Homework help grade 6
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Factor analysis of ordinal variables: a comparative study
Keywords:latent variables, factor analysis, ordinal variables, Principal component analysis, Categorical Principal Components Analysis
The paper considers different approaches to the factor analysis (FA) for ordinal data. In some studies it is necessary to find a latent variable behind the observed indicators measured on an ordinal scale. Classical factor analysis cannot be applied to those indicators as it is built on the Pearson correlation coefficient which is only applicable to interval variables. So the researcher faces a choice: to treat the ordinal variables as the interval ones, to dichotomize ordinal variables or to use special techniques for ordinal indicators such as replacing the correlation matrix or using Categorical principal components analysis (CatPCA). The study is based on a theoretical comparison of assumptions that underpin the algorithms of each applications and a statistical experiment and provides an answer to the question which of the abovementioned factorization approaches is optimal for indentifying latent variables measured by ordinal indicators on a 3point, 5point or 10point scale.


How to Get Repeat Customers for Repeat Business ?
Ever wondered if you can quantify the value of a repeat customer, specifically for your business? Can you apply a simple mathematical model to quantify the value of a repeat customer for repeat business?
Yes, you can with this simple formula. Just a few numbers to get you started! In this article, we will introduce a key concept, Repeat Customer Leverage (RCL),to help estimate the value of a repeat customer specifically for your business.
Repeat Customer Leverage
Take the case of a company, Acme. Suppose,
You can find these x and y values from your internal business metrics, or if you are an eCommerce business, it can be found in the Audience Behavior – New vs Repeat Section in Google Analytics.
First, we measure the efficiency with which repeat customers generate revenues. It is simply the percentage of repeat revenues divided by the percentage of repeat customers.
Next, we measure the efficiency with which one-time customers generate revenues. It is simply the percentage of one-time customer revenues by the percentage of one-time customers.
Then, we measure the relative efficiency with which repeat customers generate revenues compared to one-time customers. This relative efficiency is the Repeat Customer Leverage (RCL), or the relative value of a repeat customer compared to a one-time customer for that specific business.
Let’s take a look at the importance of RCL. Chances are, you may have heard about the 80-20 rule, or the Pareto Principle. The Pareto Principle is also applied to marketing where it states that 20 percent of your customers represent 80 percent of your sales. For a business whose distribution follows the Pareto distribution, the value of RCL is computed as follows.
Therefore, if the repeat customer distribution of a business follows the 80-20 Pareto principle, a repeat customer is 16 times more valuable than a one-time customer. Any one-time customer converted into a repeat customer will generate revenues 16 times more efficiently for the business. Similarly, if a business has a 90-10 distribution, where 10% of repeat customers generate 90% of the revenues, the RCL value is 81. For such a business, a repeat customer is 81 times more valuable!
A simple, easy to use value calculator can help you get started to compute the value of RCL for specifically for your business.
Less convincing required = less marketing expenditure. A study conducted by Boston Consulting Group states that the average cost of marketing to existing customers is about $7 whereas the marketing cost per new customer averages at around $34. Way too much time and money is spent on new customer acquisition without a sense of when to start focusing on customer retention.
The choice is yours, spend additional dollars on getting new customers or focus on the customers you have. But measuring Repeat Customer Leverage (RCL)is a great way to determine the value of a repeat customer relative to a one-time customer.
The higher the value of RCL, the higher the effort that should be put in to convert your one-time customers into repeat customers. Loyalty rewards is a proven strategy to convert one-time customers into repeat customers.


Dr. Jenkovszky László
Dr. Jenkovszky László, Kiev Institute of Theoretical Physics:
Memoir of Bolyai, Gauss and Lobacsevskij
Predecessors: Nearly 2300 years ago, the Greek mathematician Euclid of Alexandria laid down the basis of the geometry now known from the textbooks and used in everyday life. It was based on a number of postulates and axioms. Almost all of them were generally accepted either as obvious, or as necessary for logical consistency. A notable exception was the fifth postulate: if a straight line crosses two other straight lines, forming internal angles that equate to less than 2π radians, then these two lines—if continued infinitely — must cross somewhere on the plane where the lines and angles are located.
The fifth postulate has been something of an embarrassment ever since Euclidean geometry was first formulated. However, it became a dogma — part of European culture, supported by authorities such as Isaac Newton, Leonardo da Vinci, Galileo Galilei, Johann Kepler, Joseph-Louis (Giuseppe Luigi) Lagrange and Emmanuel Kant, among others.
It was only in the first half of the 19th century that three great men—Janos Bolyai, Carl Friedrich Gauss and Nikolai Lobachevsky, abbreviated in alphabetical order as BGL throughout the rest of this article—independently, but almost simultaneously, succeeded in generalising Euclid’s vision.
The presentation explores this journey also mentioning other notable scientist who came very close to reaching similar conclusions as BGL did: an Italian monk named Girolamo Saccheri around 1697 and Ferdinand Karl Schweikart between 1807 and 1816.


I can’t remember where I saw this but I saw it as a bar trick first. My year 8 class have been studying circles and so I showed them this starter. I got a pint glass and asked the question “Which is bigger the circumference at the top of the glass or the height of the glass?”. It is an optical illusion and most of the class voted for the height. I then put the glass on a thick book and asked again. Virtually all now agreed it had to be the height (with book). However, a bit of string showed it was still the circumference. Try it and see, you might be surprised. Filling the glass with your favourite beverage whilst testing the idea is, of course, optional.


Mastering Mathematics in KS1
Following the outstanding feedback (100% top rating) on the pilot version of this course, we are now able to offer teachers of KS1 the opportunity to spend an engaging afternoon exploring mastery of mathematics in KS1.
Delegates will leave:
- With an enhanced understanding of how the most effective teaching strategies for mathematics can be developed in the classroom.
- Techniques to modify common mathematical tasks to ensure mastery.
- Renewed confidence of how to cater for higher attaining groups of children, particularly those expected to reach greater depth.
- Confident in promoting reasoning in every maths lesson.
- Buzzing with ideas to use in their classrooms the next day.
- DATE: 26/03/2019
- Terms & Conditions:Cancellations must be received in writing or via email to firstname.lastname@example.org. The following charges will apply for cancellations: More than 15 days - no charge, 7 to 14 days - 50% charge, Less than 7 days - 100% charge. Verbal cancellations will not be accepted. Education Central reserves the right to cancel a course. Education Central is unable to accept responsibility for any costs incurred.
Thank you for trying to book this course. Currently this programme is sold out. Please email your request to Education-Central.org.uk and we will see how we can help you. We often schedule additional events and also hold a waiting list in case places become available.


- Compute the NPV and IRR for project whose initial cost is 30,000 and cash inflows are 14000, 8200, 12000, 15000, 22000. Discount Rate is 10%. Cost of Capital if borrowed is 15%. Show value of NPV at IRR as discount factor. Based on the above calculations, should the project be considered?
The calculations are done in excel where NPV is calculated using time value of money and IRR value is found which is closest to NPV =0
2. Calculate the Cash Cycle using the following information. (Assume 360 days in a year).
Raw Material 4,00,000
Finished Goods 6,00,000
Raw Material 5,00,000
Finished Goods 7,25,000
Costs Incurred during the year
Manufacturing Costs 10,45,000
Excise Duty 8,50,000
Selling and Distribution Expenses 4,20,000
Admin. Overheads 3,00,000
Total Sales 4,20,00,500
Total Purchases 3,23,00,000
30% of sales are on credit and 80% of purchases are on credit.
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We consider one-level additive Schwarz preconditioners for a family of Helmholtz problems of increasing diculty characterized by wave number k. As k increases, the solution becomes more oscillatory and thus meshes need to be increasingly refined, leading to huge linear systems. Moreover, these linear systems become more indenite and the minors of the linear systems may be even singular, which prevents the usage of many standard preconditioning techniques. We discuss additive Schwarz preconditioners, of which the action requires solution of independent subproblems with absorbing boundary conditions and (possibly) absorption added in the domain. Then the local solutions are glued together using prolongation operators dened by a partition of unity and nodal interpolation. Supporting theory for this preconditioner in the case of homogeneous Helmholtz problems is given by I. G. Graham, E. A. Spence and J. Zou in [Preprint arXiv:1806.03731.]. In this talk, we will present preliminary theoretical results and numerical experiments for this method applied to heterogeneous Helmholtz problems. We also show that the method is independent of the underlying finite element discretizations.


Claremont aims to provide a high-quality mathematics education that provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject. There is a balance of fluency, problem solving and reasoning following KCC guidance of progression. The children make use of manipulatives (equipment), working walls, the outdoors and each other in order to develop their understanding. Teachers develop children's independent higher order thinking skills. When children are able to describe, represent, explain, and create their own example of a concept, idea or technique, they are demonstrating 'mastery'. When children are able to solve problems of greater complexity, demonstrating creativity and imagination, independently explore and investigate, communicate results clearly, and systematically explain and generalise, they are working at greater depth.
Curriculum for mathematics on a term-by-term basis (Maths long term plan)
Curriculum for mathematics on a year-by-year basis
Please click on this link for a guide about learning times tables.
How can I help my child get faster at recalling their times tables facts? Watch this video:
What do I do if my child cannot memorise times tables facts? Watch these videos:
Five Ways to Share Maths with Kids
Maths has moved on since many parents were at school. It's not about mechanically answering questions, it's about enjoying maths and being curious and learning about its beauty and power.
In this TED Talk, Dan Finkel explains how maths can be playful and is something to explore and enjoy.
Freckle (Years 1-6) https://freckle.com/en-gb/
This is web-based maths learning resource. It is pupil-centred and teacher-driven. It is differentiated, engaging and effective. It continuously adapts for personalised maths practice, while offering teachers the ability to focus on year objectives. It delivers real-world scenarios, skills practice, and peer-to-peer support with pupil-friendly incentives and age-appropriate designs to provide a balance of fun and learning. It increases pupil progress and proficiency in maths through objectives-based skill development, personalised target setting, and individualised practice sessions. It is aligned to KS1 and KS2 National Curriculum standards for England. Click on the following link to read a guide about Freckle written for parents and carers.
Numbots (intervention) (Year 3) https://numbots.com
This is a web based and app based learning resource. The goal is for pupils to use efficient mental calculation strategies to add and subtract two-digit numbers, so that they leave counting on their fingers behind! Visit https://numbots.com/schools/ for more information.
Times Tables Rockstars (Years 4-6) https://ttrockstars.com
This is a web based and app based learning resource. It is a personalised times tables training programme. For information about the TTRS model, visit:
For information about the online games, visit:
Please see the parent guide for further information.
Please click on this link to find out the meaning of mathematical terms.
What is 'mastery in mathematics'?
Please click on this video link to understand what is meant by 'mastery'.
Countdown Answer Sheet
Please follow the link to the Countdown answer sheet.


How much does it cost to play at the Seven Deuce Club vs other establishments?
We encourage everyone to do the math but here are some numbers to help.
Casual poker player who drinks and plays one 4 hour session every few months
Social poker player who likes to play about 6 hours a week. (25 hours/ month)
Serious poker player who grinds 30 hours per week. (125 hours/month)
Pool payer who enjoys two 2 hour session per month. (4 hours/month)
Although we are not the cheapest place to play pool we are committed to providing the best experience possible to our members by providing a clean smoke free atmosphere where you can relax and enjoy the games you love. BYOB saves you money and helps us provide more game types to our members. *Price changes at our or other clubs may affect these numbers.


== checks if the values of the compared objects are equal
=== checks if the values and types of the objects are equal
var fiveAsText = '5'; var fiveAsNumber = 5; fiveAsText == fiveAsNumber // outputs true fiveAsText === fiveAsNumber // outputs false
This can lead to some unusual behaviour when dealing with different types. For instance, when you perform mathematical operations on a string type with a numeric value
var five = '5'; var three = 3; five + three // "53", string concatenation interprets the '5' as a string and overrides the addition operator five - three // 2, subtraction interprets the '5' as a number
Update (28th August 2015)
A perfect example of the confusion that this can cause is as below
The explanation isn''t exactly intuitive but at least shows that there is some sort of logic behind it all
+is overloaded as both the addition and string concatenation operator.
In mixed operations:
+casts numbers to strings.
-casts strings to numbers.
Both are used to denote signage. Leading either on a string will cast it to a number, with that sign.
Pretty simple stuff, right?
Number('5') - 3 '5' + (3).toString() '5' + (5).toString() 'foo' + Number('bar').toString() '5' + (-Number('2')).toString() // Long ass sign flipping, same as above Number(('5' + (3).toString())) - 3 (Number('5') - 3) + 3


MATHSBALL™ is a unique set of sessions aimed at primary schools. They are fun, inclusive and designed to demonstrate National Curriculum intent.
Implemented into the school week, they are a great way to add to your broad and balanced curriculum.
Delivered in a football environment (school field, playground, sports hall etc), they are approximately 45-minutes in duration and offer a unique way of enhancing the way mathematics and football are taught in your school.
Updated regularly, sessions are available for all primary school years; Reception with their Early Learning Goals through to Year 6's mental maths.
Football and maths in one!!


Friday, 1 February 2008
You Say a Cup, I say 2,3659 desilitres!
Some of my favorite recipes are from cookbooks or websites that use American cooking measurements. By now I have baked Heidi's holiday cupcakes enough times to know by heart approximately how many dls is 1 or 1/2 or 1 1/2 cups, but it is a pain, a small one, but a pain nevertheless, to have to do some calculations every time I want to bake something from a "non-metric recipe". And of course, the same applies the other way round, if you are used to cups and ounces and would like to bake using a "metric recipe".
Also I might be able convert cups to desilitres, but! 1 cup equals to 2,3659 dls. If your average metric measurement cup gives you wholes and halves then 2,4 or 3,6 etc. are a challenge, not to mention 2,3659!!! I know we are not talking about rocket science here! It is perfectly fine to use approximate measurements and still succeed when baking, but sometimes I've wondered whether it would make a difference to be able to be a bit more exact when using American recipes. Well, ta-daa!! Now I will be able to do just that. See what I found in a store today! I am the proud owner of this measuring cup that has both cups/oz and ml/liter marked on it. I'm one happy camper today! :) Plus it snowed, quite a lot, too! :)
Here are a couple of useful links if you need to convert kitchen measurements:


“OK, boys! Time for school! Let’s head up to the schoolroom.” Announced Mom.
“Already? We just ate, Mom. Can we get 10 minutes for our food to settle, then a little snack break?” asked Mack.
“Say what, son? That is not going to happen. Finish you juice and meet me in the schoolroom in 5 minutes.” Replied Mom.
Cheese laughed, “What were you thinking, bro? You knew better than that. Let’s go.”
Mack and Cheese drank their juice and went to go learn something.
“OK Mack. We are going to go over your math first. I have a one-minute math sheet for you. Let’s see how many you can get right in one minute… GO!” said Mom. “While he is doing that, Cheese bring me your ‘Just a Little Kid in the Woods’ book.”
Cheese thumbed through the bookshelf and found his reading book.
“Here you go Mom.” Said Cheese.
“Thank you, Cheese.” Mom turned to Mack. “OK. Stop! Your one-minute is up. Hand me your paper. Cheese. While I grade Mack’s quiz, read Chapter 6 to yourself.”
“Mom. That math test hurt my brain. Can we be done for the day?” requested Mack.
Mom replied, “If you answered every question and answered them correctly, we can take the rest of the day off.”


Gretchen Crowder has served as a campus minister and Ignatian educator for the Jesuit Dallas community for the last 15 years. She is also a freelance writer and speaker and is the host of Loved As You Are: An Ignatian Podcast. She has a B.S. in mathematics and a M.Ed. from the University of Notre Dame as well as an M.T.S. from the University of Dallas. She resides in Dallas, TX, with her husband, three boys, and an ever-growing number of pets.


GET AN EDGE
The applied mathematics major is geared toward students majoring in engineering, physics, chemistry, business or any field where advanced math skills would be an asset. You’ll tackle subjects like number theory and cryptography and delve into unique mathematical applications in your chosen field.
At ONU, you’ll receive individualized attention and become a valued member of our family. You’ll enjoy small class sizes and amazing friendships with your classmates. Our dedicated mathematics professors will get to know you. They will work closely with the professors in your specialty major to focus your learning experiences in mathematics around your future goals. They will become personally invested in your success – in college and beyond.
Large institutions just can’t provide the individualized attention and opportunity you’ll find here. Throughout your four years, you’ll assist professors with advanced research, work on independent projects, participate in mathematics competitions, and present research at regional and national conferences.
Upon graduation, you’ll have a degree with that “special something” that sets you apart. If you attend graduate school, your secondary math major will help you breeze through material other students will find challenging. If you enter the workforce, you’ll be able to problem-solve and understand material at a higher level than most of your co-workers. Either way, you’ll be on the path to success!
LET’S SOLVE IT
Match wits with talented mathematicians from around the world – and come out on top! ONU-SOLVE Problem is an intense (but fun) group of ONU students and professors that tackles the math challenges proposed by mainstream mathematics journals. The group has been recognized 10 times in recent years for submitting correct solutions and the most well-written solutions.
OH, THE PLACES YOU’LL GO
Imagine traveling to cities like San Diego, New Orleans and Atlanta – all expenses paid – with your math professors and fellow students to present research at a national conference. Or spending the summer at a large university assisting with a grant-funded research project. Maybe you can picture yourself competing with an ONU team at a national math competition or studying abroad for an entire semester. We believe a well-rounded education includes opportunities to travel, experience new places and stretch your comfort zone. These opportunities are here for you to grab.
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“MY MATH COURSES PROMOTED A RATIONAL AND DEDUCTIVE THOUGHT PROCESS THAT IS INCREDIBLY USEFUL IN THE WORK I DO EVERY DAY. ONU GAVE ME THE CHANCE TO DEVELOP NOT ONLY ACADEMIC SKILLS, BUT ALSO PROFESSIONAL AND LEADERSHIP SKILLS.”
Greg Back added an applied mathematics major to his computer engineering major because he knew math would have practical application in his future career. At ONU, he became involved in the Joint Engineering Council and ONU’s chapter of the Institute of Electrical and Electronic Engineers. He also engaged in mathematics research projects and presented his research at several local and national conferences, including the 2010 Joint Mathematics Meeting in San Francisco. Today, he’s a cyber security engineer at MITRE Corporation, where he develops software to support security analysts. He’s doing what he has always wanted to do. He writes software that helps solve complicated problems and make people’s jobs easier.
OUTCOMES AND COURSES
- Over the past five years, an average of 83 percent of our mathematics majors were employed full-time in their field or enrolled in graduate school within six months of graduating.
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- Our graduates get accepted into top-notch graduate schools, including the University of Indiana, Ohio State University, West Virginia University, the University of Florida, North Carolina State University, the University of North Carolina, Massachusetts Institute of Technology (MIT), Penn State, and Northwestern University.
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Here are some classes you’ll take as an applied mathematics major:
Foundations of Mathematics with Problem Solving
Number Theory with Applications to Cryptography
Partial Differential Equations
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WHERE COULD YOU END UP?
– Statistical Consultant – Statistical Analyst
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Main / Tools / T veerarajan probability ebook
T veerarajan probability ebook
Name: T veerarajan probability ebook
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Probability, Statistics and Random Processes. Front Cover. T. Veerarajan. Tate McGraw-Hill Publishing Company, - Mathematical statistics - pages. This book with the right blend of theory and applications is designed to provide a thorough knowledge on the basic concepts of Probability, Statistics and Random Variables offered to the undergraduate students of engineering. Detailed coverage of topics like Dispersion, Skewness. (Refer chapters 6, 9 and 10) Special Probability Distributions and their inter- relation has been explained with great clarity. Pedagogical Features: Solved.
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Advances in Consumer Research Volume 2, 1975 Pages 741-750
VALIDITY AND GOODNESS OF FIT IN DATA ANALYSIS
Donald R. Lehmann, Columbia University
[Donald R. Lehmann is Associate Professor at Columbia University Graduate School of Business.]
This paper describes the relationship between the truthfulness and usefulness of a model or hypothesized relationship between two or more variables and the goodness of fit which results from an empirical investigation of the model or hypothesis. Several obvious discrepancies between model "validity" and goodness of fit results due to such causes as measurement problems, model mis-specification, and stochastic behavior are described. In addition, a procedure for assessing the appropriateness of a linear regression model is presented.
The first step in formal research is often the specification of a model, either explicitly or implicitly. Data is then collected and the model rejected if the data and the model conflict. In the social sciences, this type of model investigation has been labeled predictive testing (Basmann, 1965) . This approach consists of deducing conclusions from the model and seeing if the data supports the conclusions.
The predictive testing approach, in spite of its obvious appeal, is not used widely in most research on consumer behavior. One reason for this is that in much research, a model is not extant prior to data analysis. Even when a model is present, it is often poorly specified. For example. A could be hypothesized to:
1) be related to B
2) be related to B in some specific mathematical way (eg., A = c + dB)
3) be related to B in some specific mathematical way with constraints on the parameters (eg., A = c + dB where c > 1 and 0 < d < 1)
Moreover, it is possible to specify the relationship between A and B to be causal or merely correlational and to assume a large random component exists. Hence lack of certainty over what the model is makes predictive testing very difficult.
Even when a model is specified, the predictive testing approach is often difficult to utilize. In the first place, it requires considerable ingenuity to deduce meaningful conclusions from models which can be falsified by data. This is especially true for models which do not assume causality. Secondly, in many cases several competing models exist, and with notable exceptions (Bass & Clarke, 1972), more than one of them can fail to be rejected. Hence because of difficulties with the predictive testing approach, as well as different training, most research on consumer behavior has tended to focus on goodness of fit measures as an indicator of useful relationships.
Goodness of fit measures have some obvious advantages in model development. First, they are quantitative indices which can be compared across models so that the best of a set of models can be selected. Secondly, there exist established statistical procedures for testing these measures for significance. Third, they can be used as a means of deducing new or modified models from a set of data. Fourth, they appear as output in canned computer programs, which is no small reason for their prominence in reported research results. Yet in spite of their advantages, goodness of fit measures have some important limitations as indicators of the truthfulness and usefulness of models.
LIMITATIONS OF GOODNESS OF FIT MEASURES
It is very easy to rely on goodness of fit measures as a means of evaluating model usefulness. Unfortunately, there are a variety of reasons why perfect goodness of fit measures should not be expected from useful models. These reasons include:
Imperfect Model Operationalization
One obvious situation where goodness of fit measures are not appropriate for estimating a model's usefulness is when the operationalization of the model does not match the model. In other words, the operationalization of the theoretical model may be due to the type of data available for examining the model. Criticisms of research on this basis are widespread (Cohen, et al., 1972; Taylor & Gutman, 1973). On the other hand, a high goodness of fit measure suggests that a useful model which is equally or even more valuable than the original model may be developed because of its empirical goodness of fit.
Goodness of fit measures' ability to estimate a model's usefulness are obviously limited to the population represented by the sample. Hence, for example, if for some reason college students fit a model differently from the general population, studies of "40 college sophomores" are potentially misleading.
A variety of measurement problems occur which will affect the size of goodness of fit measures. For example, individuals may not treat an apparently equal-interval scale as such (Stevens, 1946). In many other situations, a lower order scale (eg., ordinal) will be treated by the researcher as a higher order scale (eg., interval or ratio). In situations where non-interval scales are used in regressions as though they were interval scales, the practical maximum for R2 even assuming a valid model is often far less than 1 (Lehmann, 1972; Morrison, 1973, 1972a, 1972b). On the other hand, questionnaires may induce consistent responses across questions. When such consistency bias is present, the goodness of fit measures among the affected variables are artificially increased.
Minor Model Mis-Specification
It is possible that a model be correctly specified in the sense of containing the correct constructs and links among constructs while incorrectly specifying the mathematical form of the relationships. In such cases, reliance on goodness of fit measures may lead to inappropriate conclusions.
For example, assume attitude is determined by a multi-attribute attitude model which is multiplicative with unequal weights on the attributes: EQUATION where Bji represents the distance from the desired amount of the ith attribute possessed by alternative j, Wi is the importance weight of the ith attribute, and Aj is the individual's attitude toward the jth alternative. In this case, application of the more popular linear model EQUATION can lead to the conclusion that equal weights are superior. Assume two alternatives, M, which is one unit from the desired level on attribute l and four units from the desired level on attribute 2, and N, which is 4 and 2 units from the desired levels on the two attributes. According to the true model (multiplicative with unequal weights), M is the preferred alternative. If the linear model is applied, however, the assumption of the true weights suggests that M and N are equally preferred while the use of equal weights correctly identifies M as the preferred alternative. Hence reliance on goodness of fit would falsely suggest that the importance weights for the two attributes were equal.
A similar problem exists with comparing linear and non-linear forms of relationships. Given the degree of error or randomness in most survey research data (Hulbert & Lehmann, forthcoming), the property of a Taylor series guarantees that most of the relationship between two variables will be accounted for by the first (linear) term. Hence, unless orthogonal polynomials are employed, second and higher order terms will not appear to be very significant due to their high collinearity with the linear term. For this reason, goodness of fit measures tend to indicate that relations among constructs are essentially linear when in fact the relations may not be.
It is very possible that a model will produce a high goodness of fit while being essentially unrelated causally. This can happen because of a spurious correlation among two variables (Blalock, 1964). Hence, two variables A and B can appear related since a third variable C affects both A and B. Unfortunately, the goodness of fit level gives no indication of whether or not a spurious relationship exists.
In many cases a potentially valid model may be applied in an inappropriate way. For example, estimation of multi-attribute models across individuals by means of regression, though popular (Wilkie & Pessemier, 1973), seems an inappropriate application of an individual model (Beckwith & Lehmann, 1973). In such cases, the goodness of fit of the model does little to indicate its usefulness.
In many situations, a model is essentially developed from data rather than tested by it. Hence the goodness of fit of such a model is biased upward. This is especially true when stepwise procedures are employed.
Degree of Freedom Limitations
Goodness of fit measures are obviously biased upward whenever many parameters are estimated in relation to the size of the sample. The obvious remedies to this problem include the reporting of adjusted R2's and using split-half procedures (Morrison, 1969). Unfortunately, split-half procedures require a relatively large sample size which if extant tends to make the goodness of fit bias fairly small.
Many individuals have recently focused attention on the proposition that a considerable portion of individual behavior is random and hence in principle not explainable (Bass, 1974). This means that models with low goodness of fit indices may be both true and useful (Bass, et al., 1968). It also suggests that models with extremely high R2's may be modeling error or random behavior rather than the true underlying process.
THE APPROPRIATENESS OF R2
Probably no measure of goodness of fit receives more attention than the R2 in regression analysis. Most people feel that a good R2 is a big R2, largely due to the influence of economists who are used to dealing with aggregate time-series data. This section will proceed to argue that R2 is a poor indicator of whether a linear model is useful and appropriate.
Much of the research on consumer behavior has resulted in R2's in the .05 and .10 range. As such, it had indicated little individual-level predictive power but significant relationships among variables such as income and TV viewing time. These variables are usually measured on a 5-8 point scale. Hence it is not at all surprising that people with incomes between 5 and 10 thousand dollars vary considerably in the amount of time they spend watching TV. Figure 1 represents a typical situation in survey research.
HYPOTHETICAL REGRESSION EXAMPLE
At this point the question of exactly what model is being tested becomes important. If the model is that income is a deterministic predictor of TV viewing, then the model can be rejected. If, on the other hand, the model says TV viewing is related to income with a large random component, then the R2 becomes a poor indicator of the model's usefulness (Bass. et al.. ]968) .
Assuming the model is stochastic, the real question becomes whether average TV viewing differs across several income groups. This is essentially an analysis of variance problem. An interesting related question, however, is whether income and average TV viewing are linearly related. In order to address this question, it is useful to first partition the variance in the normal manner:
Total variance = Within variance + Between variance.
Next, by partitioning the Between variance we achieve the following situation (Lehmann, 1974):
Total variance = Within variance + Regression Explained variance + Regression Unexplained variance.
Recalling Figure 1, the within variance (which is generally by far the largest component of the variance) is essentially random and unexplainable by the independent variable. Since R2 is:
Regression Explained Variance/Total Variance, it is necessarily low. A more suitable measure of the appropriateness of a linear model would be:
pR2 = Regression Explained Variance/Between Variance
This PR2 would be 1 if the means of the segments fell on a straight line, and hence it measures how well a linear function estimates the effect of the independent variable on the dependent variable.
Two other points about pR2 are important to recognize. First, pR2 is the R2 which would result if the mean behavior for each segment were regressed against the value(s) for each segment on the independent variable(s) with the means weighted by the number of observations in each segment. Second, departures from 1 by R2 can be tested to see if a significant non-linearity exists in the relationship. The null hypothesis of a linear relationship is tested as follows:
Regression UnexPlained Variance/df / Within Variance/df2 .
Derivations for these results are available (Lehmann, 1974).
AN EMPIRICAL EXAMPLE OF PR2
This example is based on a mail survey of 513 homemakers taken in 1968. The relationship between average TV viewing (measured on a 7-point scale) and income (measured on a 6-point scale) appears as Figure 2. From this figure, it appears that income and TV viewing are linearly related. A simple regression produces an R2 of only .048, however, which is neither very encouraging nor enlightening. In order to see if the relationship is really linear, pR2 was calculated (Table 1). The results indicate a pR2 of .878, which is close to 1. The
TV VIEWING VS. INCOME
TV VIEWING VS. INCOME ANOVA
F test gives F = 18.871/4 / 2679.218/507 = 0.9, which is not significant, and and hence the relationship, as thought, is essentially linear.
VALUE OF PR2
The value of the pR2 measure, therefore, is that it indicates whether a linear model appropriately expresses the relationship among a dependent and a set of independent variables. It does not, however, indicate the relative predictive value of independent variables. For example, a pR2 Of .98 could result from variables which are not as useful in prediction as another set with a lower PR2. Hence pR2 merely indicates how well a linear and additive (non-interactive) model explains the effect of a set of discrete independent variables on a dependent variable.
A variety of reasons exist which make the correspondence between goodness of fit and model usefulness less than perfect. This does not mean that goodness of fit measures are of little value in evaluating or developing a model. It does mean, however, that care is required in using goodness of fit measures for model evaluation. This suggests:
1. Be more explicit about exactly what the model is. This would include reporting post-hoe models as such.
2. Consider exactly how high a goodness of fit measure is reasonable given random behavior and measurement error. This means being suspicious of tautologies when R2's get above .6 in cross-sectional regressions of survey data.
3. Attempt whenever possible to utilize predictive testing to examine model validity.
4. Consider using special goodness of fit measures such as pR2 for examining the usefulness of stochastic models.
In summary, goodness of fit measures are useful tools in the hands of sophisticated researchers. A high goodness of fit, however, is neither a sufficient nor even a necessary condition for model usefulness.
Basmann, R. L. On the application of the identifiability test statistic in predictive testing of explanatory economic models. The Econometric Annual of the Indian Economic Journal, 1965, 13, 387-423.
Bass, F. M. Unexplained variance in studies of consumer behavior. In J. U. Farley and J. A. Howard (Eds.), Controlling noise in marketing data. Forthcoming.
Bass, F. M. The theory of stochastic preference and brand switching. Journal of Marketing Research, 1974, 11, 1-20.
Bass, F. M. & Clarke, D. G. Testing distributed lag models of advertising effect. Journal of Marketing Research, 1972, 9, 298-308.
Bass, F. M., Tigert, D. J., & Lonsdale, R. T. Market segmentation-Group versus individual behavior. Journal of Marketing Research, 1968, 5, 264-270.
Beckwith, N. E. & Lehmann, D. R. The importance of differential weights in multiple attribute models of consumer attitude. Journal of Marketing Research, 1973, 10, 141-145.
Blalock, H. M. Causal inferences ln nonexperimental research. New York: W. W. Norton. 1964.
Cohen, J. B., Fishbein, M., & Ahtola, O. T. The nature and uses of expectancy-value models in consumer attitude research. Journal of Marketing Research, 1972, 9, 456-460.
Hulbert, J. & Lehmann, D. R. Assessing the importance of the sources of error in structured survey data. In J. U. Farley & J. A. Howard (Eds.), Controlling noise in marketing data. Forthcoming.
Lehmann, D. R. An index for assessing the appropriateness of a linear regression model. Columbia University Graduate School of Business, working paper, 1974.
Lehmann, D. R. Preference among similar alternatives. Decision Sciences, 1972, 3, 64-82.
Morrison, D. G. Evaluating market segmentation studies: The properties of R2. Management Science, 1973, 11, 1213-1221.
Morrison, D. G. Regression with discrete random variables: the effect on R2. Journal of Marketing Research, 1972a, 9, 338-340.
Morrison, D. G. Upper bounds for correlations between binary outcomes and probabilistic predictions. Journal of the American Statistical Association, 1972b, 67, 68-70.
Morrison, D. G. On the interpretation of discriminant analysis. Journal of Marketing Research, 1969, 6, 156-163.
Stevens, S. S. On the theory of scales of measurement. Science, 1946, 103, 677-680.
Taylor, J. & Gutman, J. A reinterpretation of Farley and Ring's test of the Howard-Sheth model of buyer behavior. In S. Ward and P. Wright (Eds.), Advances in consumer research. Vol. 1. Urbana, Ill.: Association for Consumer Research, 1973. Pp. 438-446.
Wilkie, W. L. & Pessemier, E. A. Issues in marketing's use of multiattribute attitude models. Journal of Marketing Research, 1973, 10, 428-441.


Jean Cavailles and Albert Lautman. La pensee mathematique. Separate papers by the two authors, with discussion by Cartan, Chabauty, Dubreil, Ehresmann, Freche, Hyppolite, Paul Levy, Shrecker, and the authors. Bulletin de la Societe Francaise de Philosophie, vol. 40, no. 1 (1946), pp. 1-39.
This is a report of a public discussion, held February 4, 1939, on views about the nature of mathematics expressed in two doctoral theses (III 167 and V 20) by Cavailles and Lautman, respectively. The brief opening statements made by the authors re-state their published doctrines; there follow critical comments by mathematicians and philosophers, and replies by the fist speakers.
Cavailles held that the attempt to reduce mathematics to logic ("ce fameux espoir de reduire les Mathematiques a la Logique") has failed - mathematics uses notions that are not purely logical, and Godel's work shows the impossibility of representing mathematics in a unique formal system. Accordingly, a philosopher of mathematics is justified in reaching the following positive conclusiions. Mathematics cannot be defined; constitutes a ,becoming ("un devenir"), a reality irreducible to anything except itself ("une realite irreductible a autre chose qu'elle-meme"); and is autonomous ("autonome") in the sense of developing unpredictably; but nevertheless subject to an inner necessity ("autonomie, donc necessite"); and mathematicians are engaged in an experiential activity ("une activite experiementale"); concerned with an object momentarily created by mathematical operations ("toujours correlatif de gestes effectivement accomplis par le mathematicien dans une situation donnee").
Lautman differs in holding mathematical truths to be objective: mathematics participates in a realm of Dialectic ("Dialectique"), constituting a higher reality of Ideas ("une realite plus haute et plus cachee, qui constitute, a mon avis, un veritable monde des Idees"). The task of mathematical philosophy is to erect the theory of Ideas, by describing the ideal structures incarnated ("incarnees") in mathematics, establishing a hierarchy of such structures, and demonstrating the reasons for their applications to the sensible universe ("les raisons de leurs applications a l'Univers sensible").
It is understandable that the comments of the mathematical participants in the discussion show them to be somewhat at a loss. I doubt, however, that their attitude of wondering respect is justified. The metaphysical conclusions presented by Cavailles and Lautman seem to the reviewer to be based upon fragmentary and incomplete evidence and to be expressed in a terminology which can only promote ofuscation. The work of Godel and Skolem, to which so much weight is attached in these remarks, by no means shows the impossibility of "reducing" mathematics to logic' nor is it ever made clear why mathematics should be in the peculiar position of resisting attempts at definition. In the absence of clear understanding of these relatively technical points, it seems premature to engage in metaphysical speculation about the ontological basis of mathematics.
Max Black, The Journal of Symbolic Logic, Vol. 12, No. 1, (Mar., 1947), pp. 21-22


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The stock market does not follow just a linear trend - it has some deviations from a linear function. Some cycles are well-known, such as, four-year presidential cycle, annual and quarterly fiscal reporting cycles, etc. In addition, some cycles are defined by intrinsic characteristic properties of the system. The stock market performance curve can be considered as a sum of the cyclical functions with different periods and amplitudes. It is not easy to analyze the repetition of typical patterns in stock market performance because cycles mask themselves - sometimes they overlap to form an abnormal extremum or offset to form a flat period. It is clear that a simple chart analysis has a certain limit in identifying cycles parameters and using them for predicting.
Addaptron Software has developed Stock Market Analyzer-Predictor SMAP, computer program, which is able not only to extract basic cycles of the stock market (indexes, sectors, or well-traded shares) but also to predict an optimal timing to buy or sell stocks. SMAP calculation mainly based on extracting basic cyclical functions with different periods, amplitudes, and phases from historical quote curve. To detect correctly major cycles, the historical price data are transformed from time domain to frequency domain (spectrum).
At the beginning, instead of calculating the prediction for the time period forward, SMAP does a simulation of forecast on relevant past data in order to estimate the accuracy of prediction with certain parameters (such simulation called "back testing" - the process of evaluating prediction on prior time periods). Then it calculates the prediction for the time period forward using internal optimized parameters. Back testing also allows finding an optimal time frame. The more difference between actual curve and forecasted, the worse prediction accuracy is.
By selecting data with different historical periods, user can identify the major cycles, which have a dominant effect in a particular time frame. To build an extrapolation (predicted curve), SMAP uses the following two-step approach: (1) applying spectral (time series) analysis to decompose the curve into basic functions, (2) composing these functions beyond the historical data. Additionally, SMAP enables finding optimal timing to buy/sell by analyzing months of year, days of month, and days of week (the calculation is based on statistical analysis). Also it can be used for intraday trading (no source data provided).
SMAP has a user-friendly easy-to-use interface. This software is intended for investors with a basic knowledge in stock investing.


My name is Kayla Ferrari. I was born and raised in Crosby, Minnesota until my graduation from Crosby-Ironton High School in 2013. I went to college at Bemidji State University and graduated in the Fall of 2016. I started teaching at Laporte in August of 2017. I have a 5-12 mathematics teaching degree, but I only teach 6th-8th grade math. I am on year four of teaching and I love it. Every day is a challenge and a new adventure!
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This causes one to go back in time to try to find the origins of mathematics, in search for an answer whether maths by itself is (eventually) inherently limited and whether there's a bound for reality, a bound for mathematics or whether both of these worlds will run parallel forever (they are complementary forever), or whether the abstract thought being developed in mathematics will eventually diverge from reality by so much, that we're now dabbling in the abstract model itself to find both problems and solutions within that model, even there's no physical counterpart that would be subject to the abstract problem.
If you look at civilizations as they develop language, at some point in their language they start to associate a "count" of something with a body part. Some civilizations evolve this further to start using more abstract tokens like sticks to count beyond the maximum number of body parts you may have. In simple societies, it is unlikely you need more than the number of parts on your body to explain some concept (you could also modify the definition of how you refer to something). Those which do evolve, eventually use abstract representations to refer to some abstract notion as a "count". This "count" has no other concept other than our visual perception of being some number of concepts.
The numbers 0-9 as we know them now have evolved over a rather long period of time and came to us from India and Arabia. The number system is base-10, which allows for relatively very easy manipulation of the numbers during calculus. For this reason, they were eventually adopted and used over the Roman glyphs that dominated, for example, in Italy at the time.
The reason why numbers became useful are related to trade. The problem with trade is that you need to figure out how much to give of this for how much of that. So the practical problem required some way to refer to some 'count' of this and some 'count' of that, also some notion eventually that 'x' of this equals 'y' of that. Hence, the bartering and trading very quickly gave birth to the notion of equality and thus the equation.
Geometry evolved after that and served to be able to make rather precise calculations about areas of land, as well as how to carve and build appealing feats of engineering, build houses, bridges, etc. Even though not all forms and shapes could be accurately described at that point, there were some basic rules that could be used already to help out in the engineering effort. It is for these purposes already necessary to think in terms of half objects or fractional objects, like a third of a pie or two-thirds down a bridge. Engineering also required the use of unknowns
As you may notice until now, the roots of mathematics are housed in the manipulation of the 'counts' of things... how many meters, how many pears, how many of that for ...
Then Newton came along and decided to use equations not just for static problems, but dynamic problems like apples falling from a tree. And here we also notice introductions of for example the differential equation. What the differential actually does is chop up some event over a larger period of time into many smaller parts, analyze their behavior in these smaller parts and develop a new equation that exhibits how the system changes over some time assuming that there is not significant deviation within that system. For a singular system, i.e. one that does not interact in the abstract model it is given with any other system, this kind of mathematics is very well suited for solving problems.
After Newton, a lot of new discoveries were made primarily on the side of physics. We do not only know how to count cows, trade land and figure out how far something is, but we can also use it to describe movement and how things move in space over time (however, with important assumptions). With Newton and the mathematics thereafter, people started to feed more abstract ideas into the language. Take into account that for every addition to this language, the deliberations have to be tested against the axioms of the language itself in order to provide consistency.
The problem with more abstract ideas is that some notions may have no reality counterparts, or that the elements that they describe in theory cannot be measured because they are either too small or too big (infinity is one such example). Just thinking about infinity itself and whether it existed or not has driven people mad (literally!).
Newtonian equations work very well for situations in which you assume a disturbance and the rest of the system is free of distortions for a certain length of time and this system has consistent and homogeneous properties (friction, etc.). But for different systems, even a very simple pendulum where you deal with oscillations, even the single system without a second interacting pendulum can only be practically computed to some degree of accuracy. That is, the real exact solution is the elaboration of some power series, depending mostly on the amplitude of the system.
So there exists already a rather simple dynamic system for which there's no real exact solution possible, because the power series extends towards infinity. If we use a supercomputer to compute the exact result, we'll never be able to calculate the result solution before the point in time we'll need it. And yet... looking at the real world and looking at a pendulum swing, there's the thing doing it. What's causing this inherent problem in mathematics, where it cannot be used with 100% accuracy on a pendulum (given some assumed mass), but can be used very precisely on the exchange of goods on a market?
There's something about mathematics that's horribly incomplete yet and it's something to do with recursion in mathematics. We need an ability to compute the outcome of recursion sets very, very quickly. The above demonstrates that the model of the real world of mathematics is really just a model and breaks down for certain practical uses of mathematics, depending on the complexity of the situation.


What is Spiritual Geometry?
How can it be assigned to a number?
Part of the answer was given by Pythagoras: a2 + b2 = c2.
This simple-looking equation can be applied to the geometric form of the triangle if you want to look at its purely physical application.
The book, Mrs. Velvet and the Blue String Theory, in fiction, takes that equation and puts the human heart into it and allows it to apply for the ascension of the human soul. The equation shows the code of the path to living daily in a higher and better conscious state than has ever been possible by humans on this planet.
As for math and geometry and putting numbers on space, here is Pythagoras’s theorm, which has gone through “the most proofs of any mathematical theorem,”* both geometric and algebraic:
a2 + b2 = c2
The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
As for spirit and heart and putting numbers on life, here is how to “do the math” according to the notes of Mary Holden done while editing the book Mrs. Velvet and the Blue String Theory, by Linda Heart: The theorem can be applied to psychology in this way: a = denial as a function of thought by the human spirit; b = acceptance as a function of thought/feeling by the human spirit; c = expansion of the human spirit.
Take a look at the little a that represents denial. Denial is a negative. To square the (a2) means a negative attitude is multiplied by itself, another negative attitude. When you multiply a negative quantity by a negative quantity, the result is a positive quantity.
A positive quantity times a positive quantity = A positive quantity result (2 x 2 = 4).
A negative quantity times a positive quantity = A negative quantity result (2 x -2 = -4).
A negative quantity times a negative quantity = A positive quantity result (-2 x -2 = 4).
So, a, or Denial (negative emotion) x a, or Denial (negative emotion) = A positive quantity of emotion surrounding denial that naturally negates denial and turns it into a neutral or even a positive emotion.
When b represents acceptance, the opposite of denial, and it is squared, it is the multiplication of two positive quantities. So, b or Acceptance (positive emotion) x b, or Acceptance (positive emotion) equals a lot of positive emotion.
To put it all together: Denial multiplied by Denial comes out as Acceptance (a2) and then Acceptance multiplied by Acceptance comes out as even more Acceptance (b2) and when they both add together, they form a perfect trinity of lines that form a triangle of acceptance, also squared by itself (c2) to create expansion of the human spirit.
The expansion of the human spirit can only happen after negativity has been changed by itself (multiplied) into a basic form of acceptance, then it is added to more pure acceptance for strength and that is when it equals expansion.
Even when you draw this as geometry (go ahead, make a triangle and then put squares on the sides)…you can see how outside of the triangle, three squares take shape into the space outside of it, and even create more space. That is expansion!
- Wikipedia sourced quotation. By the way, watch this for more fun: http://www.youtube.com/watch?v=kmAxUAh510s. It shows how the Scarecrow character in “The Wizard of Oz” misstates this theorem after being given a Ph.D. in Thinkology.This is funny to me—my degree is a B.A. in English and I got a C in Math 105 (Math for Teachers) when I was a freshman in college. I worked very diligently in that class but it was difficult for me to “see” the quality in quantities.


After hearing and watching her, I told her that she could choose to either do her math or email her teacher as to why she didn't want to do it. She chose to email her teacher. She spent roughly 30 minutes writing an email (doing the math would have taken half the amount of time... but I was curious as to what she'd write) and this is what she sent:
Dear Mrs. Neuschafer,
I am wondering when we are going to be done With reflex. Because when I am doing reflex itIs all mixed up now if you compare it to say..... IXL it is more focused on like certain Things like say I only wanted to practice 5s in multiplication I could just go into the multiplication bar and choose 5s but with reflex it does not stay on like the same fact all the time. And yes I'm kinda persuading you to go more towards IXL but... I am also trying to see if you have looked and seen something better Than reflex. By the way there is also an app for IXL and I practically use it every day And it is a great app.
Riley Nichole Grant
Sent from my iPad


Jerald J. Kovacic, a renowned leader in the application of algebraic techniques to differential equations theory died at his home of glioblastoma multiforme on April 1, 2009. He was 68. He died in the care of his loving partner and now husband of 32 years, Daniel J. Kiser.
The famed Kovacic algorithm for solving second order linear ordinary differential equations has been implemented in commercial and open source scientific software. It is a breakthrough procedure for computing the algebraic group associated with the equation,
called its differential Galois group.
Dr. Kovacic received his Ph.D. from Columbia University in 1968. He was an assistant professor at Princeton University from 1968 until 1970. He taught at Lehman College, Fordham University, and Brooklyn College. From 1981 to 2001, he was computer programmer and technical director at JYACC, a computer services firm in NYC.
Jerald was working on the challenging task of laying the foundations for a new geometry of differential equations, modeled on algebraic geometry. He was webmaster and defacto technical director of the Kolchin Seminar in Differential Algebra (KSDA), Graduate Center, CUNY.
Contributions to City College/KSDA; c/o J. Kovacic, at address below.
Memorial Service Friday, April 24, 6:30 PM, at his home, 29 East 10th Street, Second Floor, NYC.
Click here to view the original New York Times article and to leave condolences, upload photos, etc. on legacy.com


In a post I wrote about this time last year, I described a part of our day that we planned to have this year, and hoped it would be time to explore passions in school. While my dream for this time hasn’t quite come to pass, it has settled into a somewhat reasonable routine for the 40-50 kids I have each day for 40 minutes and my kids are experiencing a part of the day where they have some control over what they do and how they do it. We have some interesting activities going on, and today I decided ask my kids how they would plan out their week if they had total control over it.
I was fascinated by the responses and may go into more detail about that some other time. What surprised me was how much my kids like to play math games, as most of them asked for at least one day to play math games–either online, OR from the collection I have in my room.
(You have to understand I collect interesting math games that work on skills other than rote computation–and I have some rather hard-to-find games, as well as unusual ones, in my collection. So when kids come to my room and I give them games to play, they are strategy games-games that work explicitly on skills like understanding and using probability and/or chance, reinforcing computational skills in context or building visual-spatial orientation and envisioning.)
The kids today asked for time to play those games, such as
- Yahtzee Free For All–which adds the strategy of taking points from others to the simple reinforcement of multiplication skills
- Skip Bo Dice-which requires students to constantly assess whether it is worth the points they have earned to continue rolling the dice on a turn or whether they should pass the cup.
- Blockus Trigon, which requires kids to think Blockus with triangles and triangle-based shapes
- Shift Tac Toe-which pushes kids to think in three dimensions as they can shift the tokens left to right.
- 3-D Tetris, which is just downright HARD for me–and kids love the challenge of moving an object virtually to fit into what I refer to as a rectangular tunnel.
And there are my iPod apps as well–Math Tables has a twist, where both the problem and the answer are provided, and you just need to match the answer to the correct problem. Number Line asks players to place fractions and decimals in a continuum–from least to greatest. It’s pretty tricky, in that the line moves as you put the quantities in place. Both of these are games where kids compete against themselves–both are timed, so they can aim to beat their own time, but we talk smart moves–and how to play so that the interplay between time and increasing accuracy is maximized.
But what brought me to thinking in depth about these games was a tweet from a friend earlier today that made me think about why my kids like these kinds of games so very much. I have an advantage over most teachers in that I get to see kids from all grade levels, K-5, so I can see how kids of varying ages and levels approach the same games. I see, if you will, a progression of what kids think is important in games as they gain more sophistication and mathematical understanding.
What I see from kids who are thoughtful about their learning is a focus on the strategic thinking and sharing ways of working to play the game more efficiently or effectively. For example, as kids play Number Line on the iPod, I hear them say things like “I place the ones I know are close–like .4 and 1/2 –first . Then I scan the others quickly to make sure there are none between them.” Or, “I place the biggest one and the smallest one first, if I can see them quickly.” “I also look for ninths and single digit decimals. They’re usually easy to place because the ninths are just the double digit of the numerator.” As kids share strategies they use, they also help others with clues to converting decimals to fractions and vice versa.
Inexperienced kids playing a game are usually incredibly competitive–so they think about winning. For them, it’s not about the game, or HOW they play it, but about who wins (and loses.) Then, as they gain more experience in my room and see older kids play the games–or they are in games with the older kids, they hear the sharing of strategies. They realize the “big kids” aren’t there to beat them, but instead to try to get better at the game. So the emphasis changes from winning to figuring out how to play the game more effectively. They see that the game is simply a means to thinking.
When are we going to look at school as a means to thinking? How can we foster this kind of learning/sharing/thinking in our classrooms around content more, and more naturally? How can we support learning that isn’t about a grade or an outside evaluation, but about getting better at what you’re trying to do?


A downloadable game for Windows and Android
Learn math the fun way! Help our little character to solve math problems using only his spaceship, or shoot the right answer on the alien moon. Two games in one!
Select your language (English or Spanish), the difficulty, what kind of problem (addition, subtraction, multiplication or division), and have fun!
Available for Windows and Android.
Click download now to get access to the following files:
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So today a friend gave me a book filled with my favorite things. I'm pretty excited to read through each comic and figure out all the little puzzles Randall sneaks into the book. First nerdy thing to notice: the pages are numbered in ternary. A normal nerd would settle for binary, but Randall never settles.
He is brilliant because he writes about the things that he loves, and nothing more.
Sometimes I question why I am a math computer science major. I imagine what it would be like taking the easy way out, picking a field that I'd study less for and do better in. But then I watch the Big Bang Theory and I read xkcd and I fall back in love with life and I realize that it's all worth it.


Yes, you read that title correctly. Duolingo, the language-learning app famous for the owl that haunts my dreams every night, now has a companion app to teach you (and the kids – but mostly you, the readers) math.
To get the most pressing question out of the way: Yes, Owl is present in the new app called Duolingo Math. However, the bird appears to have been compressed into a kind of cube. I’m not sure if it continues to qualify as an owl in this new form factor. I think a mascot reclassification wouldn’t go wrong here, because owls can’t do math. Come on now. This is not fantasyland.
As someone who’s studied multiple languages through Duolingo (you’ll get my 200-day streak out of my cold, dead hands) but after the incorporation of letters that showed no natural affinity for math Lost, I consider myself an ideal candidate for the test out of Duolingo’s math curriculum, which begins with “Multiply 1”. Excellent. This is where I am.
(By the way, I assume that “magnitude” above does not refer to the field of higher mathematics that evaluates the effective diversity of point clouds, but I think it would be unbelievable if they threw it in here. Let’s say I’ll get that out after learning rounding.)
Topics that followed multiplication included division, fractions, areas, perimeters, angles, telling time, and other things that I immediately forgot how to do as soon as I graduated high school. As you progress, you get a vertical progression of lessons that build on each other.
I’ve studied a few multiplication and division units so far, and the experience is aesthetically similar to the language app. The short “ding” you hear when you get the right answer is satisfying, but not the same, that you hear in language courses. Lessons are equally quick, taking less than two minutes to complete. As you complete the lessons, you may see your little circles fill up. If you don’t study as often as you should, you get a passive-aggressive notice from a cube bird of prey.
And as is the case with language apps, no two exercises are the same. Language lessons in Duolingo typically involve you interacting with a bunch of new words in a variety of ways – you hear them, you say them, you type them, you spell them, and hopefully those Some combination of exercises will be successful to cram them into your brain. Mathematica’s approach seems similar. You connect three squares together, and then you match them to a picture of three blocks of four dots, and then you tap the block of four dots three times, and then you count from four to 12, and you hammer that foundation of multiplication really Is Remember in front of you (probably, at some point) that three times four is twelve.
You’ve even handwritten your answers to some questions in a box, which the app does a good job of recognizing and converting typed text. Unfortunately, I cannot support this feature; I got a division query wrong over and over again because it turned out that I was writing a trailing “5”. If I’ve been writing “5” on the back this whole time, I totally blame you all. One of you should have told me.
The similarities between the two apps are no coincidence, says senior software engineer Sammy Siegel, who built “everything you see” into the new math program. Siegel – who has been working on Duolingo Math for over a year and was the sole engineer on the project for most of that time – aimed to set up and practice maths to help Duolingo succeed in the language field .
“We have all these different mechanics to keep people engaged. We wanted to use everything we learned from teaching the language and apply it to another subject,” Siegel tells WebMD. ledge,
“Whether it’s calculating the tip on a check or modifying portions in a recipe, math is really essential to our lives,” Siegel says. “We also know there’s this proliferation of math anxiety where people don’t feel completely comfortable with their math skills. We think that with a fun and engaging app, we can cut down on that and make it easier for us to do it.” Can break for people.”
Of course, the methods of learning math are different from learning languages.
The primary hurdle I see is that most people starting a Duolingo language, regardless of their age, can reasonably be assumed to be beginners in that language because they are using Duolingo. Huh. While some people may get bored at the beginning (I’ve long complained about how long some Duolingo courses take to be any fun at all), the Korean I or whatever downloads with the basic alphabet are perfect. Seems like a step.
But math learners, one would suspect, come with highly variable math knowledge and ability. Figuring out where to start everyone so that the material isn’t too easy or too difficult strikes me as a difficult prospect. You can start on whatever unit you want, but there doesn’t seem to be any kind of comprehensive placement testing.
To address this, Duolingo plans to offer courses at multiple levels based on age. Currently, there’s one for elementary school students (I can only assume it’s what I’m into) and one of the functions geared more towards adults who want to brush up. (I’m too afraid to try it because as Siegel mentioned it will involve “difficult numbers, like multi-digit multiplication”. Rest of you.)
More levels, including some for high school, aren’t out of the question, though the team is currently focused on the two courses they already have.
My other concern is that the language app has been great (in my case) at helping me remember words, but I’m less good at retaining grammatical concepts in the long run – and the latter is less good at learning math. seems much more necessary. For example, in the Duolingo language app, I find myself repeating a specific sentence over and over, which helps me remember that sentence, but doesn’t necessarily help me use the words in that sentence in other sentences. be easy to do.
To address this, there is a level of randomness in the numbers that appear in every math exercise. There are some constraints, which vary by exercise, but the goal is that two people doing the same lesson side-by-side won’t always have the same problems. I haven’t seen any really funny numbers as a result of this yet, but I’m looking for one.
And to address the last thing I recently complained about with the language app: No part of Duolingo Math is behind the company’s $6.99/month paywall (at least for now). “We want to make sure we get a product-market fit before we start thinking about monetization,” Siegel says. Wow.
In case I haven’t completely clarified this fact yet: I’m not an expert in math, education, or any subject. But it was a fun app to use.
While the content is clearly aimed at people younger than me, I’ve challenged myself in parts (for which I’m not ashamed, as I have a calculator on my phone that can do this stuff for me. Kiddos, your There is much to look forward to. There’s some creativity and variety in practice that puts them a step above school on the fun scale. I can definitely see how it would speed up my mental math, and make me think more deeply about what I’m doing when I write a tip on a receipt.
The iOS and iPadOS Duolingo Math app is currently available in beta, which you can join now, with a full launch later this year.


Available in all subroutines.
x to an integer value less than or equal in absolute value.
In other words, round
x towards zero. Negative values will be rounded up
towards zero and positive values will be rounded down towards zero.
For example, 2.2, 2.5, and 2.7 all truncate to 2.0.
This is equivalent to formatting the number to base ten and removing all digits after the decimal point.
math.NAN, a NaN will be returned.
x is integral, ±0,
x itself is returned.
Otherwise, the rounded value of
x is returned.
User contributed notes
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Yale researchers were very disappointed when they tried to dispel conservative estimates of how many illegal immigrants are in the United States.
They found an average number nearly twice as high as the formerly accepted statistic.
A new Yale study, using mathematical modeling on a range of demographic and immigration operations data, suggests that the population of undocumented immigrants in the United States may be more than 22 million—more than twice what was previously believed. https://t.co/XXcMMWHqYM— Yale Insights (@YaleInsights) September 21, 2018
Using mathematical modeling on a range of demographic and immigration operations data, the researchers estimate there are 22.1 million undocumented immigrants in the United States. Even using parameters intentionally aimed at producing an extremely conservative estimate, they found a population of 16.7 million undocumented immigrants.
They were seriously trying to find a lower number!! LOL!!
The results, published in PLOS ONE, surprised the authors themselves. They started with the extremely conservative model and expected the results to be well below 11.3 million.
“Our original idea was just to do a sanity check on the existing number,” says Edward Kaplan, the William N. and Marie A. Beach Professor of Operations Research at the Yale School of Management. “Instead of a number which was smaller, we got a number that was 50% higher. That caused us to scratch our heads.”
Jonathan Feinstein, the John G. Searle Professor of Economics and Management at Yale SOM, adds, “There’s a number that everybody quotes, but when you actually dig down and say, ‘What is it based on?’ You find it’s based on one very specific survey and possibly an approach that has some difficulties. So we went in and just took a very different approach.”
The actual methodology they used is fascinating, but frankly, 11 million or 22 million, something needs to be done. That is an enormous number either way and Trump seemed to be the only candidate who took it seriously. We can still treat people humanely and with compassion, and with due process rights, but this is a problem that Americans just shouldn’t have to deal with.