preparing.html?sp=/99/108/
Jake poured 6 1/2 quarts of water into his fish tank. Each pitcher held 2 3/5 quarts of water. How many pitchers did it take Jake to fill his fish tank? If just reading that question gives you anxiety, you are not alone.
You are likely a product of the way we “used” to teach math. If you cannot remember how to solve such problems, being terrible at math may not be the reason – memorization may be the culprit. You were probably taught an algorithm and were required to practice it over and over again. If you haven’t used that algorithm for a while, well, it’s gone from the active part of your mathematical mind.
I became a National Board Certified Teacher in 2012 as a Middle Childhood Generalist because, as an elementary teacher, I teach all subjects. The Middle Childhood Generalist Standards from NBPTS states, “The knowledge that accomplished teachers have of their students is enhanced by their understanding of the social, physical, emotional and intellectual development that characterizes middle childhood. Teachers recognize that these students are maturing in their ability to progress from concrete to symbolic and abstract thinking.”
At the heart of board-certification is the understanding that teachers must have purpose in everything they do, that teaching ensures students begin to see the intrinsic value of education and that students deserve to be challenged and are short-changed if they are not. Similarly, the Common Core State Standards focus on developing critical thinking and problem solving – analytical skills that are applicable to any number of academic topics or real-world situations.
At the elementary level in mathematics, we begin by building the foundational understanding of the concrete understanding using models, number lines and drawings. Then we move toward the representation of that conceptual understanding followed by the abstract equation. Without that foundation, students are not successful mathematically.
Common Core Standards for Mathematics provide the precise structure for teachers to build that foundation. The Standards for Mathematical Practice were created by the National Council of Teachers of Mathematics back in the early 1990s. Nearly 25 years later, we have Common Core Standards for Mathematics that address these practice standards and build the foundation necessary for elementary students. As a math coach in an elementary school, I support the Common Core Standards and the Standards for Mathematical Practice wholly and completely.
With this shift in teaching and learning, some parents have expressed frustration, even anger, because the methods they learned in school are not necessarily the approach their children are learning. I remember the times my own parents became frustrated as they watched me struggle through a homework assignment, puzzled at the approach I was learning in school. Those of us now raising our own children will experience similar challenges – though having access to so many digital resources does change the dynamic a bit. The common core allows teachers and students to focus not on procedures and rote memorization but on drawing out a deep understanding of what they are learning, essentially solving for “why.”
Rather than re-creating the generational divides or repeating our errors, these higher standards will enable us to dig in and focus energy on ensuring students truly master concepts because the Common Core State Standards provide the structure necessary to build a strong foundation. We continually fall below globally in math and science. In China teachers develop the conceptual understanding of solving for “why” before they move elementary students into understanding the abstract algorithm. Singapore math devotes the majority of time and energy in building number sense. This, too, is putting the focus on building the conceptual understanding like Common Core Standards.
It is our job to ensure the children of North Carolina are given the most rigorous education possible and that we prepare – not protect them – from challenges and new ways of learning. Our children will be well-prepared for life after high school – be it college, technical school or career – if we do.
Rene Herrick of Holly Springs was the 2009 Wake County Teacher of the Year.
Friday, March 28, 2014
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.
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.
Wednesday, March 26, 2014
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.
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.
Thursday, March 20, 2014
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)
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)
Friday, March 7, 2014
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).
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:
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.
| 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% |
PS: I also really love PIE. So...if the mood strikes, feel free to drop off a slice of rhubarb or pecan pie. Seriously. :)
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