Tag Archives: Mathematics

Doing Math…





IMG_2602What is a Noetic Challenge? According to Merriam-Webster’s website, the combined definition of these two words would translate to “a difficult task relating to intellect.” This may be true, but to a group of 13 fifth-grade math students, the term Noetic Challenge means logic puzzles, math games, and a collaborative effort to determine how many pencils make up a “gross!”

Meeting weekly with our math enrichment specialist, Mrs. Feldman, these top math students finished their successful elementary school years by competing in a national math competition. The Noetic Math program is designed to challenge students’ mathematical thinking by strengthening problem-solving skills. All Lower School math classes incorporate a variety of challenging word problems to Posnack’s already advanced math curriculum in order to prepare students for an accelerated rate of learning in mathematics. Since their introduction to Singapore Math last year, students in grades K-5 are focusing on the importance of relating math to real-world situations and identifying problem-solving strategies to tackle higher-order thinking word problems. Whether it’s preparing a shopping list to stay within a given budget, or calculating the square footage of classrooms in our new Fischer High School building, students are learning to recognize that math touches everything in their lives. Posnack students have become quite adept at using bar models and algebraic thinking to solve authentic problems with a variety of strategies and as a result, have learned that success in mathematics IS something to brag about.

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When presented with the idea of a math contest, Mrs. Feldman’s fifth-grade groups were excited at the idea of showcasing what they’ve learned. Over 24,000 students from more than 600 schools in 47 states participated in this contest, and Posnack students received commendations in the categories of participation, national honorable mention, and team winner. The celebration began with the awarding of certificates by Mrs. Feldman, followed by ice cream and the presentation of mini calculators to the fifth-grade math team. Certificates, ice cream, prizes? When I asked one student what the best part of the afternoon was, he replied, “We get to do even harder math now!” This student’s enthusiasm reminded me of the accomplished mathematician Paul Halmos who was quoted as saying, “The only way to learn mathematics is to do mathematics.”  I can honestly say that it’s been a thrill and an honor to watch these talented students “DO” mathematics.

Click here for more information on the Noetic Math Challenge and to see a complete list of national winners.


The Magic of Math…


Do you believe in magic? For most, the answer would be “no.” A logical explanation can usually be found for how the rabbit was pulled from a hat, or the magician’s assistant placed inside a box and seemingly sawed in half.  But rabbits and boxes are tangible items – a careful analysis of the illusion and perhaps some help from a slow motion replay will usually reveal the secrets behind the magic. However, the magic taking place right now in the Lower School isn’t as easy to explain.

Tomorrow begins the second week of our 1st Annual “Math is Magic” Challenge. Last week, all students in grades K-5 were given a different challenging math word problem each day. Students are working together with classmates, drawing bar models and pictures, and explaining their own unique ways of solving the problems. They are comparing their answers with one another and checking for the reasonableness of their solution. The total number of correct answers are tallied daily for each grade and recorded on a bulletin board for all to see.

So what is it that makes a school-wide math challenge so magical? The magic isn’t in the problem solving or the collaboration among students. It’s not in the graphing of the number of problems solved correctly. It’s not even in the students’ ability to apply learned skills to tackle math concepts not yet encountered in class. The magic is in the excitement and engagement level of the students. Students are discussing math problems while walking through the hallways and at recess. They are talking with one another about the best way to come up with a solution. They are actually protesting when they think the problem is too easy. And they are asking for MORE – more problems, harder problems, challenging problems. Posnack students are saying that math is FUN!

The fun and excitement continue this week as we welcome the world-renowned “Mathemagician,” Dr. Arthur Benjamin, to our school for what is sure to be an incredible performance showcasing the magic of math. We invite you to join us this Wednesday evening at 7:00 p.m. in the Orlove Auditorium at the JCC. Will Dr. Benjamin pull rabbits out of a hat or saw his assistant in half? Probably not –  but he will most certainly perform other incredible feats that will surely leave you amazed.

So, will the magic of math continue to take place at Posnack? Absolutely!  There is incredible learning taking place in and around the classrooms, the hallways, and even the playground. With the discovery that math CAN be fun, the students are not likely to let that magical feeling disappear into thin air anytime soon.


It all STEMs from the beginning…


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In a previous post, I discussed the results of the 2012 Program for International Student Assessment (PISA) which compares reading, math, and science literacy around the world. In this study of 65 countries and education systems, the United States ranked 26th in the area of mathematics and 21st in science. Additional international assessments such as the Trends in International Mathematics and Science Study (TIMSS) consistently show the United States lagging behind other countries. And in yet another related statistic, the non-profit organization, ACT, Inc., reported that last year, only 44% of high school graduates were ready for college-level math classes, and 36% ready for college-level science.

So, while the statistics may not reflect positively on the U.S. educational system as a whole, you’re probably wondering what connection these numbers have with the curriculum and instruction taking place at Posnack. The answer to that question can be found in a simple acronym …. STEM. The emphasis on STEM (Science, Technology, Engineering, Mathematics) is evident across all three divisions of our school. With dedicated science rooms and hands-on STEM labs, our students are exposed to these crucial subject areas in ways that extend across the curriculum and are relevant to the world in which they live. Teachers understand that in order to engage students in the learning process, they must provide instruction which stimulates the students’ interests, builds upon their prior knowledge, and promotes an inquisitive learning environment.

This initiative cannot wait until students have entered the upper grades. Before a student can demonstrate grade-level proficiency in the STEM subject areas, he/she must have already achieved a minimum level of proficiency in the skill upon which the scaffolding begins. This approach is evident in our Singapore Math program where, for example, the study of measurement begins in Kindergarten (length, weight, capacity) and continues through fifth grade with the conversion of measures and the volume of rectangular prisms.

Another example highlighting the importance of focusing on STEM subjects at an early  age was evident during a recent first grade science lesson. Students participated in an activity focusing on nutrition in which they identified the hidden sugar found in many foods and drinks. They were introduced to food labels and read the ingredients labels to locate the number of sugar grams. They then learned to identify the various “hidden” names  for sugar such as sucrose and fructose. An introduction to the metric system and the relative weight of a gram was explored. Different sizes and shapes of food and drink containers were used as the students estimated capacity and volume, and then compared their estimates with the actual measurement. The data was collected, graphed and the results analyzed. This simple elementary school lesson developed into a cross-curricular exercise integrating math and science. Similar lessons are found across the campus as teachers pave the way for young students to make the connection from classroom lessons to real-world application.

photo 3It may be hard to believe that early exposure to these skills is going to encourage students to pursue a career in a STEM related field. However with almost 6 million STEM-related job openings just last year alone, now IS the time to give our students the foundational knowledge and analytical skills they need for future success. And just maybe….with a foundation like this, Posnack alumni may one day be responsible for boosting the U.S. rankings to the very top.

My half is bigger than your half!!


How many times have you divided a sandwich, cookie, or cupcake perfect-cookie-in-halfbetween two children only to hear them utter those dreaded words? If one cookie is divided into “halves,” can one half really be bigger than the other?

Welcome to what I like to call Misconceptions About Fractions 101. Even before your child enters school, they have already been exposed to many scenarios involving fractions, however, their understanding of fractions is usually incomplete. To complicate matters even further, students have already learned that multiplying whole numbers results in a larger whole number, but now we teach them that multiplying two fractions results in a smaller fraction! Is it any wonder that around the time when computations involving fractions occur in third grade, many children begin experiencing difficulty in math?

So aside from measuring ingredients for recipes, slicing pizza, and cutting sandwiches into equal parts, why do we really need to learn about fractions?

Recent research suggests that understanding the concept of fractions is the basis for all higher-level mathematics; from algebra to physics to chemistry. Understanding fractions is not about memorizing complex rules or teaching students to “multiply by the reciprocal” when dividing. Understanding fractions is about learning how numbers relate to one another and being able to identify fractional parts on a number line.

Since a basic tenet of classroom instruction is to build upon a child’s prior knowledge, often times a teacher may first have to clarify misconceptions and ideas a student has about fractions before proceeding with formal instruction. Students need to have a conceptual understanding of what numerators and denominators mean beyond the traditional definition of the numerator being the number on the “top” and the denominator the number on the “bottom.” Teachers at Posnack use fraction bars that correspond to Singapore Math bar models to help students understand equal parts of a whole number, and to identify the relationship between the numerator and denominator. By using fraction bars or bar models as opposed to the traditional “pizza” model, students are better able to visualize the problem and make the connection between the concrete and the abstract.

Singapore Bar Models are used for more than just solving fraction problems. Beginning in second grade, students are learning how to solve a variety of word problems using the model drawing strategy. Parent workshops will be held during the month of October to share with you the ways in which you can help your child use model drawing strategies to solve problems by developing algebraic thinking skills.

In the meantime, can you solve the following fraction problems? Send me your answer as a “comment” along with how YOU solved this problem. I look forward to seeing many of you at the upcoming workshops – the dates and times will be announced later this week.

EX #1 (Grades 3-5): Terra’s monthly allowance is $48. She puts ½ of her allowance into savings and gives ¾ of the remaining money to a local charity. How much money does Terra have left for herself each month?

EX #2 (Grades 6-8): When Erin and Amy went shopping, they started with a total of $91. Amy spent $25 and Erin spent 3/5 of her money. At that point, Amy realized her remaining money was 3 times Erin’s remaining money. How much money did Amy have when she started shopping?




If the answer is correct, why do I need to show my work?




Remember when it was enough to just come up with the right answer? While your teacher may have admonished you to “show your work,” credit was given simply based upon whether or not you got the correct answer to a given problem.  An algorithm was taught in Math class, you used that algorithm to work through several problems, and it didn’t matter if you really understood what you were doing as long as you memorized the necessary steps.

Fast-forward now to “Mathematics the Posnack Way” where the standards for demonstrating proficiency in Mathematics have changed. Research has shown that as a predictor of future mathematics success, a student’s understanding of a mathematical task is equally as important as his ability to demonstrate computational skills. But how do you know when your child “understands” a concept or idea? If your child can recite his multiplication tables without hesitation, isn’t he demonstrating proficiency in a valuable skill?

The answer is a resounding “YesANDNo!” Yes – your child has memorized an algorithm for finding the product of two numbers. This IS a necessary skill and will greatly benefit your child as he moves forward to higher levels of math. But does this ability to recite numbers based on rote memorization demonstrate a strong number sense? No – it does not. A strong number sense is not developed from memorizing facts or algorithms, but rather when children use learned strategies and an understanding of numerical relationships to solve problems.

At Posnack, our focus is on a problem-centered Mathematics curriculum. Students are encouraged to use manipulatives, drawings, games, and graphic representations to solve a variety of word problems. By discovering not just the “how” behind problem solving, but the “why” as well, your child will not only become a mathematical thinker, but will develop a life-long love and appreciation for Mathematics.

So what will your answer be the next time your child asks the question “Why do I need to show my work?” Before answering, consider this quote from Paul Lockhart, a mathematician, teacher, and author:

It is the story that matters, not just the ending.




I want to be a Doctor, a Ballerina, and a Dolphin Trainer…


mathMagicVisit any Kindergarten class and ask the children what they want to be when they grow up. Chances are great that many of them will choose one of the above professions. Chances are equally as great that none of them will say they want to be a Mathemagician.

A Mathemagician? If you had asked this question of a young Arthur Benjamin, a mathemagician is probably what he would have answered. Several years later and true to his word, Arthur Benjamin has become a well-known and highly respected Mathemagician. Dr. Benjamin is also a Professor of Mathematics at Harvey Mudd College in California, and an author and frequent talk show guest. But the “magical” part about Dr. Benjamin is that he possesses the unique and amazing talent of being able to mentally calculate three digit squares in less time than it takes using a calculator. He can also tell you the day of the week on which you were born when given just the date. He has even memorized the decimals in Pi to 100 digits using a mnemonic device based on a phonetic code. In short, his mathematical skills are so fantastic that it is difficult to imagine that they are anything less than magic. But what do these amazing feats have in common? They are all accomplished through the use of mental math.

So by now you are (hopefully) intrigued and possibly wondering how this anecdote relates to Posnack students. This year, Posnack students are focusing on strengthening their mental math skills. More than just rote memorization, mental math requires students to develop a greater understanding of math concepts so that they may think critically and approach problem solving in a logical manner. Mental math also provides a foundation by which students can more easily progress to higher-level mathematics and strengthen their ability to gauge the reasonableness of their solutions.

As the school year progresses, you will be seeing how our students are developing their mental math skills by demonstrating a strong understanding of number sense. From the Kindergarten and First Grade students using their Soroban Abacus, to the Third Grade students multiplying multi-digit numbers from left to right, to the Upper School students using bar models to break down the intricacies of word problems – these are all ways in which the Posnack teachers are building upon the relationship between critical thinking and mental math.

Are you still wondering if there is really such a thing as a Mathemagician? Watch Dr. Benjamin’s TED talk at http://www.ted.com/talks/arthur_benjamin_does_mathemagic.html . He will turn you into a believer before you can say Abracadabra!




Batman, Superman, Wonderwoman, and Edward Frenkel





Are you wondering if this is a puzzle where you have to figure out which one doesn’t belong?

Well, it’s not a puzzle and all of the names do have something in common…. They’re all Superheroes!  Never heard of Edward Frenkel? Until recently, neither did I – but nonetheless, Edward Frenkel is now MY personal superhero.

Mr. Frenkel is an inspiring mathematician; one who recognizes that the way math is frequently taught in the classroom is creating a generation of children who think they are really “bad” at math. A tenured professor at the University of California at Berkley where his work includes research in algebraic geometry and mathematical physics, Mr. Frenkel was recently profiled in an article in The Wall Street Journal. Referring to what he sees as a deficiency in elementary math instruction, he is quoted as saying,

“It’s like teaching an art class where they only tell you how to paint a fence but they never show you Picasso.” He goes on to say “People say, ‘I’m bad at math,’ but what they’re really saying is ‘I was bad at painting the fence.’ ”

So, as educators and parents, how can we teach our children that there is more to mathematics than simply “painting the fence” or rote memorization of multiplication facts? Posnack teachers are following a pedagogical approach commonly referred to as the Socratic Method of teaching. Utilizing this approach, teachers continually ask questions of the students, facilitate discussion, and eventually reach a class consensus as to the correct answers. Students are also encouraged to produce and explain their own methods of thinking and problem solving in order to show that there is more than one way to reach the “right” answer.  This process of discovery not only results in a classroom where students are more engaged, but it requires the development of higher-order thinking skills. Both of these traits are vital to a student’s success in all academic subject areas, not only in mathematics.

Last May, in a well-publicized conference call, a senior IRS official announced that she was “not good in math” when unsure as to the answer for “one fourth of 300.” Political correctness aside, this statement reinforces the need for a change in our attitudes regarding math.

At Posnack, we are focused on being the catalyst for this change with new instructional methods being introduced into all classrooms. Obviously this change will not happen overnight, but if we can eliminate the “I’m not good at math” from our vocabulary, your child may one day join the ranks of Superman and Frenkel and be regarded as YOUR Math Superhero!

To see the Wall Street Journal article, please click here.

A Sensational Second Day…



Kindergarten and First Grade teachers have introduced Abacus Math to the children. The students were intrigued by the teachers’ abacus and excited to get their hands on one of their own. It was an amazing sight as they discovered the difference between the “heavenly” beads and the “earthly” beads and began adding and subtracting using the abacus as their calculator. Mrs. Liroff took the children on an imaginary journey outside so they could “visualize” the terminology by looking upward and down below. Visualization plays such an important role in mathematical thinking, and is emphasized beginning in Kindergarten as a precursor to mental math skills.

Teachers will begin sending the abacus home this week for the children to “practice” and feel comfortable manipulating the beads.


Grades 2 through 8 are all beginning to incorporate Writing in Math as an important component of their Math classes. Research has shown that in order for students to effectively use mathematics in solving everyday problems, they must be able to explain the procedures they use and reflect on their thinking. Communicating through writing in a Math journal also affords the student the opportunity to maintain an ongoing record of what they are learning and keeps them actively involved in thinking about mathematics.

Today in 2nd Grade, Mrs. Spier gave each student a “target” number and let him or her explore the many different ways in which they could reach their target. They recorded this activity in their Math journal so they can refer back to it as well as to keep a running record of their progress over time. The creativity and mathematical thinking in class today was awesome! Thank you Mrs. Spier, Mrs. Liroff, and all of the “K through 8” teachers for starting off the school year on such a mathematical high note!