The “One of Many” Fallacy

Nice one!


I’ve been on book tour for nearly a month now, and I’ve come across a bunch of arguments pushing against my book’s theses. I welcome them, because I want to be informed. So far, though, I haven’t been convinced I made any egregious errors.

Here’s an example of an argument I’ve seen consistently when it comes to the defense of the teacher value-added model (VAM) scores, and sometimes the recidivism risk scores as well. Namely, that the teacher’s VAM scores were “one of many considerations” taken to establish an overall teacher’s score. The use of something that is unfair is less unfair, in other words, if you also use other things which balance it out and are fair.

If you don’t know what a VAM is, or what my critique about it is, take a look at this post, or read my book. The very short version is that it’s…

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Denis Ian: Competency Based Education and the End of Education

Kids need hammers.

Diane Ravitch's blog

Denis Ian warns that “competency based education,” online teaching and assessment, spells the end of education and of childhood. It is not just a threat to public education. It is a mortal threat to education of any kind.

He posted this comment:

Competency based education isn’t a mirage anymore. It’s here.

Beyond the view of skirmishes now underway across an array of states, is an emerging reality that … in a very short while … this destroying reform will have razed an American institution to a mound of rubble.

And in its place … for as far as the eye can see … will stand drive-thru learning centers offering kiosk-educations from a B. F. Skinner touch-screen that will supply the finger-pointer with all they need to succeed in a life of rich monotony.

That’s what your now titling schools are going to look like. And that’s your child’s purgatory. Dante…

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Split a length into 5 equal pieces – fractions

The parallel equally spaced lines

and the  desired  length HI, of ribbon, wood, anything non-elastic.



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1/5 is one fifth of the length of a line segment of one unit – but how?

This comes from the Common Core

Develop understanding of fractions as numbers.
1. ……
2. Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the
interval from 0 to 1 as the whole and partitioning it into b equal
parts. Recognize that each part has size 1/b and that the endpoint
of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off
a lengths 1/b from 0. …….

….but how do you do it ?????

The mystery is solved…….

Here is the line, with 0 and 1 marked. /You chose it already !


Here is a numbered line, any size, equally spaced, at intervals of one unit.

It only has to start from zero.


Now construct the line from point 5 to the “fraction” line at point 1, and a parallel line from point 1 on the numbered line.


The point of intersection of the parallel line and the “fraction” line is then 1/5 of the distance from 0 to 1 on the “fraction” line.


1/5, 2/5, 3/5, 4/5 and 1 are equally spaced on the fraction line.

L cannot be moved in the static picture.


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Adding fractions – phew!

Who needs LCM ?

First, three views of LCM with no comments :

1: Change them to equivalent fractions that will have equal
denominators. As the common denominator, choose the LCM of
the original denominators. Then the larger the numerator, the
larger the fraction.

2: Jun 26, 2011 – If b and d were same it was easy to find LCM
since if denominators are same, we just need to find LCM of
numerators, hence LCM of (a/b) and (c/b) would be LCM(a,c)/b.
So we have to first make denominators of both the fractions same.
Multiply numerator and denominator of first fraction by LCM

3: The GCF and LCM are the underlying concepts for finding
equivalent fractions and adding and subtracting fractions, which
students will do later.


Now we can do fraction addition without LCM. It just needs the use of the distributive law, and the result shows the way in which the divisors combine.


And now using 3/4


But the best one is via multiplication ……


Now for multiplication and division.





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The Sun Also Rises

 What an inspiring country!!!! Guess where I’ve been the last couple weeks. Let me give you a hint: There is virtually no litter, no profanity, no violent crime, no graffiti, and no public smoking. And I’m not talking about just those civil middle class folks (although this country claims that 90% of its people are… Read more »

Source: The Sun Also Rises


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Pokemon Go – a new purpose for Edtech

I found this on Medium

Gracias to Junaid Mubeen

Oxford Mathematician turned educator. @HGSE ’12. Head of Product @MathsWhizzTutor. Long-distance runner. Anagrams.
18 hrs ago6 min read

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George Lakoff: Understanding the Appeal of Donald Trump

The linked article is superb.Share it please.

Diane Ravitch's blog

George Lakoff, the psycholinguist, is expert in explaining how people respond to verbal messages. His book “Don’t Think of an Elephant,” was a best-seller.

I met Lakoff a few years back and asked him about how to frame issues in the education debate. We spent two hours talking. He left a lasting lesson with me: liberals think that people are persuaded by facts; conservatives persuade with narratives, not facts.

In this important article, he explains the reason for Trump’s success: Trump is the Father, the strong authoritarian father who will protect us and keep us safe from all threats.

“In the 1900s, as part of my research in the cognitive and brain sciences, I undertook to answer a question in my field: How do the various policy positions of conservatives and progressives hang together? Take conservatism: What does being against abortion have to do with being for owning guns?…

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What are negative numbers?

The question really is “What are the positive and negative numbers ?”

First of all we have “the numbers”. 1, 2, 3, … and zero for completeness sake, and also that “none” is a number, and “nothing” is not a number.
“The numbers” are actually very different from one another. The counting numbers, 0, 1, 2, … , are very different from the measuring numbers. Measuring numbers have “quantity”, counting numbers have “counts”.
Measuring numbers are cakes, pizzas, watts, feet, mass, volume, area, and so on, where the quantity is “some” or “none”.
Examples of measuring numbers are “half of a foot”, “2/3 of a pizza”, “0.05 square feet”, and they are just “numbers”, attached to units of measurement.

Now there is a problem.

“Apply and extend previous understandings of numbers to the system of rational numbers.” (quote from CCSS)
Just like that !!!!!!
There is NO easy extension, “half of a foot” is NOT extendable to “three feet below sea level”.
The meaning of “positive and negative numbers”, or the “signed numbers”, is not a “some or none” situation at all.
The “signed numbers” are abstractions of “relative position” and “change of position”, and the position of “zero” is often, if not always, arbitrary.

A temperature scale has a zero, and temperatures above zero are “positive”, temperatures below zero are “negative”.

A different temperature scale has a different zero, and, worse still, the scale factors (scales) are different as well.

“Feet above sea level” and “meters above Mount St Helens” are similarly “different”.

An electrical circuit can have chosen a voltage value of zero at any point in the circuit.

In these and all similar situations the zero is chosen by a human, and not as the “none or nothing” value.
The value 9, or the value -5, is marked on the scale as a position relative to the zero on that scale.
Marks on the left, or the “down side”, are conventionally the “negative” marks, and marks on the right, or the “up side” are the “positive” marks.
The positive marks. “+”, are conventionally ignored, but at the start one should put them in.

… to be continued, when the confusion between “negative” and “subtraction” is resolved.

And this CCSS bit is so stunningly superficial.
Grade 6
Apply and extend previous understandings of numbers to the system
of rational numbers.
5. Understand that positive and negative numbers are used together
to describe quantities having opposite directions or values (e.g.,
temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts,
explaining the meaning of 0 in each situation.
6. Understand a rational number as a point on the number line. Extend
number line diagrams and coordinate axes familiar from previous
grades to represent points on the line and in the plane with negative
number coordinates.
a. Recognize opposite signs of numbers as indicating locations
on opposite sides of 0 on the number line; recognize that the
opposite of the opposite of a number is the number itself, e.g.,
–(–3) = 3, and that 0 is its own opposite.

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