NAEP (National Assessment of Educational Progress)

Teachers,

Please click here: NAEP

Thanks and have fun!
Larissa

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.

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.

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.

Read more here: http://www.newsobserver.com/2014/05/26/3883639/the-role-of-rigor-in-preparing.html?sp=/99/108/#storylink=cpy