Tuesday, July 20, 2010

Session 3 - Question 1

How does Edward Zaccaro create rich and in-depth avenues for gifted students to learn and experience mathematics? Site page numbers to support your response.


  1. Zaccaro creates rich and in-depth avenues for gifted students by
    • Creatively introducing a concept and then leading students through a series of increasingly harder exercises, in the form of the Level 1-3 and Einstein Level problems, to apply and practice what they’ve learned by solving real-life and life lesson problems
     Introducing a calculator, p. 66, as a exploration tool for the more complex problems that students may not have the computational ability to solve
     Creating problems where students reverse steps in the mental process such as the changing machines on p. 195 that involve the opposite operation
     Using algebraic symbols such as in Chapter 20, p. 200, demonstrating the gifted student’s ability to use numbers and symbols to communicate
     Playing up a gifted quality such as a strong sense of fairness on p. 211 to help students identify with being fair to both sides of an equation
     Making math meaningful, challenging, mysterious, humorous, engaging, and just plain fun.

  2. Zaccaro appeals to the gifted students:

    1. He picks up on their offbeat humor and sense of whimsy. If you tell a kid he's going to learn "algebra" you'll most likely get the look of death, if you tell him he's learning a secret language (Chap.20 ) he's all yours.

    2. He uses "differentiation" - it's easy for a student to find out what they already know and then they can move on to the level 2 & 3 problems.

    3. The cartoons are cute and fun but not "dorky".

    4. He doesn't "talk down" to the students. At my school we seem to own multiple games that claim to make "fractions fun". Fractions are not "fun" and no matter how many dorky games one plays, they still aren't "fun". Chapter 10 "How much will I need?" is fun!

  3. Zaccaro creates rich avenues for gifted students to learn and experience math by presenting problems in a manner that appeals to kids. The illustrations are fun without being condescending (like the fiddle playing ant on p.31, a circling centipede with a “you are here” sign on its tail end on p. 80, and a goggled dog flying ace on p. 165). His recurring characters of Einstein and the amoebas give helpful hints throughout the chapters to keep the kids on track. The text never really “instructs” the child about the math concept being addressed, but rather gives them an approach for solving the problem that doesn’t require prior knowledge of the mathematical computations involved. For example, in Chapter 9, Zaccaro introduces the concept of percentages by explaining how his Percent Machine cuts everything into 100 equal pieces, so to find 35%, you just take 35 of the pieces. In another example, on page 53, Einstein assures kids, “As you do the problems on the following pages, don’t worry if you haven’t studied fractions in school yet. Most of the problems can be solved by just thinking about them a little. ”Zaccaro introduces each chapter with a model problem and then Einstein and his buddies walk the student through different ways to think about solutions in a casual conversational way that would hold a kid’s attention. On page 66, Einstein advises, “This chapter is trying to help you learn how to use multiplication, so it is okay to use a calculator with some of the problems.” There are no “drill and kill” exercises that would bore a gifted child to tears. The problems are presented in engaging scenarios and word problems. Zaccaro provides four different levels of questions so that every child can find an appropriate challenge. Each subject is covered by a chapter with five questions per level – just enough for a gifted child to get their feet wet and move on to the next challenge. Overall, I think it’s a hit!

  4. Session 3-Question 1

    You ask, how does Zaccaro create such rich and in-depth avenues? It prompts me to ask about Zaccaro’s childhood math experiences. Did he have an elementary math teacher who was creative, practical, magical, and made math “sing”? It is obvious he loved math, but was Zaccaro exposed to uninspired math teachers and ended up being bored-to-tears? Did his childhood “bucket list” include writing an elementary math book that would be operatic? Each and every page supports this goal.

  5. Session 3-Question 1a

    Oops! I posted too soon. I feel we have answered this question in depth in various ways in the previous two sessions.
    Revisiting the chapter headings for an example, I feel they are all excellent hooks. For example: Chapter 2-page 9: Don’t Let It Break. What child has not been parley to this command from an adult? His little mind is probably working overtime to learn some “prevention” tools!
    Chapter 12-page 121 offers: Magic of Math-Making Smart Guesses. Our young student questions: Can math really be magical…and am I really given permission to guess? WOW!!!!!
    Even I fall for that one!

  6. kHarrell -
    First - the dedication page - to his students: his passion is shared! What more could make students want to learn, but feel the passion of their teacher!
    Secondly - from what I can tell, the book is not necessarily sequential - what a great thing for a GT student - you can start anywhere...the Table of contents is very detailed, so you can pick the topic you want to study - perhaps as we were able to pick, we should allow our students to select - can you be GT in different areas of math? And then - each topic is dealt with in a simple way explaining it step by step...but in a fun exciting way.
    thirdly, I think it reads more like a Novel than a non-fiction book- most inticing for a student to pick up...the print is big, there are graphics..and they are fun.
    As you read the book, you almost can hear him talking to his students. (didn't give page numbers)

    In response to of life....I wish I had written your statement..I agree with them all...I think he has almost made fun...at least as fun as I can take for math!

    In response to fMoore - Creatively introducing a concept and then leading students through a series of increasingly harder exercises, in the form of the Level 1-3 and Einstein Level problems, to apply and practice what they’ve learned by solving real-life and life lesson problems - I so agree...I can see my students stepping directly to the Einstien level as they all think they are Einstein!

  7. The greatest thing I’m taking away from this book is the way he looks at math as a whole. For me, the entire book is like relearning math. I really wish my teachers had had this book when I was in school! I never “got” the whole percent thing, but after reading and working through the problems in chapter 9, I get it! The explanations on pp. 82-89 make it make sense to me. The concept of negative numbers baffled me as well. Chapter 13 spells it out and makes it perfectly clear to me. I don’t know if my tired old brain was finally developmentally ready or what, but I got that, too! I must confess that I put up a mental roadblock when I read the word “algebra” on page 211. I struggled so with that in middle school and high school. But after reading pp. 211-216, it doesn’t seem like such an obstacle. The ways that Zaccaro presents the information in kid-friendly form without talking down to them is refreshing. He also provides adequate practice without it being skill and drill. It makes so much more sense to present the information the way that he does. I can’t wait to try it out on my kids in a few weeks

  8. In response to NanetteG - I hadn't thought of the way he was taught math! Maybe he had some of the same uninspired math lessons that I did! He has done a great job of putting some inspiring ones together for us!

  9. In response to Of Life, Education... - I agree with all four of your points. The humor part is important, but only because the kids will love it! I also think it's so important to talk to kids on their level, but not down to them. That is an instant conversation killer with them. And he does really make learning these skills fun. What a novel idea!

  10. In response to FMoore, you mention Zaccaro’s introduction of the calculator in Chapter 7 as a strategy for solving multiplication problems. He actually goes on to recommend the use of calculators on several additional occasions (e.g. page 88 for solving percentages, page 115 for division, page 150 for speed of light problem, page 157 for time machine division, page 236 for square roots, etc.). At first, I was surprised to see the calculator offered up as a problem-solving tool in the context of this creative problem solving book. But the more I think about it, the more I believe the calculator is an authentic, real-life tool and students should have the opportunity to see how it can be applied to every day problem solving. With calculator tools on our cell phones, iTouches, lap tops and the like, who in real life is going to sit down to solve long-hand the answer to 55 miles per hour divided by .621 kilometers in a mile to get 88.56682 kilometers per hour for question #3 on page 199? Obviously, students need to learn the basic concepts of multiplication, division and square roots, but let’s not torture them (or us!) by banning the use of calculators on the more complex mathematical equations.

  11. He creates rich and in-depth avenues for gifted students to learn and experience math in several ways. One is that he Does Not explain everything -- the leaves some things for students to figure out. For example, in chapter 21, the sample problem on p.214: Einstein states that 5n = 25 and "Nicky is 5 years old." Another way Zaccaro succeeds is by the kind of problems he offers. Students gifted in math love logic problems, especially where they can figure things out in their heads. He also offers multi-step problems (and not just 2 steps, but several at times!). Chapter 25, p.256, (I can answer that question) is full of multi-step and/or logic problems. He also offers advanced problems, the kind that are not usually offered to this age level. For example, Chapter 23, p.233, is all about exponents and square numbers. Gifted students like to be ahead of the pack.

  12. In response to FMoore, I hadn't thought about the fairness concept for gifted students (Zaccaro's explaination for solving algebraic problems). But I think you're right -- gifted students have a tremendous sense of fairness, and this will appeal to them.

  13. There are lots of ways the author uses rich and in-depth experiences to hotwire the students into having fun and being challenged, and consequently learning that math almost as a by-product. In this last section, algebra was being pushed heavily. It's great that is happening because algebra can be such a high anxiety subject for many students. Introducing the concepts early and in a fun manner reduces a lot of that potential stress. On page 201, the author compares learning algebra to learning a new language. This truly hits the nail on the head, because it truly is the most different math that most students encounter. It takes a lot of the scariness out of it.. I also liked the square root chapter, page 233. Again the author makes it a game, with a little visual: a cute machine that makes square roots and the reverse. So much more appealing than that dreary square root formula I learned in seventh grade. On page 257, chapter 25, boolean logic appears. As I recall, I did not encounter this until 7th or 8th grade. The author introduces it early, and in the usual entertaining way. Chapter 26: what kid has not asked why do I have to learn this? Most of us don't find the answer to that until we are adults and actually find the occasion to use some of that math we learned! Here, students see a connection immediately between math and the real world. Chapter 27 piques student interest with its title (it certainly intrigued me) and then proceeds to slip in percents, decimals and fractions.

  14. I think Zaccaro does a wonderful job of creating rich mathematical experiences for gifted students. I find that while gifted students are the most intelligent in a classroom, they'll also sometimes be the most reluctant. A huge part of the PGP program, to me, is allowing those reluctant students time to really jump in to a subject or an experience that is not the norm of the every day classroom. This book is a perfect example of taking a subject that can be mudane (yes, I do think teaching math in the classroom is a bit mundane sometimes) and blowing that ho-hum attitude out of the water with real life and full immersion examples. For example... p. 189 starts Chapter 19: Changing Machines. Instead of just giving students some items and a scale along with a conversion chart, the student working this chapter is allowed to think WAY outside the weight-related box and work problems on the metric system, conversion, length, and much more in a very different way. However, in this book, the material isn't just black-and-white problems... it's completely expanded and engaging.

  15. In response to Eleanor:
    You mention that he leaves some things to be discovered by the student... that's what drives me crazy (in a good way) about this book! I'm a pretty "black and white" person, and I like to know that I'm on the right path and moving in the right direction as I move through a process. Zaccaro allows for lots of "gray area" within these math concepts, which is just brilliant. He's taken a very "black and white" subject and turned it into "gray". Great idea and perfect for the gifted learner!

  16. I agree with Head Squirrel - you don't have to do the book in sequence, so student who is fascinated by percent or algebra can just work on that chapter. I also much appreciate that he allows calculators - because that's what people do in the real world!

  17. Edward Zaccaro creates avenues for depth and complexity in his book, Primary Grade Challenge Math. For depth he adds the Einstein section to make the child think on a deeper level. Example is on pg. 185 when the question ask how much the magic frong weighs on a given day. The next question as if the frog will ever weigh zero pounds. This has added another layer of thinking for the child to go deeper. He adds complexity by causing the child to think across disciplines. Weather, thunder question pg. 152, Zoology, pg. 158, 159, and astonomy, pg. 250. He adds complexity by adding logic questions, pg. 259 "Which drawing best shows that all cats are animals, but not all animals are cats?" This is early SAT preparation for goodness sakes!

  18. I agree with KHarrel with the format of the book not necessarily being sequential. GT students need choice and this book is set up for flexibiity and choice. A GT student can follow interest and make choices on what area they are interested in and want to be challenged in. I can see even a menu being created for activities in the book utilizing the strategy of choice for the GT child.

  19. I agree with Millie and others on the child friendliness of the illustrations in the book. Even though they are GT and will think on a much higher level, they are still kids and respond to fun formats like all kids their age. He knew this in creating the book for this appeal factor.

  20. While noted in the session 1 comments, the simple reference to "Einstein" as a question level is an instant GT connection.

    The language used through the book also enriches the math experience. His references to different types of "machines" (pgs. 82, 191) riddles (p. 221), common feelings (pgs. 104, 222), mistakes (p. 211) are just a few examples of hooks that will draw the PGP student to the material.

    Given freedom and some allowance to experience challenge without frustration, his problems will empower many of the PGP students. All? I think it is a little too soon to tell. But I think the students will identify with his presentation.

    Of Life, Education...I liked the point you brought up about the book not "talking down" to our students. I think Zaccaro presents the ideas in such a way that it's okay to say "this is hard" or "this involves work." Sometimes math (or anything) is fun, but often it involves making mistakes and trying again.

    FMoore, you summed it up with your first statement about "apply[ing] and practic[ing] what they’ve learned by solving real-life and life lesson problems."

  21. The author, Ed, has been in the same situation as each and every one of us. I don't have the materials I need to work with my students. We TRUELY need to KNOW our students strengths and interest to catch, hold and develop their skills and "passions" that are aching to come out. We all agree that the different LEVELS help differentiate teaching the objective to a wide range of math skills. But at each level and pretty much EACH problem, the drawings are excellent to assist the student. I can see this (drawings) helping second language learners. The students are encouraged to use a variety of resources to solve problems. Multiplication charts, calculators, PAPER and PENCILS. These can be the kids "machine". Might as well start adding the iTouch, iPhone, cell phone, etc to the MACHINE list. Ed does well is introducing a problem and then RE-LOOKING at a problem a different way. Page 97.
    Do we all not do that every day and sometimes turn to another student to try and explain it to the student? The student sees that there are many ways to look at a problem. The GT student is given a wide variety of problems and levels and each level is not seen as Drill and Kill. There might be 20 questions, BUT 5 per level. It is like the electronic games. The mission is to get to the next level.

  22. I definitely agree with the term LANGUAGE. Many of the GT students are curious with other languages and will be the first to ask someone how to say something in their language. Introducing ALGEBRA as a language and teaching them how to TRANSLATE (turn it into p.200). I never saw Algebra as a language and always just wanted to cry when I saw those x’s and y’s. I still do not have that love yet, but I sure do wish little Einstein was on my shoulder in Mr. Carpentar’s math class.