This is a professional development blog primarily for teachers in Spring Branch ISD.
I know one of my "ah-ha's" was that I still remembered some formulas from my highschool math! I first realized this when I was working the example problems on p. 172 & p.174. I still love relating probability to fractions. That helps many students build a stronger foundation in fractions, but also gets them more interested in fractions AND probability. Another "Ah-ha" occurred on p. 223. "Statistically Significant-" this will be a great tool to incorporate into science when students are creating a hypothesis and whether considering the factor more will cause them to adjust their hypothesis before testing things out. Factors and variables are very important in math AND science.
In Chapter 14 on page 219 it presented Predicting by Using Samples. This I had forgotten. We have been teaching the frequency charts but then creating the relative frequency in a decimal form this makes predicting easier.I am also glad to see Occam Razor covered. If you are searching for an explanation for an event that is perplexing or unusual the correct explanation is usually the most basic or down to earth explanation. I love that it takes real situations and put the most likely explanation. Middle school students are so often tempted to fall into the wild explanation and training them to find the down to earth will help them.
I really enjoyed Chapter 14, which begins on page 210. Zaccaro does a great job of explaining several concepts in statistics. I wish I had this book many years ago. It would have certainly helped me. Pages 211-215 gives very good defintions and thorough, yet simple explanations for several important terms in statistics. Another Ah-Ha moment that I had was when I got to page 260, the chapter on graphing equations. Again, it begins with a bit of historical background and takes the students from the very simple to increasingly more complex. It also gives real-world instances of how to apply graphing of equations. In math, it's especially important to show students the daily application of math concepts.
"Ah-Ha's" and "Golden Nuggets" include:1) (p.202) an interesting application of the probability of something <> happening;2) (p.215-216) defining the term "relative frequency" (predictor numbers);3) (p.238) defining the term "average speed", along with an interesting application where distance is divided into "blocks" to solve a problem.
nlopez in response to MW on July 8th- elementary students fall into the same trap sometimes- trying to explain things in the most outlandish way possible. For a few of my kids, I tell them to pretend that they are having to try and explain the idea to a kindergartner. That helps them simply their thoughts and answers so they are more able to focus on the basic truth instead of possibly causing more confusion b/c they are trying to incorporate thoughts that are not needed at the time.
nlopez in response to tatumt on July 10th- I appreciated distance being divided into "blocks" to solve a problem as another strategy that can easily be used. this should definitely help our more visual learners.
p. 197- Adding probabilities.. Didn't have a clue that you could add probabilities! Who knew it was so simple.p. 210- Statics, not really an ah-ha, but I know it still scares me!p.238- Different Kind of Average Speed, I thought this was really interesting b/c I never really thought about finding the average speed of two different speeds.
I really enjoyed Chapter 13 pg 191. The chapter on Probability starts from the basic and moves to the more difficult thinking in finding the probability of more than one event. I would say my ah-ah moment in this chapter was the way Zaccaro connected the tree diagrams to multiplying fractions. Even though multiplying fractions is not a TEK in elementary math it allows for GT children to make that connection and allows for middle school and high school teacher to connect multiplication of fractions to probability.
In response to Tatum...I, too, liked how he broke down the average speed down into blocks. The visual does help make it easier to understand the concept.
In response to Ratliff on July 13th. I agree with your pg 238- Different Kind of Average Speed. I too thought this was really interesting and I had never thought about finding the average speed of two different speeds to solve the problem. I did several problems in this section that made this work so much easier!
My "A-Ha" moments were on pages 223 to the end of Statistics Chapter 234. I really liked the way author explained concepts of "Statistically Significant", "Experimental Bias" and "Occam's Razor". The way he relates statistics to scientific thinking and perplexing events is very impressive.
My ah-ha moment was in the probability chapter. The simplicity that the author uses (pg 197,200) to combine and multiply the conversions is great. Very easily understood.
In rsponse to nlopez--I also love relating probability to fractions. It really helps the student internalize the concepts.
In response to Tatumt and Ratliff-I agree that breaking the speed down into blocks is great way to do this problem. Once again, model drawing simplifies a difficult concept.
In response MW on July 8th, I agree that using frequency tables and using relative frequency in decimal form is a great way to make predictions. I was thinking, changing relative frequency to percents can also be used to show parts of a whole and reinforce percent concept.
My biggest AHA moment was on page 202 when it shower thestrategy for figuring out the probability of something not happeningand then subtracting that answer from 1 to get the probabilityof sonething happening. I had never seen this strategy explained before this reading.
In response to ReneeR, i agree that p. 197-200 give clear examples and added to the simplicity. Oh, to have had this bookwhen i needed it in the past would have been great!!
My "A-HA" moment was in reading chapter 14's (pgs. 210-233) lesson on statistics. I appreciate how simple Zaccaro's explanations are, espcially difficult topics like prediction by samples, statistical significance, experimental bias, and occam's razor.
I like the way that the author explains the use of median and range on page 213. He explains that if a range of numbers is too large your data will not be useful. He also explains that you need to determine what measurements of data would be useful and you don't always have to do all six for each set of numbers. We usually teach all of them, but don't really explain which would be useful for the data that is being analyzed.
In response to Tatumt on July 10th, I too find the average speed calculation with the blocks. It sure make sense and will definitely help GT children understand speed, time and distance relationship from a different perspective.
I had a couple of thoughts when reading these sections:1) I really liked the discussion of Occam's Razor included on pages 230+. Several of the questions allow students to journal what they think could have happened. This is a great way to integrate their creativity, logic skills, and focus on writing skills as well.2) pg. 194. After teaching probability many times, I find one of the hardest concepts for students to understand is the number of outcomees for rolling two dice. Unless students see the list of 36 outcomes, they also answer that there are only 12 outcomes. I was somewhat dismayed that the author accepts as fact the 36 outcomes without going into a discussion of why. Even GT students benefit from making a list or creating a tree diagram on this concept.
In response to SDawson:I agree with you about the author's inclusion of scientific evidence. I like that the author also uses examples where miscalculations can have devastating results (Challenger) or lead to great discoveries (sanitation during cholera epidemic).
My “Ah-Ha” was how deep students can go into probability and statistics. I loved the statistics information on page 212 where they have to analyze the data to determine mean, median, mode, range, and frequency. This is a great concept that G/T students can dig very deep into because it isn’t introduced in my grade level.
In response to NDeans on 7/15, I too liked the author's explanation of the statistical terms. I really thought that after learning about these terms, gifted students can deeply analyze data. This is an area that they could really grow intellectually!
I really enjoyed the chapter on probabilities. More specifically, the information on pages 197-198 on adding probabilities and then the tree diagrams on pg. 199 for multiplying them. It has been so long since I have seen this type of math, but it was so much easier to follow than the way I learned in high school!
In response to PKassir:I completely agree with you when you said you wished you had this book when you were learning the material. The way Zaccaro presents the information is useful, quirky and fun and I think it would have made high school math much more enjoyable.
My aha moment was when reading about Chapt. 14 pages 211-213 referring to mean, median and range. These are always so confusing to keep track of for kids, and I really like the way they are explained. I believe if I present these concepts like Zaccaro did then the kids will truly understand them better.
I agree with nlopez and ReneeR's comments about the connection between probability and fractions helping students internalize these concepts more. I think that if they have a solid foundation of fractions- then probability will be much easier for them to understand.
.One of My “Golden Nugget” moments was on page 199 about the Tree diagrams or multiplying ot determine probabilities. I really enjoyed how the author explains this concept and the problems it uses. An “aha” moment is that I had forgotten in how many ways you could use trigonometry. I could problaby use this book as a quick set with my 4th graders to stimulate their brains and to think further than just the numbers/answers.
I had several ah-ha's. I really enjoyed the chapter on probability. I find probability to be a very interesting topic, and I think it can also be fun and motivating for students. I use tree diagrams frequently for teaching combinations, and I love the way that it has been extended to probability such as on page 199. I also enjoyed the information regarding mean, median, range and mode on pages 212-214. I am always looking for new ways to present this information and help my students to remember these terms once and for all. I find it interesting that on page 223, Zaccaro uses the word "factor" to refer to what we would typically consider "variable".
I agree with MW on 7/8 and nlopez on 7/11 about students frequently wanting to go for the outlandish or extreme explanations sometimes rather than for the simple one that's right under their noses. And of course I agree with many including sluther, nlopez, reneer, and others about the important connections to be made between probability and fractions...I find that students that have difficulty understanding fractions often also struggle with probability. Likewise, those students that understand fractions seem to build their understanding even further by using them to work with probability.