Equal or Equivalent?



CZ Romani girl solving math problem in special school

Does 2 + 2 = 4?

Are the group of numbers on the left side of the “equal sign” really “equal to” the number on the right side? By presenting this problem to students and using the word “equal,” we are telling them that those two little parallel lines mean there is only one possible answer, and by doing so, we may be missing an important opportunity to introduce algebraic thinking at an early age.

Research has shown that teaching elementary age children to think algebraically is a precursor for success as they move toward advanced levels of mathematics. By introducing the idea that the “equal sign” represents a relationship between two mathematical expressions, students begin the important process of relational thinking.

Let’s look at the following example:

7 + 3 = 2 + 8

If we say that 7 plus 3 equals 2 + 8, we are essentially limiting the ability of primary grade students to comprehend this unique relationship. Seven plus three does not ONLY equal two plus eight; it also can equal nine plus 1, and one plus one plus eight, etc. By using instead the words “equivalent to,” parents and teachers can emphasize this relationship and begin a math dialogue with their children to encourage a better understanding of mathematical expressions.

Understanding this concept can also help the student who struggles with memorizing addition facts. Using mental math, students can explore the concept of “friendly numbers,” those numbers like 10 and 20, which are easy to compute mentally. Asking a 1st grader to tell you the sum of 19 plus 28 may create a sense of panic in those who rely strictly on memorization. But when young children begin the process of decomposing numbers and understanding that there are numerous ways to create any given number, their computational fluency increases dramatically. Some students may decide to increase the number 19 to 20, by reducing the 28 to 27. So the problem 19 + 28 becomes equivalent to 20 + 27! Others may change the 28 to 30, thus creating the equivalent expression, 17 + 30.

At Posnack, we have encouraged all elementary level teachers to adopt the phrase, “equivalent to” in their math classes as a means of emphasizing this relationship and starting the journey to algebraic reasoning at a young age. Watching the Kindergarten and 1st grade classes in the morning is an amazing display of mathematical wonder. Students use the number of days they have been in school to develop word problems, number bonds, and mathematical expressions. And while many may still have a little difficulty pronouncing the word “equivalent,” they have no problem at all explaining how it works using two- and three-digit numbers. Watching the excitement in the classrooms as the students discover more than one “right” answer, is a thrill that is “equivalent” to none!



I’m Inspired…




I have a new favorite word this week. It’s a word we’ve all used from time to time, and in a variety of contexts. It’s a word that brings to mind visions of hard work, dreams realized, and hopes for greatness in future endeavors. It’s a word that makes me smile broadly when I think about it being used as teachers teach and leaders lead. I like to think of my word as a super-hero action word – a verb that soars above buildings and goes where others dare not venture.

I have a new favorite word this week and that word is INSPIRE.

The dictionary definitions alone create a feeling of wonder and the determination to plow ahead. Entries found in Dictionary.com include “to fill with an animating, quickening, or exalting influence,” “to guide or control by divine influence,” and “to fill or affect with a specified feeling or thought.”

WOW! Just reading these definitions makes me want to go out and conquer the world. So what does this word have to do with education? EVERYTHING!

A misconception is that anyone can be a teacher given a college degree and an affinity for working with children. It’s true that oftentimes, important skills like classroom management, assessment design, and lesson planning may not really be perfected until well after entering the classroom. But it takes much more than a teaching degree and a state certificate to turn a teacher into an inspiring force. It takes teachers who INSPIRE students to take on challenges that they might not think they are ready to face. It takes teachers who INSPIRE students to not worry only about the right answer, because as history has taught us, what’s right today may actually be obsolete tomorrow. And it takes teachers who INSPIRE and nurture an innate curiosity in students while cultivating the seeds of learning with attention and care.

Hopefully in our lifetime, we have all encountered teachers who have INSPIRED us to do our best, be our best, and bring out the best in others. I see many of those teachers every day, as they arrive at school early and leave well after the last bell has rung. And as an educator, I too continue to be INSPIRED by others who embrace change, think “outside the box,” and create an environment that questions the status quo.

So by now you may be wondering what happened this week to make me feel especially “INSPIRED.” It was a series of events, but in particular, a TED Talk video by a chemistry teacher, Ramsey Musallam. His three simple rules for teaching provide a pedagogy that can be used by educators in all grades and subject areas. Watch Dr. Musallam in the clip below and prepare to be INSPIRED.


Climbing the Ladder…


I am excited to announce that this blog has been ranked by the Teach100 website!

 Blog rankings are updated daily, so with your assistance, we can watch together as Curriculum Corner moves up the ladder. Please share this blog with your friends and family so that they too can see what sets Posnack’s curriculum apart from all the others.

Turning Mirrors Into Windows…



I truly believe that everything we do in life must have a goal or purpose. Look around and you’ll see that most of what occurs has a specific outcome in mind. During this holiday season, the airports and roads are filled with travelers. Their goal? They want to arrive safely at their destination. Football, basketball, hockey teams? Their goals right now are to have winning seasons.

But how do we define the goal or purpose of intangible and wide-reaching activities like education? When asked why they come to school, an informal survey of elementary students turned up answers like “so we can learn,” “so we’ll get a good job,” and even “so we can make friends!” These all sound like worthwhile and lofty goals, but are these the only outcomes we expect when we ultimately see our children walk across a stage and receive their diplomas?

I recently came across a quote attributed to the well-known journalist Sydney J. Harris, in which he says, The whole purpose of education is to turn mirrors into windows.”

I was reminded of this quote as I walked the hallways of our school and listened to the youngest students telling me that their favorite subjects were lunch and recess. I was reminded once again when I observed a fifth grade student ask his teacher why he had to learn how to multiply fractions because he was sure he would never use it again.

And it suddenly occurred to me that these students were simply looking into the mirror – they only saw education for what it provided them during one isolated moment in time. As educators, our role requires us to take students beyond their own personal reflection, and help them to make the connection to the world around them. In essence then, our goal really is to “turn mirrors into windows.”

We accomplish this task through the introduction of authentic learning – providing opportunities for students to use critical thinking skills, and creating a bridge between that which is provided in textbooks with the experiences and ideas that constitute the “real world.” We encourage collaboration among peers, for in the “real world” we rarely work alone without input or guidance from others. We insist on driving questions and asking things like “I wonder what would happen if…” because we know that problems in the “real world” seldom come with a neat set of multiple choice answers.

We are in the midst of an exciting time at our school. While students are busily working in the lower school, construction workers are even busier pouring tons of concrete as they prepare to erect the walls of the new high school. Every day, students look out the windows of the lower school building and watch the progress being made on the construction. I too look out the windows, but instead view it as progress being made for the future.

So as the first half of the school year comes to an end and we approach a well-deserved Winter break, please don’t forget that the learning doesn’t stop when our students leave the building. In fact, I would like to think that the learning becomes “real” when students are given an opportunity to apply what they’ve learned by moving beyond the mirror and exploring the world outside the windows.

What Do Teachers Make?


This year, during Thanksgiving dinner, my family began what I hope will be a family tradition for many years to come. I’m sure many families already partake in a similar ritual and it’s played out many times on Thanksgiving episodes of television shows. What we did was very simple….we each took turns sharing specifically what it was that we were thankful for. One at a time, we spoke up, without the aid of notes or technology, without prior warning or preparation, without fear of reprisal or criticism. Grandparents, children, cousins, and friends – one by one each of us spoke while the rest listened. An occasional “l’chaim” or clinking of glasses, but the moment belonged to the one holding, for those few minutes, the attention of those sitting around the table.

So, what were we thankful for? Good health, the love of family and friends, being able to spend time with the ones we love. I was most thankful for having my children sitting beside me. Later, I began to think not just of what I was thankful for today, but of the life events that brought us all together to the table that evening. I focused on my children; the youngest starting her trek as a college freshman, and my first-born beginning an even longer journey through graduate school.  I remember like it was yesterday when they first began to read books to ME before bedtime, when they discovered that multiplication was really just addition on steroids, when the science homework became much too complicated for me to assist. And it dawned on me that there was an entire group of people still left to thank.

I could take some credit for providing the books, the study space, and what I believed to be their inherited drive to always strive for their best. But what about their content knowledge, the creativity in their presentations, and the ideas that inspired them to forge ahead? Well, I believe that much of that credit goes to the teachers who touched their lives in the past and continue to do so in the present.

And so, I would like to extend my “thanks” and gratitude to all teachers – teachers who make a difference every day, teachers who make this profession their life calling, teachers who make students reach for the stars. And for those who may question why someone would become a teacher or what teachers “make”, I share this amazing video performance by the renowned Slam Poet, Taylor Mali.

For more information about Mr. Mali’s book and ways in which teachers make a difference, simply click on the link to the right.


Follow the Map…



What do RIT scores, Percentiles, and the Zone of Proximal Development all have in common?

Please join me this Wednesday, October 22, to discuss how Posnack teachers use MAP testing results to set student goals and differentiate learning in the classroom.

Location: Middle School Multi-Purpose Room
Time: 9:15 am (immediately following the PTO meeting) and 7:00 pm

RSVP to jsilverman@posnackschool.org

College Bound…



Tonight, as I am packing the last few items in the duffels that will accompany my daughter to college tomorrow, I remember each “first day” of her progression from kindergarten through 12th grade. The rites of back-to-school were always the same; trips to the uniform store convincing her that polyester skorts with elasticized waists were definitely cool, finally accepting that the vinyl hot pink Hello Kitty backpack could not possibly be used two years in a row, and trekking to Target to find the college-ruled 5-subject spiral notebook that wouldn’t be used for more than one subject. I remember lining up for carpool an hour before dismissal so that I could get the first glimpse of her face in an effort to know ahead of time whether or not today was a good day. Did she sit with her friends at lunch? Did she get chosen to be line leader? These simple questions soon gave way to other questions that seemed at the time to be so much more important. When will she learn multiplication? Is she reading at a 5th grade level while still in 1st grade? Did she ace her spelling test again? The questions continued into high school. How many AP classes can she take? Should she start her college essays in 10th grade? How many times should she take the SATs or the ACTs?

So now as I look at the empty hangers in the closet and the drawers so vacant that they actually close, I realize that none of those “important” questions really mattered. She learned multiplication, at some point read at a 5th grade level, and had a high school transcript impressive enough for more than one college to offer her a spot in their class of 2018. Is she ready for college because of accelerated math skills or the diorama she painstakingly glued and painted as part of a book report? No… she’s ready for college because sometimes the math wasn’t easy, and sometimes the literary elements were confusing, and sometimes the test scores were not in the 99th percentile. She’s ready for college because she learned to persevere whether it was learning a new skill or navigating the halls of a new school. She’s ready for college because she has character, values, and a sense of what’s right and wrong. She’s ready because she has what is referred to as “grit.” Grit isn’t found in a curriculum or textbook – grit comes from realizing that life may not always be fair and it may not always be easy. Grit is developed by setting goals that may at times seem insurmountable, but striving for them anyway. We don’t give grades for grit, but maybe we should, for if we did, it would be interesting to see which students really earned straight A’s.

A new chapter in our lives begins tomorrow. Although I won’t be there to pick her up next week on another first day of school, I have a feeling that she’ll be okay without me asking the important questions about her day or paving the way with gifts for her teachers. And as for me? I’ll have to work on developing grit too as I learn to make my way in a house that will be just a little too quiet.


Read more about the characteristics of grit


Leading the Way…





It’s hard to believe the summer of 2014 is already coming to an end. Back-to-school specials fill the stores and children are returning home from summer camps and family trips around the world. Posnack families kept in touch through social media sites, and our very own Ram News featured students and their family members boasting of their Posnack connection by wearing Ram gear and holding up signs proclaiming #posnackeverywear.

So while Chaucer’s well-known proverb, “All good things must come to an end” seems fitting during these last few days, I prefer to think more along the lines of “The best is yet to come.” Posnack is proudly entering it’s 40th year with a wide array of NEW additions to the curriculum, NEW technology, and the ground-breaking for a NEW high school building and gym! Computer coding, a 3-D printer, personal finance courses, innovative engineering programs – these are just a few of the exciting changes students will find when they return to our school next week.

For those readers outside of the Posnack community, I wish you a school year filled with learning that is engaging, challenging, and interactive. This is a wonderful time to be a student in a school rich with technology, as well as for teachers with a passion for their profession. Experiential and interactive learning give students the chance to see their vision become a reality and to make the connection from classroom learning to real-world living. With all the tools and technology available to our students today, I truly believe that the best is yet to come. Thank you, Posnack, for leading the way.