This is a professional development blog primarily for teachers in Spring Branch ISD.
The overall A-HA moments for me in chapters six through eleven were the speed at which Zacarro covers THE WEALTH of concepts, for which so so so many skills are needed, and the lack of pertinent information included, but needed, to solve many of the problems. The following are examples: On pg. 87 (#9) the children are required to know how to add mixed numbers, but no explanation is given. On pgs. 84-86, Zacarro doesn’t explain that when adding/subtracting fractions, one must only find the sum/difference of the numerators- not the denominators. Though he does include examples to derive this, he does not explicitly state this important fact. On pgs. 88-89 he explained one method to subtract fractions, however it might have been easier to just convert the mixed numbers into improper fractions right off the bat. On pg. 90 he writes how to convert improper fractions into mixed numbers, but he doesn’t supply a visual aid to help explain that the remainder of the division problem becomes the fraction’s numerator and that the divisor becomes the fraction’s denominator. Zacarro should have supplied a mnemonic here. For example, tell the students that “bottom goes to top and left goes to right.” In chapters 7-9, Zacarro should have shown the actual mathematical formulas for finding perimeters/circumference, areas, and volume, instead of just writing them in word form. And, after he briefly explains the skill(s), he requires the children to use algebra in random problems, like on pgs. 107 (#4- you have to know how to manipulate the equation) and 110 (#4). On pg. 110 (# 5, which also comes up again on page 111 in #8), the kids should have been shown a strategy to make separate polygons out of the overall, original one. On pg. 109 (#3), you must use the radical symbol on a calculator to solve, because 20 isn’t a perfect square, unless you know the algorithm! On pg. 111 (#7), the kids should be reminded that d=2r. While I still feel this book is an exceptional tool, realizing these “A-HAs” reinforces my choice to use it as a supplemental resource.
My Ah-Ha moment came when I saw the introductory information given at the beginning of each chapter. For example, on page 128, there is a wonderful and entertaining explanation about Archimedes and his water displacement theory. Kids find this type of story-telling very interesting. I also think that reminding students to look at what the "Gumby-like" characters say to help them with understanding problems. Also, usually the boxes with the Einstein drawing will give important rules or information to solve problems.
I noticed that Zaccaro introduces a concept such as area in Chapter 8 and without much repetition or directions he moves onto area of complex figures in Level 1 questions (pg120). So my A-ha moment here was, Zaccaro addressed GT students ability of understanding principles at a fast pace and forming generalizations, and using them in new situations. In Chapter 6, he used almost no modeling (except the pizza division problem on page 95), gave the rules for fractional operations but provided more examples before going onto Level 1 problems. My a-ha moment after that chapter was, Zaccaro addressed GT students capability of understanding and handling abstract information.
Since fractions tend to be a really difficult concept for my students, including the G/T kids at times, I really like Chapter 6. While it is challenging, i was able to take some ideas away from it that I think will help all of my students. I like how Zaccaro refers to the idea of the scale factor (a term I use to talk about how much both the numerator and denominator are scaled up or down for an equivalent fraction) as "fair" (p. 86). I think my students would really buy into that. Something else I find interesting is that he is still using the term "reducing" (p. 91) where most of the more recent books refer to that task now as simplifying...to reduce is to make something smaller, but the value is actually the same so that's a misnomer. Another explanation Zaccaro uses that I really like is the idea that the fraction bar means you are comparing (p. 84 and again in Chapter 10, p. 145, in his discussion of percent.) Seeing as this book is entitled Challenge Math, I think it is very appropriate, as mentioned by others here, that Zaccaro doesn't expound on every last little detail of every topic, leaving some analysis and higher-level thinking for the student.
In response to SadloK posted on 6/16, I think that it is Zaccaro's intent to be somewhat vague in some of his explanations, allowing for the gifted learner to come to some conclusions of his/her own. For example, you mentioned that he never specifically says that when adding or subtracting fractions, only the numerators are added or subtracted. However, if the learner analyzes the example, he /she will (hopefully) figure that out. PKassir, I really like the little dialogue boxes with the "Gumby" characters as you referred to them. I agree with you that these are important points for the reader to help further thinking.
In response to SusanM on June 18th, I also have some concerns about math vocabulary used in the fraction chapter. As you mentioned reducing the fraction is one of them. Another term I am not very fond of is "flip the fraction over" (page 96, dividing fractions). I agree that it helps students to remember but it is very important to use the correct vocabulary such as reciprocal, simplify or least common multiple from the very beginning.
My A-ha moment in chapters 6-11 is the very simplistic way he presents very complex problem solving. Most specifically, Zaccaro explains Pythagorean Theorem using steps(1,2,3)on page 109 which gives the student a process to follow while solving the problem. This is excellent, it makes a potentially itimidating situation much more approachable. It reminds me of a student I once tutored. He would often complain that his teacher didn't say it the way I did and that is why he didn't understand it. The saying--"It's all about the presentation", doesn't just pertain to food!!!
In response to PKassir on June 17th, I agree that the introductory stories are an excellent way to get the student interested in the concept. I especially liked the story about Eratosthenes in Chapter 7, it really emphasizes how amazing math and mathematicians are.
Reply to SDawson from PKassir: I agree with your last sentence: "My a-ha moment after that chapter was, Zaccaro addressed GT students capability of understanding and handling abstract information." I think this statement explains why Zaccaro leaves out basic steps in the solving of problems. He truly lets the GT student figure it out in a manner that makes sense to them. While there is some guidance given, there is a lot of room left for the GT mind to analyze and formulate a manner for solving a problem.
In response to SusanM, I agree that it is good that Zaccaro does not expound on every detail. It can sometimes confuse students needlessly when they learn something new. Just for fun I had my son (5th grader) read the section on dividing fractions in chapther six, he thought Zaccaro's explanation was much easier to understand.
My "a-ha" moments were very similar to what others have posted- that SOME of the basics are given, but then the gt students must rely on their other skills to solve complex problems. Yes, this is what we want our students to do because in the real world,situations are not always as basic as "let me just turn the decimal into a percent." We need that basic knowledge and then must apply or modify it to the specific situation at hand. I also had a more simple "a-ha" when I read the volume anectdote on page 132. I liked reading about another way to determine volume based on displacement of water. I think the students will find this process more fun since they have to collect the spilled water to determine the volume of the object. Kids seem to enjoy things spilling:) They can also compare their results using 2 methods such as finding the difference of the liquid levels when using a graduated cylinder or other measuring contatiner with collecting the spilled water and then measuring that.
nlopez in response to ksadlo's comment on 6/16- I agree with you. As I was reading through these chapters, I kept making notes "but why?- because you are multiplying by 100." It was somewhat bothering me that certain ideas were not really being explained, but were just given an algorithm to be used. At the same time I had to remember that if the focus is gt, we're actually fast forwarding a bit because they either understand the basics already or will grasp them quickly enough to move on to more challenging applications.
My A-HA moment was that even though Zaccaro covers lots of concepts in a few short pages he builds sequential concepts in order. For example in fractions pages 84 to 104, he starts with the basic vocabulary and meaning of fractions then moves on to adding & subtracting. Then eventually moves on to multiplying and dividing. Even though multiplying and dividing fractions is not a K-5 TEK it allows for differentiation and if a child has the concepts taught at an applied level the child may be able to conceptualized the process through problem solving. The other A-HA moment I had was when Zaccaro used “borrowing” on page 89….when will everyone be on the same page ?????….it’s called “regrouping!!” Our children learn from us and in order for them to understand and make use out of math vocabulary we must you the correct words.
An “A-HA” moment for me was the use of the term cross reducing (page 94). I was aware of the term cancellation, removing common factors from the numerator of one fraction and the denominator of the other fraction before multiplying them, but I had never heard the term cross reducing before.(eg. ⅔ • ¼ => ⅓ • ½ = 1/6),
In response to SashaLuther: I hate to admit it, but the term BORROWING is just a really, really (oops, I'm showing my age) old fashioned term which means 'regroup'.
In response to Sadlok on June 16th. I agree with you about how there Zaccaro covers a wealth of concepts and there may be a lack of pertinent information that a student needs to be successful in a concept. However, a “true” GT student who really understands a concept will be able to move and develop hypothesizes and opinions and knowledge with very little instruction and maybe just feedback so step by step information and teaching may not apply to that student. Just a thought to think about as we use this book and who we should be giving these types of questions to.
My A-Ha Moment was on pg 94 when they were talking about cross reducing. I have had children in the past be able to do this complex process in their head with a given set of fractions but not be able to explain it very well. I think by students having the freedom to explore different ways of thinking about how this process is done- it will help build up their math skills. Fractions and reducing them has always been tricky to some- I think by showing them this explanation- it will help many kids.- Sharon G.
I agree with SDawson's comments related to what PKassir said about the basic steps missing. I think the fun and interest will come when everything is not given to them and they have to "think" of different ways the problem is solved. It is through this thinking and discussion that kids learn from each other. I encourage them to find as many different ways possible to solve problems.
In response to nlopez's respond to ksadlo's comment on June 21st, I agree that these are GT kids and they will grasp the logic behind the algorithm on their own even if the "why?"s are not given. I think this is part of the "intellectual frustration" mentioned in the first chapter. If every step of the algorithm was explained there would be no intellectual frustration and no fun left for GT students.
I had several AHA moments at the literal connections the author makes.On p.84 the connection of the fraction (fracture) to a body fracture is one I have never made but it does make sense and one that students would understand. Explaining that it from the Latin "to break" also helps clarifythe concept as well.My second was the story about volume on p.128 with the story about Archimedes proving whether the king's crown was solid gold or not also was a clear connection to the displacement concept. That story gives the students a point of reference and I think would help the them remember the concept when using it in math or science at later points.The third AHA was the clarification in the Einstein box about the "amount" of increase or decrease being the top number on p. 146 is an important one mathematically.All of these gave me a mental pause and clear connections as I was reading this selection.
In response to P Kassir's AHA moment comments, I agree that the water displacement story is entertaining and will click with the students interest. I like the idea of the cartoon characters being "Gumby like" although not all of my students will know who Gumby is-this had surprised me in the past. But it's a nice comparison.And I agree that the clarity of the Einstein drawings lets the students realize that is important information as I carefully read them for that purpose myself.
I was struck in these sections by how often the author provides a formula for the student to use with very little explanation as to where the formula comes from. For example, on page 118, the author gives the formula for the area of a triangle without any explanation of why division by two is necessary. On page 130, there is no reason given for why this formula for the volume of a cylinder works. I've found that my GT students very rarely like having a formula thrust at them with no explanation. They like the AHA moments that come when the understand where the formula comes from. I don't see a lot of hope for long term retention of material when formulas are just given to students who are capable of much more understanding of their origins.
In response to SadloK,I agree that this book does omit a lot of pertinent information that students will need to know. It's great to encourage the students to seek multiple ways to solve the problems, but I'm often surprised by how little information is included in the instruction. Most of my GT students need really solid basic instruction before they can run with a topic on their own. I agree that I would use this merely for supplemental material and problems.
There were a few "ahas" for me: I liked the way Zaccaro introduced fractions and think that is a good way to introduce them to my students as well: "The word fraction comes from a Latin word that means to break. If you fracture your leg, it means you broke your leg..." he continues on to say that fractions are needed to measure these fractured parts. I like this, as it connects something abstract to something tangible.I also really enjoyed the section (Chap 11) on ratios and proportions. The story of the dad cutting down the tree and the daughter asking him to wait for 5 minutes so she can double check was another real-world example that the students can relate to, before you get into the more complex percentages and formulas.
I agree with Sluther about Zaccaro building on concepts. It was much easier to read the beginning parts and now I am having to re-read some of the harder parts with harder formulas! :)
The “A-HA” moment that I had was the explanation of finding the volume of an odd-shaped object on page 132. This would be a great lesson to demonstrate to the students especially since we are so used to teaching volume using only rectangular objects. Also , I have never used the concept of “cross reducing” on page 94 before, but I think the GT kids will understand and want to use this method.
In response to PKassir, I agree that the stories Zaccaro includes at the beginning of each topic are wonderful and entertaining and that the kids find this type of story-telling very interesting. They offer us teachers great ideas for excellent class discussions both leading into and wrapping up the lessons. Which leads me into my next comment...In response to Katie Kavanagh, I also enjoyed Zaccaro's story about the girl's dad cutting down the tree, to get into ratios and proportions. This story could be a set into shadows and an "A-HA" for the children, for them to realize that we, too, can do the same thing with shadows in the field.
I agree with SDawson on June 17. The GT student usually understands the math concept after explaining/modeling the first time and needs very little repetition. They can learn the basic information quickly and move on to the more challenging problems.
In response to kohlerj on June 23rd...I agree that most gifted students want some background explanation of a formula before being turned loose with it. It's part of their internalizing the concept- understanding it fully. This just creates an opportunity for a small group mini-lesson on the explanation of the formula.
I really had an “A-Ha” reading the chapter on fractions. Students really have a hard time with fractions, so it is a topic that we spend a lot of time covering. The book really goes deep in easy ways to extend fractions for students who have mastered the grade level material and need extra challenging. I really liked the steps on pages 87-88 covering subtraction of fractions. What a great way to apply what you’ve learned in a different way. I really like how the word problems invoke a lot of thinking on the learner’s part. It’s not just a simple one step problem.
There were a couple of "ah-ha" moments for me. The first was on p.94, cross reducing. I guess I have never seen this since we don't teach that far onto multiplying fractions in 5th grade. I thought is was kind of fun and interesting that you could cross reduce with any numerator and denominator that had common factors. The other moment was finding the volume of objects other than rectangular shapes on p. 130-131. Again we don't teach anything other than rectangular shapes in 5th grade. Last was finding the percentage of a number on p.142. I have always solved these problems one way, now I have another way!
In response to ndeans...I agree with you about stretching the minds of GT kiddos with finding the volume of odd shaped objects. I think this would be a fun challenge or them. That cross reducing was really fun also again another way to step up the lesson for the gt kiddos
An "A-Ha" moment for me was in the Volume section of chapter 9 when it talked about finding the volume of an odd-shaped object like a crown or a rock. This would be a great lesson to demonstrate to the students when talking about displacement concept in Science and relating it with volume in Math. The story of page 128 about Archimedes and the crown of Heiro would help the students remember and relate the concepts better.
In response to ndeans comment on June 24, I agree that most GT students learn the basic information quickly and can move on to the more challenging problems, but I also agree with Kohlerj that most of my GT students need really solid basic instruction before they can run with a topic on their own, and that is why this book Challenge Math is an excellent supplemental book for my GT small group after I have model the lesson and gone thru Ck fr undrs, and GPrct.
Just like Sadlok I am surprised at the speed that fractions are covered and wonder if other materials are needed even with advanced students while contemplating this chapter I received this web sitehttp://www.conceptuamath.com/fractions.html#TopicsIt would help the students visualize fractions if they were having trouble. I also like the way Zaccoro helps the students make sense of the hard number it is suggested that the student substitute and easier number they work it with the difficult one. This is done on page 101 problem 1.