This is a professional development blog primarily for teachers in Spring Branch ISD.
I think that kids that are gifted in mathematics appreciate this approach because it is different from how they typically study math. The interesting introduction moves them into the concept explanation and practice. There is not a lot of practice, I imagine, because the gifted only need 1-2 repetitions of new material to remember it. Furthermore, the varying levels of difficulty are also attractive to a student who would enjoy being challenged in mathematics. The degree of difficulty increases with each level, and that is also appealing to the GT student. This book also allows students to solve problems anyway they want. They are not forced to show their work using a specific problem solving model. A child who is gifted in mathematics is able to solve problems in ways that are inherent to them, and that make sense to them. Sometimes, they just get it. This book is also appealing to the GT identified student because it can serve as enrichment in math. Many of the topics and math concepts presented in the book are relevant to everyday life, and are not topics that students would necessarily see on their everyday math. For example, the chapter on Statistics (chapter 14) plants the seed for future manipulation of statistical information.
I think GT students would love the way Zaccaro introduced mathematical concepts using physics, astronomy and finance. The historical facts about Eratosthenes, Archimedes, and Kepler would definitely provoke interest for deeper understanding of these famous mathematicians’ studies. Other scenarios used to introduce concepts such as the credit card interest calculation, or tree and its shadow are also very effective in making those real life connections.
I think Zaccaro's approach in this book would appeal to the gifted student because:1. the many topics included in the book, such as astronomy and algebra, provide variety aside from what is being discussed in an average math classroom and are topics that students tend to find interesting2. Zaccaro gives the child just enough information and leaves some element of exploration and analysis up to the student3. the book provides for differentiation with the different levels of questioning4. it is challenging, appealing to most gifted students because it provides them with an opportunity to learn and grow on their level
In response to PKassir posted on 6/17, I agree with you wholeheartedly that we need to respect the various solution methods our gifted students (and all students really) use to tackle problems, especially those that are less routine and present more of a challenge to the student. It's funny that you mention the problem solving model, because I find that my gifted students can and often do find it difficult to expand on their process and share their thinking in detail, despite the fact that they normally do have the correct answer. But, alas, it is one of those things the district says we must do! :-)
Response to SusanM from PKassir:I have also heard quite a few GT kids (and their parents) express concerns about the problem solving model that is used in our district. I am curious--how do you handle the students' hesitation in expounding on their mathematical thinking? Do you know if the students will continue using the problem solving model, or another similar organizer in middle school? Thanks for the information.
I feel all children would appreciate Zaccaro's methods of teaching math. G.T. students would especially like it becuase it is independent learning and it offers challenges through the varied levels of difficulty.
In response to SDawson--I agree with your observation about real life applications. Zaccaro consistently provides examples and situations of how math can be used and applied in everyday life. We do not always do a good job of relating math back to everyday life and that is when the concept is actually internalized.
In repsponse to Pkassir, your comment, "There is not a lot of practice, I imagine, because the gifted only need 1-2 repetitions of new material to remember it." You are probably right that a lot of practice may not be needed, however if it is needed you can always supplement the material. I always wonder why my sons teacher would send home 40 math problems when it was obvious that he undersood them after the first TEN. It usually frustrates him and makes him dislike math as he has to continually do the same thing over and over again. On the other hand, some of the other children may need all that repetition.
I think students would like Zaccaro's methods for a few reasons. He starts off each chapter with an engaging story of some sort that let's the student know what they will be focusing on. His comic characters add some humor to the learning, but also bring up some relevant concerns or common mistakes at times. There are opportunities for practice, application, and challenge allowing students to move at their own pace and to select the level of difficulty they feel comfortable with.
nlopez in response to PKassir's response to SusanM on 6/20- I feel the district's problem solving model is a good graphic organizer for helping students "think about their thinking." However, since they should have had experience with this model since first grade, by grade 4 it should be internalized and they should not be so restricted by the boxes. As a 4th grade teacher, I follow the model, but don't use the graphic organizer. It's posted in my room and students have a copy of it in their math folders, but I prefer not to use the boxes for assignements. No one thinks in the same patterns so students should not be told what order they need to think in and that they MUST think in that order. I prefer to give my students a question (or let them choose one)and then the remainder of the page is blank so they can work however they want- even if my eyes have to wander all over the page to find everything I'm looking for. They also know that they can add more pages as well. The communication portion is very important in my classroom. Students need to be able to communicate their thinking in writing. You can not always just look at their math work and tell if they actually understand what they are doing. The same goes if they don't even show any math work. If I just see a 12 on a page and nothing else, I have to wonder: did they understand, how did they get this answer, was it a lucky guess, did they copy? I feel that if students can't write about their thinking, they don't necessarily have a full understanding. If it's the fact they can't be "bothered" with explaining themselves, that's too bad for them. My strong feeling about this may also come from me being a 4th grade teacher where writing is stressed so much due to the writing TAKS, but I find writing in math to be a very important skill and tool.
The author provides ample tips, formulas, and reminders (e.g. Einstein ‘boxes’ on pages 90, 92, 108, 130, 144, and 146); ‘cookie men’ clarify concepts by exchanging animated remarks (e.g. pages 86, 87, 107, 119, and 131); and there are numerous worked examples for students to study (e.g. pages 86, 109, 118, 130, and 139). These things and plenty of engaging real world practice problems are appealing strategies.
In response to SusanM: I agree that the elements of exploration and analysis fuel the imagination and a create a desire to learn more about a topic.
I think the GT student would like the approach of Zaccaro because he connects it to other interests and factual information for students such as astronomy. His methods for asking questions isn’t just one way to get the solution, but it allows the GT student to explore and solve the problem in multiple ways. Also, he allows children to start at different differentiated levels so that all GT students can feel success and explore at their own rate.
In response to SusanM on June 18th. I agree with you on all your reasons why a GT student would benefit from this approach. I completely agree and support #2- Zaccaro gives the child just enough information and leaves some element of exploration and analysis up to the student. I feel he also allows the student not to just answer the question but to question the question.
I think that GT students would enjoy the challenge of the way Zaccaro uses different strategies/methods to get children to think about the problem. The openness of ways to solve it would be challenging and intriguing to gt students. I think they would enjoy the challenge and freedom to come up with different ways of solving problems and learning how other students think and reason.- Sharon G.
I agree with nlopez's response that gt kids would enjoy the comical character and the way that the problems are presented. I think it is scaffolded in its delivery and many of my students would love the challenge with problems are entitled "Einstein Level"- they will soon find out!!!
In response the ReneeR on June 21st, I agree that GT students love challenges through the varied levels of difficulty. Author also provided "Super Einstein" problems which worth three Einsteins on page 127 and 151. I can see GT students working on those first once they reach the Einstein level.
I think the author's approach is refreshing. Students could work on this independently or with a partner by the topic or use it as a resource when looking for particular type of problem almost like looking for a recipe in a cookbook. It is very concise and clear and with the cartoons imbedded it makes it very student friendly. I am anxious to try it with some of my talented math students to see what their responses are. It should be interesting and I suspect very positive.
In response to nlopez's comment I agree that the students would like the author's methods for the very same reasons. The format and organization with the story at the beginning, the humor and the relevant connections or common mistakes are very appealing and would work for a student or students working their way through the math concepts.
I think that most of my GT students would like the level of difficulty of the questions instead of the usual worksheets or problems from the math textbooks. Some of these questions are ones that you can really sink your teeth into for awhile and come up with multiple approaches. So many of our GT students love this problem-solving freedom and use it freely in OM competetions. Yet in the classroom, they are often confined to the one method presented by the teacher or the book.The lack of structure is definitely refreshing, but I think I would use this book more as a supplemental for problems than just allowing kids to use this book. There are a lot of concepts that students would miss if they focused solely on the methods presented in the book.
In response to PKassir,In middle school, there isn't a lot of consistency in which problem solving problem method is used. Although most teachers have the posters in their rooms, I find it not implemented very consistently. We used to try Problems of the Week where students did one step of a more complicated problem each day, but time constraints and schedule changes have made this more difficult. In addition, I have noticed that some grade level teams make up their own strategies that are more TAKS focused.
I think the kids will enjoy Challenge Math because Zaccaro includes interesting introductions to each topic that are rich with neat facts not typically taught, Zaccaro's topics are relevant to the kids' lives, Zaccaro includes humor within the characters' dialogue, Zaccaro requires the students to independently make many connections in order to successfully solve the problems, and Zaccaro made the problems quite challenging to solve.
I think students identified as gifted in Mathematics will enjoy Zaccaro's methods because they are quirky, connected to real-world situations and stories, and they have several different levels of questions, so they can start small and build up.
I agree with PKassir's comment about only needed a few examples for practice b/c the gifted kids don't need repetition once they have mastered a concept.
My GT students would like this author’s methods because he makes it real world and challenging. I like the way he gives some history and background information for the various concepts. The examples provide the students with enough information to build a foundation to solving the various levels of problems.
I agree with PKassir's comments on June 17 that GT kids like not being forced to use a specific problem solving strategy. They will often come up with a method that makes sense to them and many will only show the amount of work necessary to solve the problem.My own daughter who is a GT student often complains about having to solve problems the way the teacher wants it done and not the way she would do it.
I think that students that are gifted, and have mastered on grade level math material, would love the author’s methods. The author breaks down upper level mathematical concepts and processes to a level that younger students can understand. Of course they need to have mastered the basic part of the concept to go further. Math is used everyday, so why not go deeper into concepts with students that are completely capable of doing so. To students who do well in math, problems are just a game. Even though challenging at times, there is so much satisfaction in solving the problem correctly. The author’s methods also push the gifted student to look at math at much more complex points of view. It is not a single approach method, which they have to do over and over. Once they understand the concepts, gifted students want to find creative ways to solve problems.
In reponse to ReneeR on June 21 (who was responding to PKassir)I know that all gifted kids get frustrated doing the same basic math problems over and over- the greatest thing about this book is the levels. Students are more likely to do multiple problems at differing levels. Gifted students thrive more on the quality of the problem, not the quatity.
Response to SusanM and PKassir on June 18 & 20- SBISD problem solving modelThis is in reference to the discussion about the problem solving model... When I first taught primary- we did the model step by step with the boxes. We were later encouraged to not tie any of our kids to the boxes. We still discussed the steps, but they were just listed to the side. Too many students were getting too wrapped up in the model and weren't growing as problem sovlers. I could see this as a huge disadvantage to intermediate gifted math students. While I still discuss the steps of the process and the importance of the answer statement, I am no where near as tied to the model's steps. I don't know if that was just from our former SIS, or the direction that we're all going.
In response to CKohl on June 24th, I agree with what you said about not having a single approach to solve problems. GT students are natural problem solvers. Problems given in the book give them the opportunity of using different methods, and get creative.
The author's methods of teaching or explaining concepts is very unique and fun for the gt kiddos. His methods are fun with all the illustrations to go along with the problems. The author's approach to taking these math concepts a step farther than we would teach them to the non-gt kiddos is really great and I think gt kiddos would really jump on the boat with the processes. The extensions are very well thought out and applicable to everyday life.
In response to ndeans... that is a good point about the author giving background and history for the math concepts. I think that gt kiddos would really like to know why and how these concepts came to be. I think he also leaves them wondering and wanting more of the problems.
2) I think Zaccaro’s method would be appealing to students gifted in mathematics for several reasons: The chapters are short and sweet with various level of difficulty that would satisfy and challenged their curiosity. The author begins each chapter with an engaging story that gives a focus of the concepts to be taught, then it moves into the concept with a brief illustrative explanation given some room for intellectual frustration and then into different levels of practice. I love that the problems are very well thought and applicable to everyday life (I agree with bratliff’s comment on 6/24)
I agree with Katie Kavanagh's comment about GT students only needing a few examples for practice b/c the gifted kids don't need repetition once they have mastered a concept and that's why this book is perfect for their small group challenging them to go deeper.
In response to PKassir's response to me, and the conversation that followed it about the district PSM by nlopez and kohlerj, I no longer give my students the model with the separate boxes, it is simply one large box with the five steps outlined down the left of the page. However, since I teach 5th grade and still have students that initially learned the model with the boxes, many of them draw them in anyhow, even though I've stressed to them that it's a cyclical process and as long as the five steps are all present, they are fine. The rubric used requires that a student expand on their thinking and communicate using good math vocabulary. I find that the more challenging a problem is, the better opportunity I've provided the students with to be able to do this. If I give too simple or straighforward a problem, the communication piece typically is lacking. I just wish the grading guidelines weren't so tied to the problem solving model and corresponding rubric and allowed for more variety, although I guess it is a good way for the district to standardize us from campus to campus. I know that there is a "Secondary PSM" that is basically the same five steps as ours in a little more advanced wording, but I don't beleive they are tied to using it the way we as elementary teachers are.
I agree that our GT students and our high achievers that that the Pre AP math classes do not need a lot of practice the same problem. Some of them do need to learn what to do when the solution steps are not given straights out. They need practice t with how to stick with the hard problem and not give up and to try some alternate solutions. I like the problem like on page 114 number 5. Where some of the information is missing and new way has to be found before the problem can be solved. I also think the students will enjoy the way those solutions build on each other. Like problem 8 on page 115 for perimeter then problem 9 on pages 127 with area and number 2 on pages 163. These skill will be used again in high school SAT and AP test.