Like Galielo and his thought about the earth, he believed that "understanding came from an observation that others had not seen because they did not have or had not experienced the technology of his telescope."  If given time the correct observation will appear.  Now we are finding that there are nonlinear dynamics to math and this is revolutionizing thought.  The calculator, computer, and other technology like this are responsible for making the "new nonlinear math."  The older curriculum of math still dominates.  This is different from Galielo because linear math is also true.  The direction that the math NCTM wants teachers togo is for educators to ask. "Are students' discussion and collaboration encouraged?  Are students expected to justify their thinking?  If students are to learn to make conjectures, experiment with various approaches to solving problems, construct mathematical arguments and respond to others' arguments, then creating an environment that fosters these kinds of activities is essential."
  This incorporates the idea that conceptual understanding is an important component of proficiency.  There is a large set of curriculum nonlinear that is waiting to be integrated because the math we have now can only describe a fraction of our world.  What needs to be considered is the organic and non organic world is interactive, iterative, recursive and scalable.  There is no way to predict  long term physical or biological systems.
    One new way of thinking that has come out is spiral curriculum by Bruner.  The spiral is an inter linked and basic ideas presented in a brief and simple ways with a circular strand or thread.  When a learner completes circle and returns to the first in which it becomes extended and new ideas are addressed more deeply.  This leads to good questions and right questions.  CROP "problem solving ability passport for the future."   This happens through the discovery and clarification of questions.  The quality of answers you get is dependent on the quality of question.  What we produce is effected by our questions.  Models for these questions are recall, analysis, comparison, inference, and evaluation from Bloom's taxonomy.  For every linear way the chart shows there is a nonlinear type of math.

Bibliography
EDELCompEduc/Ch4/Ch4-MathsprdsheetThink.html

    Calculators and computers are tools and can help by creating visual images of mathematical ideas.  It is a good source for organizing and analyzing data because it is efficient and accurate.  They provide a boost which helps kids make decisions, reflect, reason, and problem solve.  Technology can not replace basic math.   If used wisely and responsibly it will enrich students learning of math.  Ideas are that all students should have access to technology.  It enables students to examine representational forms than are feasible by hand.  They can make and explore conjectures and provides access to visual models.    
       The main source of ownership of abstract math can be used with technology, because it enables students to view math in multiple ways.  Technology provides feedback and focus, and offers adaptations for special student needs.  It is a teaching tool and there fore can be used well or poorly.  Things that technology is good at are graphing, visualizing, and computing.  This opens up more variables because data and resources are available form the internet and the web.  The teacher of a technology rich environment must decide if, when and how technology will be used.  The teacher should observe children using and focusing thinking this aids in assessment which allows teachers time to examine the processes used by students.
    The benefits of technology are that now students can explore and solve larger problems.  They can see shapes in new way.  This helps them to organize and analyze.  Middle school students can study linear relationships and revolutionary change at the same time through creation of computer based experiments.  This helps blend math out of artificial boundaries of algebra, geometry, data analysis.

Bibliography

    Teachers and school administrators effect the future by their choices regarding content and character of mathematics.  This six principles put in place to guide educational systems are
1.  Equity- "high expectations and strong support for all students."
2.  Curriculum- "...more than a collection of activities:  it must be coherent, focused on important mathematics and well articulated across the grades."
3.  Teaching-  "Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.
4.  Learning-  "Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge."
5.  Assessment- "..should support the learning of important mathematics and furnish useful information to both teachers and students."
6.  Technology- "...is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances student learning."

Bibliography
http://standards.nctm.org/document/chapter2/index.htm

1.  How can look be translated across curriculum?  If you are looking at the map on this web page, you would see look in math, science, and technology.  In math, Look translates to identifying the problem and understanding it.  In Science, Look translates into Problem Posing or observation.  In technology it can translate into Local actions through calculators, spreadsheets, databases, word processing, outlining, drawing, painting, video, audio, animation or virtual reality.  Otherwise in technology it can translate into Global tools such as a network of computers.

2.  How can Evoke be translated across the curriculum?  If you are looking at the map on this web page, you would see evoke in math as strategies like guess and check multiple choices, Make an organized list like logic problems, look for patterns like repetition problems, draw a diagram like word problems, act it out like collaborating, work backwards like long problems, and simple related problem if haven seen before, analyze merits, analyze merits and choose strategies or strategy.  In science, it could translate to problem solving, developing hypothesis, conducting experiments, draw conclusions. Technology local actions could translate into typing and recording. As a global tool this could translate into computer conferencing, Net news, LISTSERVs, collaborative screen use of programs, writing, talk, chat, and email.

3.  How can Assess be translated across the curriculum?  If you are looking at the map on this web page, you would see assess in math as execute strategy, and check the problem and the correctness of the answer. In science it would be seen as persuasion.  In local actions it would be seen as check, lookup, read, and listen.  In global tools it would look like, email, groupware, net phone, video conferencing, CU-SeeMe, and private networks.

4.  How can publish and perform be translated across the curriculum?  If you are looking at the map on this web page, you would see publish perform in math as share findings.  In science as sharing findings, in local actions as a table of contents, index, format, and print, and as a global tool in net tools, FTP, audio, video streaming, world wide web, virtual reality, wireless palmtops, and PDA's, or legacy systems such as TV, radio stations, publishing houses.

Humans are not computers they add the value system.

Bibliography

EDELCompEduc/Ch4/mathsciLEAPtable.html

    Bloom created a taxonomy with other educators so that they could categorize questions.  Since you can not have higher order thinking without recall, you can have recall without higher order thinking.  The five categories are Recall, Analysis, Comparison, Inference, Evaluation.  Trigger questions look as follows.

Recall "Do you want someone to recall information?"  what does this translate into the learner recognizing and remembering key facts, definitions, concepts, rules, and principles.  Which is found through verbatim or paraphrased answers, and the students learn through rehearseand practice and then associate with other concepts.  This could be seen as repetition and translation into own words.  Key words that go with recall are define, repeat, identify, what, label, when, list, who, and name.  Examples of questions are define the word dog? , What is a car?,  Label the following tooth with its parts.  Identify the triangle in this picture? Who did the doctor give a shot to in the story?

Analysis "Do you want to analyze a situation?"  This translates for the student as the division of the whole into component elements. This involves part/whole relations and cause/effect relationships that leads to being more complex in objects or ideas, basic actions of procedures or events.  In science this looks like separating the components of the process.  Identifying the features of animate and inanimate objects.  In social science could look like analyzing the components or elements of an event.  In literature it looks like identifying components of literary, expository, and persuasive discourse.  Key words for analysis are subdivide, categorize, breakdown, sort, and separate.  If you are breaking a story down into different parts.  Example questions are what are the basic elements in a short story?  What is/are the functions of  this mathematical problem.  Inventory the parts of this spreadsheet.  Categorizethe hero element of Odysseus?  Sort the blocks by color and dimension.  What is the order of steps in the mystery?

Comparison Thinking "Does someone need to make comparisons?"  Learners need to recognize and explain similarities and differences.  If it is a simple comparison need one or a few very obvious attributes or component process.  A complex comparison requires identification and differentiation among many of more complex attributes or components.  In science, social science, and literature this can be seen as the comparison of the properties of objects or events, the comparison of causes and effects of separate events, the comparison of social political, economic, cultural, and geographic features, the comparison of meanings, themes, plots, characters, settings and reasons.  Key words are compare, differentiate, contrast, distinguish, relate.  One example is to compare the theme of two stories.  Compare the Trojan army before and after the trojan horse.  Contrast the attitudes of the Gods to the attitudes of Odysseus men.  Differentiate between the Suitors who stole from Odysseus house, and Odysseus who stole from the cyclops.

Inference thinking "Should you speculate or make an inference?"    This could be either deductive or inductive reasoning.  Deductive reasoning is a type of reasoning where students take a generalization and recognize and explain evidence that relates to it.  This could be as simple as seeing if an if then rule applies.  Inductive reasoning is a type of reasoning where students have evidence or details and they come up with the generalization.  Skills required to integrate information like hypothesizing, predicting, concluding, and synthesizing require or are related to inference learning.  Application of a rule is deductive, wheresynthesis can be both.  Science, social science, and literature showinference thinking when students draw conclusions, make predictions, posehypotheses, tests, and explanations, conclude, Infer character motivation or cause and effect.  How do I make this character more believable.  Hypothesize what will happen if I take this metal stripe off of a diskette.  Predict what would be true if I did not save my documents to the internet.  Conclude what the result will be if I forget to download one of my web pages. What if I had saved my spreadsheets to a different disk instead of saving them to the already full disk?

Evaluation "Is it time to help someone judge or evaluate?",   When a task requires judgment, quality, credibility, worth or practicality.  This represents students using established criteria to explain how theseare being met or not met.  The criteria is a standard, a rule, or atest on a judgment or decision.  This can be established by rules of evidence, logic or shared value.  In science, social science, and literaturethis is reflected when soundness and significance of findings are present.  This is also present when credibility of arguments, decisions and reports have are evaluated.  If the significance, form , believability, completeness, or clarity are evaluated.  Key words are evaluate, argue, judge, recommend, assess, debate, appraise, critique, defend.  If you were to evaluate a story you would need to tell if the story were well written and why orwhy not this is so.  Sample questions are What you would do if you lostyour job?  Why?  Judge what would be the best way to solve the problem of world hunger.  Why did you select that solution on questionfive?  Evaluate whether you would steal from the Cicones or go hometo your beloved wife Penelope.  Odysseus poked the Cyclops eye out inthis situation.  Why?

NC has confidence that end of grade testing will move to a more open ended approach and that higher order thinking skill are vital to the future of the students.

Bibliography
http://www.ceap.wcu.edu/houghton/Learner/think/thinkhome.html

http://www.ceap.wcu.edu/houghton/Learner/think/recall.html

http://www.ceap.wcu.edu/houghton/Learner/think/analysis.html

http://www.ceap.wcu.edu/houghton/Learner/think/compare.html

http://www.ceap.wcu.edu/houghton/Learner/think/inference.html

http://www.ceap.wcu.edu/houghton/Learner/think/evaluate.html

http://www.ceap.wcu.edu/houghton/Learner/think/NCthink.html

    The old spreadsheets were grade books and accounting journals which consisted of tables or rows and columns that were used for the organization of data and record keeping analysis.  The first spread sheet was in 1978 by Dan Bricklin a harvard boy and Robert Frankston a MIT boy called Visicalc.  This was one of the first things that legitimized the personal computer industry.  Spreadsheets illuminate errors but can not eliminate all errors.  If you put bad information in, you get bad information out.
    Things to consider when selecting a spreadsheet are platform compatibility, will this work with my hardware.  Can this run on different computers.  Second features teacher needs to test to see if it will meet student need and competencies.  Third ease of use, can this be used easily for this age group.  This will confuse students if they are given information before they can understand it.  Fourth, Help does this have a help program built in with walk through or assistants and wizards. Finally graphing, does this spreadsheet create a variety of charts and graphs. Can data in a cell be automatically changed to a graph.
   Spreadsheets can be reviewed by going to Yahoo and typing in spreadsheet review or spreadsheets reviews.
      Bibliography
      EDELCompEduc/Themes/Spreadsheets/spreadsheets.html

   1.  "Use the Muffins spreadsheet:  What is the formula for determining how much Honey is needed to make muffins?"
2. "Use the U.S. Debt spreadsheet:  If the population in 1990 had been 1 billion greater, what would be the per capita debt in 1990 dollars?"
3. "Use the Bowling Scores spreadsheet:  If Sam Thomas had bowled a 300 in Week 5, how much would this have brought up the team average for the week?  What formula is used to determine team average for week 5?  How much would it have brought up their average from all games?  What formula is used to find this overall average?"
4.  "Use the Nutrients spreadsheet:  How milligrams of Vitamin C are in 5 servings of Cooked Fresh Carrots and 5 servings of Apples?"
5.  "Use the Nutrients spreadsheet:  What is the total number of milligrams of Protein in a meal for the hungry campers:  4 servings of hamburger patty, 8 servings of chicken drumsticks, and 4 servings of backed potato?"
6.  "Use the Nutrient spreadsheet:  In addition to 1 serving of Baked potato, one serving of which food would provide the greatest total amount of Vitamin C?"
7.  "How many servings of Chicken Drumsticks will give the same amount of protein as 72 servings of cooked fresh carrots?"
8.  "The trail guide says that you need a meal that provides 162 grams of calcium. Which combination of two items of two servings will provide you at least with these nutrient totals?  You can choose from Spinach, cooked fresh carrots, and raw cabbage."
Bibliography
      EDELCompEduc/Themes/Spreadsheets/assignSSpractice.html

4.1  "Interpret and communicate information in an existing spreadsheet."- Yes I can
4.2  "Enter data in an existing spreadsheet."-  Yes I can
4.3   "Create a spreadsheet with rows, columns and headings"-  Yes I can
4.4 "Create /copy formulas and functions to perform calculation."- Yes I can
4.5 "Create a graph from spreadsheet data."- Yes I can
4.6 "Insert spreadsheet into word processing document"- Yes I can
4.7 "terms such as spreadsheet, cell, data entry bar, formula, and function"-  Yes I understand terms