Top Eight Tool Kit





1) Hyper Space: Michio Kaku.

           This book is a wonderful example of how math and science can have beautiful applications. Michio Kaku explains time travel, alternate dimension and travel through interstellar space. He explains these topics without the over use of technical jargon and he uses analogies accessible to the average person. Mathematicians don't just sit around and think about solving math problems: they also think about what awesome applications the math they work on can have. This book will allow students the opportunity to learn that the topics that would interest them also interest mathematicians as well. This book also will allow students to learn that most math majors don't work in pure math. Mathematicians are people too and they want to work on interesting projects just like everyone else does. 

           This book also gives a very brief introduction to the history of mathematics. This history of mathematics is not taught to students until there junior year in undergrad. Providing students with an easy introduction can help prepare them for their training. This book and others like it can provide some amount of scaffolding for interesting topics in mathematics and the history of mathematics. This book is meant for the non math fluent reader so it is accessible to most readers. 

           Hyperspace also offers a broad overview of cosmology. Michio Kaku goes into detail about scientist best theories on what the universe was like before the Big Bang, why there was a Big Bang and how our universe will end. Most people do not have access to these theories without going to grad school for astrophysics but Michio Kaku and other writers like him explain these theories before we get to grad school so we can know we are interested in them before we decide math has no interest for us. 

 Kaku, Michio. Hyperspace: A Scientific Odyssey through Parallel Universes, Time Warps, and the 

            Tenth Dimension. New York: Oxford UP, 1994. Print.                          


2)  The Man Who Knew Infinity: Robert Kanigel

            This book is about a  self taught mathematician who grew up in India during the late eighteen hundreds and turn of the century named Srinivasa Ramanujan. This mathematician was a devoutly religious Hindu and believed that a specific Hindu deity acted as a muse and gave him his mathematical theorems. Ramanujan developed a mathematical style that was extremely original and unorthodox but also very prolific: he discovered 3900 theorems during  his short life. This work would give young math students the opportunity to learn about a man who thought of math as a form of creativity. He believed math was a form of spiritual enlightenment.  In India, he flunked out of every subject except math, but single handedly discovered mathematics so advanced that we are still finding applications for his math almost a one hundred years later. 

             While it is true mathematicians work hard on the memorization component of their work, this book will allow students to see a mathematician who saw himself as a creative artist. When Ramanujan was admitted into Cambridge and they tried to show him some mathematics he didn't know he would become inspired by what they were telling him and create new theorems based on what they were telling him. Literally watching him most have been like watching Mozart improvise at the piano on a given theme. 

             Ramanujan often would create mathematics impulsively. Inspiration would hit him at odd times and he would spend hours working out the details in his math notebooks. He filled his notebooks with thousands of theorems. He once discovered 100 new theorems in one day. Even when he was death bed he the ideas were coming to him and he was still jotting them down. 
               
              We are used to thinking of math as a subject we do to get a good grade. This book would give students an example of a man who did math because it was an art form for him. No one ever told ramanujan math wasn't fun so it became a spiritual quest for him. 


Kanigel, Robert. The Man Who Knew Infinity: A Life of the Genius, Ramanujan. New York: C. 

           Scribner's, 1991. Print.            

3)    Women in Mathematics


           This book contains biographies of 6 of histories most influential female mathematicians. This book will allow students to see that not only can great mathematicians be women but they can lead interesting lives outside of mathematics. Many women believe that women are not as good at math as men are and this is really a myth. Most scholars who ascribe to that theory believe the effect of testosterone on the developing fetus gives men an advantage at the more advanced concepts so this effect shouldn't the k-12 grades. The differences in performance and in career choice for men and women are complicated, but there is a large social component influencing young girls feelings toward math. This book does not provide proof that males and females are equal in math ability but it can be a god start in convincing some girls. Some people like examples better than theory. Those people will respond to the claim that females and can do math with the same level of expertise as males with the question name a great female mathematician. After a student reads this book they could name 6.

           This book also gives some detail how historically men believed they were better than woman at math and how slowly woman are closing the achievement gap. For example it is believed by some researchers that woman are better at algebra and men are better at calculus. This belief existed in the 1800's when women were not given the opportunity to study math in college and could only earn their respect by doing the long difficult and tedious calculations that we now use computer programs to do. Male mathematicians took advantage of female mathematicians that wanted to build a reputation by making them do the repetitive and boring parts of their job and then said it suited women better.

Osen, Lynn. Women in Mathematics. Cambridge, Massachusetts. Mit Press. 1974. Print.

4)    Math Games for Middle School: Challenges and Skill-Builders for Students at Every Level

              This book would allow students to enjoy recreational mathematics. Math games typically only require basic arithmetic but high amounts of practice builds fluency and all people are more likely to engage in activities that they enjoy. No one wants to do math homework for half an hour every day but playing a math game 15 minutes , twice a day is an easy objective to meet. Success in these games would build confidence. The type of studying most likely to achieve results are repeated short intervals and not long cramming sessions. Practicing 15 minutes every day is more likely to have have positive results than practicing 2 hours in one day once a week. The game allows for repetition which will build speed and ease. Math games also allow students to actually enjoy math and not just see math as a way to get a good grade. We educators have to foster the love of learning in our students. 

                This book is specifically designed for middle grade students. The book is part of a series with books for every grade level. Therefore the book contains techniques that the students will actually do in class and is directly useful to their education. These types of books would make good supplemental material for any school text book. This is an easy non threatening way to introduce some of the jargon specific to mathematics to the student. If the student uses these books as introduction to the material there ability to relearn the math in class will be greatly enhanced. 

Salvadori, Mario. Wright, Joseph. Math Games for Middles School: Challenges and Skill-builders at 

             Students at Every Level. Chicago Review Press 1998. Print.

5)     A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

This book is an important addition to the list because students must understand that if they decide to either major in math or a math intesive field that it is possible to enjoy math. The author makes the analogy between math and art. He believes that math is more like art than most of us realize. If art where taught the way we teach math no one would think that art is a method of self expression. The author makes the claim that math can be just as creative an expression of individuality as art. In math class we are drilled on technique and there is no room for creative self expression or discovery. All creative fields rely on route practice but the point of practicing musical scales or dance routines is to do something creative with them. Once we know the techniques of math we can begin to attempt problems that we do not know how to solve. These problems will require us to think creatively. 

The author says that when we are at our most creative we must not be afraid to fail. A math class is the exact opposite of a creative environment. Students are only concerned with the right answer and therfor not interested in taking the kind of risks that creativity requires. It is important that students learn to arrive at correct computations and to do it quickly but we as a society have forgotten that there is another side to math; the side that looks for patterns symmetry elegance and even beauty. Mathematics are used to make computer graphics increase the beauty of computer graphics to create the shimmering effects of the mandelbrot set. 

Lockhart, Paul. A mathematicians lament: How Schools Cheat Us Out of Our Most Fascinating and Imaginative             

             Art Form. Belleview Literary Press. 1, April 2009. Print.

6)    The Cartoon Guide to Statistics

Statistics is an intimidating field of math. Students are required to master many difficult concepts and to understand many mathematical techniques requiring two or three dozen steps. Also, the students that are asked to learn these techniques are not math majors or in math intensive fields so they are thrust into the world of learning a lot of math very quickly. This is a very stressful situation. The Cartoon Guide to Statistics is an easy low stress introduction to statistics and a wonderful primer for the topic before taking the class. So many college majors require statistics class that it is a good idea for any high school student aspiring to go to college to read this book. Since I believe that all students with normal IQ should go to college almost every high school students should read this book before they go to college. 

When and if it is appropriate I would incorporate some of the text in this book into some of my class work for the students. It can be difficult to teach material that is not required by the state but if the class has a section of work from statistics I would allow them to use parts of this book. Either I would print hand outs or scan the needed pages and email the pages to them. Not all students have regular access to email so I might provide the text in more than one way. I would use this work to alleviate some of their math and statistics anxiety. 

Gonick, Larry. The Cartoon Guide to Statistics. Harper Perennial. 1993. Print.

7)      Math Fables Too: Making Science Count 

This book is the easiest of the texts on this list. This work is also intended for the youngest on audience on the list at about 1st grade. This book uses children's stories to teach math and to give examples of the usefulness of basic arithmetic. These stories require students to count and learn the basics of addition subtraction in order to follow the plot. I included this book on the list because I wanted to be prepared to provide some supplemental reading for students of any age in the Kindergarten - twelfth grade system. 

If I included this work for students in middle grade maths I would have them read the stories and create their own stories and share them with each other. In our modern times we don't even need to wait for a publisher to recognize the merit of the students creative work but we can publish it ourselves on a Blogster. This would give students the opportunity to think critically about when does a scenario require a specific math skill. Reading and rereading their classmates stories would allow students to become experts at solving simple word problems. This book would serve as great practice for students who have problems knowing when to do a specific technique while working on word problems.     

Tang, Gregg. Math Fables Scholastic Press (September 1, 2007)

8)    Fear Of Math: How to Get Over It and Get on With Your Life! 

Math anxiety is a documented phenomenon that impedes math success for students. Math anxiety is completely learned and it is possible to unlearn it. It is generally harder to unlearn a behavior than to learn a behavior. To be specific we do no unlearn behaviors we only learn to more nuance to a behavior when under different circumstances, in a different context, or in response to different environmental cues. Millions of Americans have some amount of Math Anxiety. Fear is an emotion we learn to associate with specific situations from our environment. It is true that humans are predisposed to develop certain fears with more than others and that is why certain phobias are more common than others. However, math anxiety can come from a variety of sources. Social Shame and embarrassment is one of the key factors that allow for the possibility of math anxiety. No one likes to be ridiculed and we are made to do a task that we feel we are not good at in front of everyone it is easy to fell that our fears will be realized. 

I would allow my students to read short excerpts from this book and tell them it is my intention to create a safe place where the students do not need to feel intimidated and can learn without the added stress of ridicule. However, once I tell this to my students I must show this to them as well. My students and I will then spend the school year improving their math abilities increasing the sense of self efficacy and alleviating their math anxiety.

Zaslavsky, Claudia. Fear of Math: How to Get Over It and Get on With Your Life! Rutgers University Press.  

          1994. Print.

1)   Think-Aloud Strategy:

To utilize this strategy a teacher will read a text out loud and periodically pause to explain and summarize key concepts of the text. Teachers can give definitions, compare and contrast to other texts or supplement a text with more detail. Implementation of this strategy allows the teacher to model the desired behavior and gives the teacher the class activity to include the students when they are ready to participate. This strategy is very good for students who have trouble thinking critically while reading the text. This strategy will make them take a break from the reading and think critically. 

Example:

I would choose a word problem and read it aloud and then stop and think outloud. For example, “ Tommie is 19 and rick is 41 at what age will Tommie be when he is exactly half of ricks age”.  I would then stop and say I need to know what age Tommie and rick are every year and a chart is good for that. Also it seems as if the text insinuates that this half age point hasn’t occurred yet so I assume it is a future date.

2)  Jigsaw Strategy:

This strategy consists of braking the text up into smaller units and handing out each smaller unit to small groups in the class. This allows students to practice summarizing texts and linking the larger text together. This technique will allow students to learn that long problems can be divided into a group of smaller problems. This will help to alleviate math anxiety when working on word problems and allow students to communicate with the other students to emulate some of their success.

Example:

This strategy works best for long texts. I would break up the text and pass out the smaller texts to many smaller groups in the class. If the text can be broken into four sections I would have four groups in the class. For example, if presented with a table of different schools basketball scores over a period of a week and asked what was the team with the highest increase in average score. One Team could add the scores one group could do the averages and one team could compare the average scores and choose the best answer.

. 

3) Incorporating Aspects of Speaking, Listening and Viewing

This strategy requires the use of more than one sense. The higher number of senses involved when learning the easier it is to remember the learned information. While it is true that all students could benefit from this strategy this strategy is especially useful for special needs students of many kinds. Some intellectual impairments interfere with short and long term memory. This strategy improves both the recognition and recall aspects of memory.  The use of manipulatives can be thought of as a strategy of its own but an abacas would allow students to use the sense of sight sound and touch to while learning basic multiplication if they say the problems as they work on them.

Examples:

There are many different ways to incorporate this technique. When learning basic multiplication, the students can make flash cards and use them with each other. They can read the flash cards to one another and one of the students would answer. Some students could recite the first five multiples of numbers under ten, and the rest of the students could repeat what they hear. Students can also repeat the steps to a process out loud and then right them down and read them out loud. For example, when adding fractions students can say out loud if the denominators aren’t the same use the Lowest common multiple as denominator. Change the numerators to keep the same ratios of original. Just add numerators.

4) Short Answer:

 This strategy is useful for making sure students understand how they came to the correct or incorrect answer. In order to ensure that the successful computation of problems is reproducible we have to make sure it is luck or a fluke that the students arrived at the correct answer. It is also true that if  students’ consistently arrives at incorrect answers, we want to know they thinking process behind this phenomena as well. It is easier to change an erroneous thinking process once the student tells what it is.

Example:

I would just ask a student to explain how they got their answer. What computations did the students perform and why did the student perform that strategy. If the student can not say how they got the right answer then they must be reminded how to get the right answer.  Could even do a matching activity. Given 5 word problems match the right operation to the word problem and give a short answer explaining your choice. The idea is that a student may use some sort of context clues to reject initial wrong answers.

5) KWL Chart

This strategy is an acronym . K is for what I Know. W is for What I Want to know. L is for what I learned. This chart allows students to take a long word problem and distill it down into is most important points. It is a method for summarizing that allows students to navigate the details of difficult word problems. Part of the challenge of a word problem is also know when to ignore irrelevant information and the chart could serve as a tool to allow students to learn this skill. The internet has many examples of variations of this chart where students can add more spaces in the chart to allow for hypothesis comparisons etc. This allows students to quickly figure out what their goals are in a word problem.

What I know	What I want to know	What I learned
Rick is 19 Tommie is 41	When is Rick exactly half of Tommie’s age	When Rick is 22 Tommie is 44 How to make a table.

6) It Says -- I Say -- And So (Beers, 2003, p. 165 - 168)

This strategy allows students the opportunity to practice making inferences. Sometimes A word problem is testing more of the critical reading skill than the math skill. Word problems are an opportunity for students to incorporate and synthesis many techniques while working on one problem and critical reading skills is one of those skills. Problems like these are closer to real world examples. In the real world sometimes even the way we describe a problem can assist or hinder our attempts to solve it successfully.  This strategy allows students to move beyond doing basic arithmetic and really move into the many fields of science.

Example

A baker has 66 eggs left in his egg refrigerator. Each egg containers that hold 12 eggs. How many containers does this person need to put all the eggs in a container?  The answer is 6. 6 This is a division problem and 66 divided into 12 parts is 5 with 6 left over. However, those six leftover eggs must go into a container so they would go into a sixth, half full container. This question may seem a bit unfair to a student at first but in the real world the baker will not throw away 6 perfectly good eggs just because 60 divided into 12 is 5. The baker could make two more cakes with those eggs.

7) Anticipation Guides

In a math class this is a short document that either has an outline of what the objectives and goals are for the class or the analogous pertinent information. This can also be achieved in a question response format. This shorter text serves as a light introduction to the material so the students receive an easy to follow guide for the text. This strategy is like a vaccine. It presents the problem in an easier to solve manner so that a student may already know how to deal with this type of problem in the future. This strategy can help students to be prepared for more difficult tasks.  

Examples

This could be a hand out outlining the class activities and its objectives.  The anticipation guide should be very concise and easy to understand.  I like to use solving system of simultaneous equations as an example because it is a technique with a lot of steps. A student must solve these quickly in order to solve them on a test. In the anticipation guide the student could see all the individual techniques necessary to solve the problem and it would be obvious that they know how to perform each individual step so the real challenge is learning to link them together.

8) Retelling.

This strategy is a short summation of a text. This technique is an assessment to ensure the student has a high enough level of comprehension to even understand the text of a word problem. After all, it is difficult to analyze a text that one does not understand. If the student can give a synopsis of a plot then the student can engage basic level of reading that is needed to do more advanced types of reading of a text such as critical analysis, compare and contrast, or create a text of their own following the same format. Retellings can become very interesting when students create a creative retelling such as a modern hamlet or modern Romeo and Juliet

Example

This strategy is one of the simplest strategies that doesn’t split a text into smaller parts. If a student can not do this strategy, then the educator must increase the students reading comprehension. Once a student can do this well the challenge will be to limit the amount of detail the student gives until they only give the most important facts about a text. After all, students must also learn to be concise when it is necessary.