text:
GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 16 Oct 2019, 12:41

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

For which of the following functions f is f(x) = f(1-x) for all x?

Author Message
TAGS:

Hide Tags

Manager
Joined: 05 Oct 2008
Posts: 217
For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

25 Oct 2009, 02:34
22
177
00:00

Difficulty:

45% (medium)

Question Stats:

68% (02:03) correct 32% (02:04) wrong based on 2791 sessions

HideShow timer Statistics

For which of the following functions f is f(x) = f(1-x) for all x?

A. $$f(x) = 1 - x$$

B. $$f(x) = 1 - x^2$$

C. $$f(x) = x^2 - (1 - x)^2$$

D. $$f(x) = x^2*(1 - x)^2$$

E. $$f(x) = \frac{x}{(1 - x)}$$
Math Expert
Joined: 02 Sep 2009
Posts: 58381
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

28 Oct 2009, 16:17
140
141
study wrote:
For which of the following functions f is f(x) = f (1-x) for all x?

A. f (x) = 1 - x
B. f (x) = 1 - x^2
C. f (x) = x^2 - (1 - x)^2
D. f (x) = x^2 (1 - x)^2
E. f (x) = x/(1 - x)

I've already seen quite a few GMAT questions of this type. They are quite easy to solve, since you understand the concept.

$$f(x)="some \ expression \ with \ variable \ x"$$, means that the value of $$f(x)$$ can be found by calculating the expression for the particular $$x$$.

For example: if $$f(x)=3x+2$$, what is the value of $$f(3)$$? Just plug $$3$$ for $$x$$, $$f(3)=3*3+2=11$$, so if the function is $$f(x)=3x+2$$, then $$f(3)=11$$.

There are some functions for which $$f(x)=f(-x)$$. For example: if we define $$f(x)$$ as $$f(x)=3*x^2+2$$, the value of $$f(x)$$ will be always positive and will give the following values: for $$x=-5$$, $$f(x)=3*(-5)^2+2=77$$; for $$x=0$$, $$f(0)=3*0^2+2=2$$. Please note that $$f(x)$$ in this case is equal to $$f(-x)$$, meaning that for positive values of $$x$$ you'll get the same values of $$f(x)$$ as for the negative values of $$x$$.

So, basically in original question we are told to define the expression, for which $$f(x)=f(1-x)$$, which means that plugging $$x$$ and $$1-x$$ in the expression must give same result.

A. $$f(x)=1-x$$ --> $$1-x$$ is the expression for $$f(x)$$, we want to find whether the expression for $$f(1-x)$$ would be the same: plug $$1-x$$ --> $$f(1-x)=1-(1-x)=x$$. As $$1-x$$ and $$x$$ are different, so $$f(x)$$ does not equal to $$f(1-x)$$.

The same with the other options:

(A) $$f(x)=1-x$$, so $$f(1-x)=1-(1-x)=x$$ --> $$1-x$$ and $$x$$: no match.

(B) $$f(x)=1-x^2$$, so $$f(1-x)=1-(1-x)^2=1-1+2x-x^2=2x-x^2$$ --> $$1-x^2$$ and $$2x-x^2$$: no match.

(C) $$f(x)=x^2-(1-x)^2=x^2-1+2x+x^2=2x-1$$, so $$f(1-x)=2(1-x)-1=1-2x$$ --> $$2x-1$$ and $$1-2x$$: no match.

(D) $$f(x)=x^2*(1-x)^2$$, so $$f(1-x)=(1-x)^2*(1-1+x)^2=(1-x)^2*x^2$$ --> $$x^2*(1-x)^2$$ and $$(1-x)^2*x^2$$. Bingo! if $$f(x)=x^2*(1-x)^2$$ then $$f(1-x)$$ also equals to $$x^2*(1-x)^2$$.

Still let's check (E)

(E) $$f(x)=\frac{x}{1-x}$$ --> $$f(1-x)=\frac{1-x}{1-1+x}=\frac{1-x}{x}$$. $$\frac{x}{1-x}$$ and $$\frac{1-x}{x}$$: no match.

But this problem can be solved by simple number picking: plug in numbers.

As stem says that "following functions f is f(x) = f (1-x) for all x", so it should work for all choices of $$x$$.

Now let $$x$$ be 2 (note that: -1, 0, and 1 generally are not good choices for number picking), then $$1-x=1-2=-1$$. So we should check whether $$f(2)=f(-1)$$.

(A) $$f(2)=1-x=1-2=-1$$ and $$f(-1)=1-(-1)=2$$ --> $$-1\neq{2}$$;

(B) $$f(2)=1-x^2=1-4=-3$$ and $$f(-1)=1-1=0$$ --> $$-3\neq{0}$$;

(C) $$f(2)=x^2-(1-x)^2=x^2-1+2x+x^2=2x-1=2*2-1=3$$ and $$f(-1)=2*(-1)-1=-3$$ --> $$3\neq{-3}$$;

(D) $$f(2)=x^2*(1-x)^2=(-2)^2*(-1)^2=4$$ and $$f(-1)=(-1)^2*2^2=4$$ --> $$4=4$$, correct;

(E) $$f(2)=\frac{x}{1-x}=\frac{2}{1-2}=-2$$ and $$f(-1)=\frac{-1}{1-(-1)}=-\frac{1}{2}$$ --> $$-2\neq{-\frac{1}{2}}$$.

It might happen that for some choices of $$x$$ other options may be "correct" as well. If this happens, just pick some other number for $$x$$ and check again these "correct" options only.

Similar questions:
for-which-of-the-following-functions-is-f-a-b-f-a-f-b-for-93184.html
for-which-of-the-following-functions-is-f-a-b-f-b-f-a-124491.html
let-the-function-g-a-b-f-a-f-b-143311.html

Hope it helps.
_________________
SVP
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1824
Concentration: General Management, Nonprofit
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

03 Jul 2010, 23:15
13
5
Okay, so you need to check if f(x) = f(1-x)

Let's do this one by one.

Case A: - Wrong
$$f(x) = 1-x$$
$$f(1-x) = 1-(1-x) = x$$

So we clearly see that f(x) is not f(1-x)

Case B: Wrong

$$f(x) = 1-x-x^2$$
$$f(1-x) = 1-(1-x)-(1-x)^2 = x - 1 - x^2 + 2x = 3x-1-x^2$$

Case C: Wrong

$$f(x) = x^2 - (1-x)^2 = x^2 - 1 +2x - x^2= 2x-1$$
$$f(1-x) = (1-x)^2 - (1-(1-x))^2 = (1-x)^2 - (x)^2= 1-2x$$

Case D: Right

$$f(x) = x^2 (1-x)^2$$
$$f(1-x) =(1-x)^2 (1-(1-x))^2= (1-x)^2(x)^2=f(x)$$

Let's check E just to be sure.

Case E: Wrong

$$f(x) = \frac{x}{1-x}$$
$$f(1-x) = \frac{1-x}{1-(1-x)}=\frac{1-x}{x}$$
General Discussion
Kaplan GMAT Instructor
Joined: 21 Jun 2010
Posts: 67
Location: Toronto
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

04 Jul 2010, 22:33
4
Bunuel:

sometimes, I almost die from the awesomeness of your posts.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9705
Location: Pune, India
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

29 Dec 2010, 21:38
22
4
metallicafan wrote:
Is there a faster way to solve this question rather than replacing each "x" by (1-x)? Thanks!

For which of the following functions $$f$$ is $$f(x) = f(1-x)$$ for all x?

A. $$f(x) = 1-x$$
B. $$f(x) = 1-x^2$$
C. $$f(x) = x^2 - (1-x)^2$$
D. $$f(x) = (x^2)(1-x)^2$$
E. $$f(x) = x / (1-x)$$

Tip: Try to first focus on the options where terms are added/multiplied rather than subtracted/divided. They are more symmetrical and a substitution may not change the expression. I will intuitively check D first since it involves multiplication of the terms.
_________________
Karishma
Veritas Prep GMAT Instructor

Manager
Joined: 12 Oct 2011
Posts: 154
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

23 Jan 2012, 00:40
3
This one is quite simple. We have to substitute (1 - x) for x in each of the options and see which option yields the same result for both f(x) and f(1 - x).

However, careful observation of the options will easily tell you that only D will fulfill this condition because in D, the x now becomes (1 - x) and (1 - x) now becomes x. The overall outcome will remain the same. You don't even need to expand or do any calculations whatsoever.
_________________
Consider KUDOS if you feel the effort's worth it
Math Expert
Joined: 02 Sep 2009
Posts: 58381
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

12 Oct 2012, 03:41
1
1
Similar question to practice: http://gmatclub.com/forum/functions-pro ... ml#p717196
_________________
Senior Manager
Joined: 13 Aug 2012
Posts: 401
Concentration: Marketing, Finance
GPA: 3.23
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

13 Dec 2012, 00:40
1
A.
$$1-x ? 1-(1-x)$$
$$1-x ? 1-1+x$$
$$1-x ? x$$NOT EQUAL!

B.
$$1-x^2 ? 1-(1-x)^2$$
$$1-x^2 ? 2x + x^2$$ NOT EQUAL!

C.
$$x^2 - (1-x)^2 ? (1-x)^2 - (1-(1-x))^2$$
$$x^2 - (1-x)^2 ? (1-x)^2 - x^2$$ NOT EQUAL!

D.
$$x^2(1-x)^2 ? (1-x)^2(1-(1-x))^2$$
$$x^2(1-x)^2 ? (1-x)^2(x)^2$$ EQUAL!

_________________
Impossible is nothing to God.
Manager
Joined: 12 Mar 2012
Posts: 79
Location: India
Concentration: Technology, Strategy
GMAT 1: 710 Q49 V36
GPA: 3.2
WE: Information Technology (Computer Software)
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

23 Mar 2013, 22:01
5
You need to check for every option.
Either you can replace x in every option with 1-x or you can take x = 1 and then check for which option f(1) = f(1-1) = f(0).
In A f(1) = 1-1 = 0. f(0) = 1-0 = 1.
In B f(1) = 1 - 1^2 = 0. f(0) = 1 - 0 = 1
In C f(1) = 1^2 - (1 - 1 )^2 = 1. f(0) = 0 - (1-0)^2 = 0 - 1 = -1
in D f(1) = 1(1-1)^2 = 0. f(0) = 0.(1-0)^2 = 0. ( Right answer )
And for option E you can take x = 2 because if you take x = 1 in denominator the denominator becomes zero, which makes the f(x) undefined.
f(2) = 2/(1 - 2) = -2. f(1-2) = f(-1) = -1/(1-(-1)) = -1/2.

Please give a kudo if you like my explanation.
Intern
Joined: 23 Jan 2014
Posts: 2
Location: India
WE: Consulting (Consulting)
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

27 Mar 2014, 10:06
1
Thank you Bunuel.

I didn't understand till the time i read about the option of picking numbers.

Even then it took a while for me to understand that i have to check both the values (-1 and 2) and check if i get the same answer.

Please do let me know if there are more similar questions besides the ones you mentioned.

Thank you

Rajat
Math Expert
Joined: 02 Sep 2009
Posts: 58381
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

27 Mar 2014, 10:47
rajatsp wrote:
Please do let me know if there are more similar questions besides the ones you mentioned.

Thank you

Rajat

Here are several more:
for-which-of-the-following-functions-does-f-x-f-2-x-155813.html
for-which-of-the-following-does-f-a-f-b-f-a-b-164979.html
for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html

Hope this helps.
_________________
SVP
Joined: 06 Nov 2014
Posts: 1873
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

12 Jun 2016, 22:33
Given: f(x)= f(1-x)
This means if we replace x by 1 - x in the function, still the result should be same.

Checking each option:

A f(x)=1-x
f(1 - x) = 1 - (1-x) = x. Not equal to f(x)
INCORRECT

B f(x)=1-x^2
f(1-x) = 1- (1-x)^2 = 1 - (1 +x^2 - 2x). Not equal to f(x)
INCORRECT

C f(x)=x^2-(1-x)^2
f(1-x) = (1-x)^2 - (1 - 1 +x)^2 = (1-x)^2 -x^2. Not equal to f(x)
INCORRECT

D f(x)=x^2(1-x)^2
f(1-x) = (1-x)^2(1 - 1 + x)^2 = (1-x)^2*x^2.
This is equal to f(x)
CORRECT

E f(x)= x/(1-x)
f(1-x) = 1-x/1-x + x = (1-x)/x. Not equal to f(x)
INCORRECT

Correct Option: D
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8069
Location: United States (CA)
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

30 Mar 2017, 16:24
1
study wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

A. f(x) = 1 - x
B. f(x) = 1 - x^2
C. f(x) = x^2 - (1 - x)^2
D. f(x) = x^2*(1 - x)^2
E. f(x) = x/(1 - x)

Since we are not given any restrictions on the value of x, let’s let x = 1. Thus, we are determining for which of the following functions is f(1) = f(1-1), i.e., f(1) = f(0). Next, we can test each answer choice using our value x = 1.

A. f(x) = 1 - x

f(1) = 1 - 1 = 0

f(0) = 1 - 0 = 1

Since 0 does not equal 1, A is not correct.

B. f(x) = 1 - x^2

f(1) = 1 - 1^2 = 1 - 1 = 0

f(0) = 1 - 0^2 = 1 - 0 = 1

Since 0 does not equal 1, B is not correct.

C. f(x) = x^2 - (1 - x)^2

f(1) = 1^2 - (1 - 1)^2 = 1 - 0 = 1

f(0) = 0^2 - (1 - 0)^2 = 0 - 1 = -1

Since 1 does not equal -1, C is not correct.

D. f(x) = x^2*(1 - x)^2

f(1) = 1^2*(1 - 1)^2 = 1(0)= 0

f(0) = 0^2*(1 - 0)^2 = 0(2) = 0

Since 0 equals 0, D is correct.

Alternate Solution:

Let’s test each answer choice using x and 1 - x.

A. f(x) = 1 - x

f(x) = 1 - x

f(1 - x) = 1 - (1 - x) = x

Since 1 - x does not equal x, A is not correct.

B. f(x) = 1 - x^2

f(x) = 1 - x^2

f(1 - x) = 1 - (1 - x)^2 = 1 - (1 + x^2 -2x) = 2x - x^2

Since 1 - x^2 does not equal 2x - x^2, B is not correct.

C. f(x) = x^2 - (1 - x)^2

f(x) = x^2 - (1 - x)^2 = x^2 - (1 + x^2 - 2x) = 2x - 1

f(1 - x) = (1 - x)^2 - (1 - (1 - x))^2 = 1 + x^2 - 2x - x^2 = 1 - 2x

Since 2x - 1 does not equal 1 - 2x, C is not correct.

D. f(x) = x^2*(1 - x)^2

f(x) = x^2*(1 - x)^2

f(1 - x) = (1 - x)^2*(1 - (1 - x))^2 = (1 - x)^2*x^2

Since x^2*(1 - x)^2 equals (1 - x)^2*x^2, D is correct.

_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager
Joined: 31 Dec 2016
Posts: 68
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

18 Jul 2017, 17:37
Most people seem to suck at explaining these function related questions. Here is a good video explaining some function skills that helps on other questions. https://www.youtube.com/watch?v=T6-Zdr5w_bE

The key to these questions is understanding that f(x) is the function. Meaning that f(3) would mean everytime you see an x you sub in a 3. Or in this case we sub in an X-3. So for instance the first question is F(x) = 1-x............. Imagine the X is a blank or a blank parentheses waiting to be filled in by a number. so F(3) = 1-3 or F(x) = 1 - X or F (blank)= 1-(blank) or F(1-x) = 1-(1-x)
Attachments

Function question 6.png [ 417.34 KiB | Viewed 11580 times ]

Intern
Joined: 20 Aug 2018
Posts: 25
For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

24 Nov 2018, 17:24
Think of functions in terms of inputs and outputs. We want a function for which the input of x will lead to exactly the same result as the input of (1-x). Below is a video explanation

For a function question, it is almost always best to pick numbers, and to choose numbers that are small and manageable. Lets chose 1 for x. The quantity (1-x) would therefore equal 0.

If you put each of those inputs into the function in answer choice A you'll see very quicky that the two outputs are not equal to each other. You'll see very quickly that answer B is also incorrect. Answer C takes a bit more time. The correct answer is D.

The question if easiest to answer if you create separate columns for f(x) and f(1-x). Once again, a video explanation can be found here:
_________________
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4006
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

29 Dec 2018, 09:53
1
Top Contributor
study wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

A. $$f(x) = 1 - x$$

B. $$f(x) = 1 - x^2$$

C. $$f(x) = x^2 - (1 - x)^2$$

D. $$f(x) = x^2*(1 - x)^2$$

E. $$f(x) = \frac{x}{(1 - x)}$$

Let's try plugging in an easy value for x. How about x = 0.
So, we can reword the question as, For which of the following functions is f(0)=f(1-0)
In other words, we're looking for a function such that f(0) = f(1)

A) f(x)=1-x
f(0)=1-0 = 1
f(1)=1-1 = 0
Since f(0) doesn't equal f(1), eliminate A

B) f(x) = 1 - x^2
f(0) = 1 - 0^2 = 1
f(1) = 1 - 1^2 = 0
Since f(0) doesn't equal f(1), eliminate B

C) f(x) = x^2 - (1-x)^2
f(0) = 0^2 - (1-0)^2 = -1
f(1) = 1^2 - (1-1)^2 = 1
Since f(0) doesn't equal f(1), eliminate C

D) f(x) = x^2(1-x)^2
f(0) = 0^2(1-0)^2 = 0
f(1) = 1^2(1-1)^2 = 0
Since f(0) equals f(1), keep D for now

E) f(x) = x/(1-x)
f(0) = 0/(1-0) = 0
f(1) = 1/(1-1) = undefined
Since f(0) doesn't equal f(1), eliminate E

Since only D satisfies the condition that f(x)=f(1-x) when x=0, the correct answer is D

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Non-Human User
Joined: 09 Sep 2013
Posts: 13210
Re: For which of the following functions f is f(x) =  [#permalink]

Show Tags

09 Oct 2019, 04:26
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: For which of the following functions f is f(x) =   [#permalink] 09 Oct 2019, 04:26
Display posts from previous: Sort by


text:
• ### Domain wall QCD with physical quark masses(1411.7017)

May 16, 2016 hep-lat
We present results for several light hadronic quantities ($f_\pi$, $f_K$, $B_K$, $m_{ud}$, $m_s$, $t_0^{1/2}$, $w_0$) obtained from simulations of 2+1 flavor domain wall lattice QCD with large physical volumes and nearly-physical pion masses at two lattice spacings. We perform a short, O(3)%, extrapolation in pion mass to the physical values by combining our new data in a simultaneous chiral/continuum `global fit' with a number of other ensembles with heavier pion masses. We use the physical values of $m_\pi$, $m_K$ and $m_\Omega$ to determine the two quark masses and the scale - all other quantities are outputs from our simulations. We obtain results with sub-percent statistical errors and negligible chiral and finite-volume systematics for these light hadronic quantities, including: $f_\pi$ = 130.2(9) MeV; $f_K$ = 155.5(8) MeV; the average up/down quark mass and strange quark mass in the $\bar {\rm MS}$ scheme at 3 GeV, 2.997(49) and 81.64(1.17) MeV respectively; and the neutral kaon mixing parameter, $B_K$, in the RGI scheme, 0.750(15) and the $\bar{\rm MS}$ scheme at 3 GeV, 0.530(11).
• ### Continuum Limit of $B_K$ from 2+1 Flavor Domain Wall QCD(1012.4178)

Dec. 19, 2010 hep-ph, hep-lat
We determine the neutral kaon mixing matrix element $B_K$ in the continuum limit with 2+1 flavors of domain wall fermions, using the Iwasaki gauge action at two different lattice spacings. These lattice fermions have near exact chiral symmetry and therefore avoid artificial lattice operator mixing. We introduce a significant improvement to the conventional NPR method in which the bare matrix elements are renormalized non-perturbatively in the RI-MOM scheme and are then converted into the MSbar scheme using continuum perturbation theory. In addition to RI-MOM, we introduce and implement four non-exceptional intermediate momentum schemes that suppress infrared non-perturbative uncertainties in the renormalization procedure. We compute the conversion factors relating the matrix elements in this family of RI-SMOM schemes and MSbar at one-loop order. Comparison of the results obtained using these different intermediate schemes allows for a more reliable estimate of the unknown higher-order contributions and hence for a correspondingly more robust estimate of the systematic error. We also apply a recently proposed approach in which twisted boundary conditions are used to control the Symanzik expansion for off-shell vertex functions leading to a better control of the renormalization in the continuum limit. We control chiral extrapolation errors by considering both the NLO SU(2) chiral effective theory, and an analytic mass expansion. We obtain $B_K^{\msbar}(3 GeV) = 0.529(5)_{stat}(15)_\chi(2)_{FV}(11)_{NPR}$. This corresponds to $\hat{B}_K = 0.749(7)_{stat}(21)_\chi(3)_{FV}(15)_{NPR}$. Adding all sources of error in quadrature we obtain $\hat{B}_K = 0.749(27)_{combined}$, with an overall combined error of 3.6%.
• ### Kl3 semileptonic form factor from 2+1 flavour lattice QCD(0710.5136)

Dec. 22, 2008 hep-ph, hep-lat
We present the first results for the Kl3 form factor from simulations with 2+1 flavours of dynamical domain wall quarks. Combining our result, namely f_+(0)=0.964(5), with the latest experimental results for Kl3 decays leads to |V_{us}|=0.2249(14), reducing the uncertaintity in this important parameter. For the O(p^6) term in the chiral expansion we obtain \Delta f=-0.013(5).
• ### Physical Results from 2+1 Flavor Domain Wall QCD and SU(2) Chiral Perturbation Theory(0804.0473)

April 3, 2008 hep-lat
We have simulated QCD using 2+1 flavors of domain wall quarks on a $(2.74 {\rm fm})^3$ volume with an inverse lattice scale of $a^{-1} = 1.729(28)$ GeV. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617 and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter $B_K$ and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulae from both approaches fit our data for light quarks, we find the higher order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the $\Omega$ baryon, and the $\pi$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find $f_\pi = 124.1(3.6)_{\rm stat}(6.9)_{\rm syst} {\rm MeV}$, $f_K = 149.6(3.6)_{\rm stat}(6.3)_{\rm syst} {\rm MeV}$ and $f_K/f_\pi = 1.205(0.018)_{\rm stat}(0.062)_{\rm syst}$. Using non-perturbative renormalization to relate lattice regularized quark masses to RI-MOM masses, and perturbation theory to relate these to $\bar{\rm MS}$ we find $m_{ud}^{\bar{\rm MS}}(2 {\rm GeV}) = 3.72(0.16)_{\rm stat}(0.33)_{\rm ren}(0.18)_{\rm syst} {\rm MeV}$ and $m_{s}^{\bar{\rm MS}}(2 {\rm GeV}) = 107.3(4.4)_{\rm stat}(9.7)_{\rm ren}(4.9)_{\rm syst} {\rm MeV}$.
• ### 2+1 flavor domain wall QCD on a (2 fm)^3 lattice: light meson spectroscopy with Ls = 16(hep-lat/0701013)

Oct. 23, 2007 hep-lat
We present results for light meson masses and pseudoscalar decay constants from the first of a series of lattice calculations with 2+1 dynamical flavors of domain wall fermions and the Iwasaki gauge action. The work reported here was done at a fixed lattice spacing of about 0.12 fm on a 16^3\times32 lattice, which amounts to a spatial volume of (2 fm)^3 in physical units. The number of sites in the fifth dimension is 16, which gives m_{res} = 0.00308(4) in these simulations. Three values of input light sea quark masses, m_l^{sea} \approx 0.85 m_s, 0.59 m_s and 0.33 m_s were used to allow for extrapolations to the physical light quark limit, whilst the heavier sea quark mass was fixed to approximately the physical strange quark mass m_s. The exact rational hybrid Monte Carlo algorithm was used to evaluate the fractional powers of the fermion determinants in the ensemble generation. We have found that f_\pi = 127(4) MeV, f_K = 157(5) MeV and f_K/f_\pi = 1.24(2), where the errors are statistical only, which are in good agreement with the experimental values.
• ### Light meson masses and non-perturbative renormalisation in 2+1 flavour domain wall QCD(hep-lat/0610119)

Oct. 26, 2006 hep-lat
We present results for the light meson masses, the bare strange quark mass and preliminary non-perturbative renormalisation of B_K in 2+1 flavour domain wall QCD. The ensembles used were generated with the Iwasaki gauge action and have a volume of 16^3 x 32 with a fifth dimension size of 16 and an inverse lattice spacing of 1.6 GeV. These ensembles have u and d masses as low as one quarter of the strange quark mass. All data were generated jointly by the UKQCD and RBC collaborations on QCDOC machines.
• ### Light meson masses and decay constants in 2+1 flavour domain wall QCD(hep-lat/0512009)

Dec. 8, 2005 hep-lat
We present results for light meson masses and psedoscalar meson decay constants in 2+1 flavour domain wall QCD with the DBW2 and Iwasaki gauge actions, using lattices with linear sizes in the range 1.6 to 2.2fm and $u$ and $d$ quark masses as low as one quarter of the strange quark mass. All data were generated on the QCDOC machines at the University of Edinburgh and Brookhaven National Laboratory. Despite large residual masses and a limited number of sea quark mass values with which to perform chiral extrapolations, our results agree with experiment and scale within errors.
• ### Baryons in 2+1 flavour domain wall QCD(hep-lat/0511011)

Nov. 4, 2005 hep-lat
We present results for some of the light baryon masses and their excited states in 2+1 flavour domain wall QCD. We considered several lattice spacings, with the DBW2 and Iwasaki gauge actions and different sea quark masses on a volume of $16^3\times32$ and a fifth dimension of size 8. All data were generated on the QCDOC machines. Despite large residual massses and a limited number of sea quark mass values with which to perform chiral extrapolations, our results are in reasonable agreement with experiment and scale within errors. Finite size effects on most ensembles appear to be small.
• ### Lattice QCD with mixed actions(hep-lat/0411005)

Aug. 3, 2005 hep-lat
We discuss some of the implications of simulating QCD when the action used for the sea quarks is different from that used for the valence quarks. We present exploratory results for the hadron mass spectrum and pseudoscalar meson decay constants using improved staggered sea quarks and HYP-smeared overlap valence quarks. We propose a method for matching the valence quark mass to the sea quark mass and demonstrate it on UKQCD clover data in the simpler case where the sea and valence actions are the same.
• ### Exploratory spectrum calculations using overlap valence quarks on a staggered sea(hep-lat/0408043)

Aug. 27, 2004 hep-lat
We present exploratory results for the hadron mass spectrum and pseudoscalar meson decay constants using mixed actions. We use improved staggered sea quarks and HYP-smeared overlap valence quarks. We obtain good signals on 10 configurations at one lattice spacing and two different sets of sea quark masses.
• ### Progress in building an International Lattice Data Grid(hep-lat/0309029)

Sept. 16, 2003 hep-lat
We report on progress in setting up the International Lattice Data Grid. We describe the aims and objectives of the ILDG, what has been achieved during its first year of activity and invite feedback from the community.
• ### The spectrum of D_s mesons from lattice QCD(hep-lat/0307001)

July 1, 2003 hep-ph, hep-lat
The spectrum of orbitally excited $D_s$ mesons is computed in the continuum limit of quenched lattice QCD. The results are consistent with the interpretation that the narrow resonance in the $D_s \pi^0$ channel discovered by the BABAR Collaboration is a $J^P=0^+$ $c\bar{s}$ meson. Furthermore, within statistical errors, the $1^+-1^-$ and the $0^+-0^-$ mass splittings are equal, in agreement with the chiral multiplet structure predicted by heavy hadron chiral effective theory. On our coarsest lattice we present results from the first study of orbitally excited $D_s$ mesons with two flavors of dynamical quarks, with mass slightly larger than the strange quark mass. These results are consistent with the quenched data.
• ### Effects of non-perturbatively improved dynamical fermions in QCD at fixed lattice spacing(hep-lat/0107021)

Dec. 14, 2001 hep-lat
We present results for the static inter-quark potential, lightest glueballs, light hadron spectrum and topological susceptibility using a non-perturbatively improved action on a $16^3\times 32$ lattice at a set of values of the bare gauge coupling and bare dynamical quark mass chosen to keep the lattice size fixed in physical units ($\sim 1.7$ fm). By comparing these measurements with a matched quenched ensemble, we study the effects due to two degenerate flavours of dynamical quarks. With the greater control over residual lattice spacing effects which these methods afford, we find some evidence of charge screening and some minor effects on the light hadron spectrum over the range of quark masses studied ($M_{PS}/M_{V}\ge0.58$). More substantial differences between quenched and unquenched simulations are observed in measurements of topological quantities.
• ### Quenched QCD with O(a) improvement: I. The spectrum of light hadrons(hep-lat/9910022)

March 6, 2000 hep-ph, hep-lat
We present a comprehensive study of the masses of pseudoscalar and vector mesons, as well as octet and decuplet baryons computed in O(a) improved quenched lattice QCD. Results have been obtained using the non-perturbative definition of the improvement coefficient c_sw, and also its estimate in tadpole improved perturbation theory. We investigate effects of improvement on the incidence of exceptional configurations, mass splittings and the parameter J. By combining the results obtained using non-perturbative and tadpole improvement in a simultaneous continuum extrapolation we can compare our spectral data to experiment. We confirm earlier findings by the CP-PACS Collaboration that the quenched light hadron spectrum agrees with experiment at the 10% level.
• ### Recent results on light hadron and quark masses(hep-lat/9810054)

Oct. 26, 1998 hep-lat
Recent results for the spectrum of light hadrons provide clear evidence for the failure of quenched QCD and encouraging signs that simulations with dynamical sea quarks rectify some of the discrepancies, although string breaking has not yet been observed. The use of perturbation theory to match lattice quark masses to continuum schemes remains questionable, but non-perturbative methods are poised to remove this uncertainty. The inclusion of dynamical sea quarks substantially reduces estimates of the light quark masses. New results for the lightest glueball and the lightest exotic hybrid state provide useful input to phenomenology, but still have limited or no treatment of mixing. The $O(a)$-improved Wilson quark action is well-established in quenched QCD for $\beta\geq 5.7$, with most parameters obtainable non-perturbatively, in which range scaling violations are small. Progress has also been made with high-order improvement schemes for both Wilson and staggered quarks.
• ### Light hadron spectroscopy with O(a) improved dynamical fermions(hep-lat/9808016)

Aug. 17, 1998 hep-lat
• ### Quenched Hadrons using Wilson and O(a)-Improved Fermion Actions at beta=6.2(hep-lat/9205016)

May 21, 1992 hep-lat
We present the first study of the light hadron spectrum and decay constants for quenched QCD using an O(a)-improved nearest-neighbour Wilson fermion action at \beta=6.2. We compare the results with those obtained using the standard Wilson fermion action, on the same set of 18 gauge field configurations of a 24^3 times 48 lattice. For pseudoscalar meson masses in the range 330-800 MeV, we find no significant difference between the results for the two actions. The scales obtained from the string tension and mesonic sector are consistent, but differ from that derived from baryon masses. The ratio of the pseudoscalar decay constant to the vector meson mass is roughly independent of quark mass as observed experimentally, and in approximate agreement with the measured value.

format: plain


text:
## Birds and Worms

research

Pay attention to your thought process and how you use expert knowledge as you answer the question below. How do you think very young students would think about it?

Here are some birds and here are some worms. Suppose the birds all race over, and each one tries to get a worm. Will every bird get a worm? How many birds won’t get a worm?

Hudson (1983) found that, among a small group of first-grade children (mean age of 7.0), just 64% completed this type of task correctly. However, when the task was rephrased as follows, all of the students answered correctly.

Here are some birds and here are some worms. Suppose the birds all race over, and each one tries to get a worm. Will every bird get a worm? How many birds won’t get a worm?

Interpret the Results

Still, what can we say about these results? Is it the case that 100% of the students used “their knowledge of correspondence to determine exact numerical differences between disjoint sets”? That is how Hudson describes students’ unanimous success in the second task. The idea seems to be that the knowledge exists; it’s just that a certain magical turn of phrase unlocks and releases this otherwise submerged expertise.

But that expert knowledge is given in the second task: “each one tries to get a worm.” The question paints the picture of one-to-one correspondence, and gives away the procedure to use to determine the difference. So, “their knowledge” is a bit of a stretch, and “used their knowledge” is even more of a stretch, since the task not only sets up a structure but animates its moving parts as well (“suppose the birds all race over”).

Further, questions about whether or not students are using knowledge they possess raise questions about whether or not students are, in fact, determining “exact numerical differences between disjoint sets.” On the contrary, it can be argued that students are simply watching almost all of a movie in their heads (a mental simulation)—a movie for which we have provided the screenplay—and then telling us how it ends (spoiler: 2 birds don’t get a worm). The deeper equivalence between the solution “2” and the response “2” to the question “How many birds won’t get a worm?” is evident only to a knowledgeable onlooker.

Experiment 3

Hudson anticipates some of the skepticism on display above when he introduces the third and last experiment in the series.

It might be argued, success in the Won’t Get task does not require a deep level of mathematical understanding; the children could have obtained the exact numerical differences by mimicking by rote the actions described by the problem context . . . In order to determine more fully the level of children’s understanding of correspondences and numerical differences, a third experiment was carried out that permitted a detailed analysis of children’s strategies for establishing correspondences between disjoint sets.

The wording in the Numerical Differences task of this third experiment, however, did not change. The “won’t get” locutions were still used. Yet, in this experiment, when paying attention to students’ strategies, Hudson observed that most children did not mentally simulate in the way directly suggested by the wording (pairing up the items in a one-to-one correspondence).

This does not defeat the complaint above, though. The fact that a text does not effectively compel the use of a procedure does not mean that it is not the primary influence on correct answers. It still seems more likely than not that participants who failed the “how many more” task simply didn’t have stable, abstract, transferable notions about mathematical difference. And the reformulation represented by the “won’t get” task influenced students to provide a response that was correct.

But this was a correct response to a different question. As adults with expert knowledge, we see the logical and mathematical similarities between the “how many more” and “won’t get” situations, and, thus we are easily fooled into believing that applying skills and knowledge in one task is equivalent to doing so in the other.

## Entia Successiva

The term ‘entia successiva’ means ‘successive entities.’ And, as you may guess, it is a term one might come across in a philosophy class, in particular when discussing metaphysical questions about personhood. For instance, is a person a single thing throughout its entire life or a succession of different things—an ‘ens successivum’? Though there is no right answer to this question, becoming familiar with the latter perspective can, I think, help people to be more skeptical and knowledgeable consumers of education research.

Richard Taylor provides an example of a symphony (in here) that is, depending on your perspective, both a successive and a permanent entity:

Let us imagine [a symphony orchestra] and give it a name—say, the Boston Symphony. One might write a history of this orchestra, beginning with its birth one hundred years ago, chronicling its many tours and triumphs and the fame of some of its musical directors, and so on. But are we talking about one orchestra?

In one sense we are, but in another sense we are not. The orchestra persists through time, is incorporated, receives gifts and funding, holds property, has a bank account, returns always to the same city, and rehearses, year in and year out, in the same hall. Yet its membership constantly changes, so that no member of fifty years ago is still a member today. So in that sense it is an entirely different orchestra. We are in this sense not talking about one orchestra, but many. There is a succession of orchestras going under the same name. Each, in [Roderick] Chisholm’s apt phrase, does duty for what we are calling the Boston Symphony.

The Boston Symphony is thus an ens successivum.

People are entia successiva, too. Or, at least their bodies are. Just about every cell in your body has been replaced from only 10 years ago. So, if you’re a 40-year-old Boston Symphony like me, almost all of your musicians and directors have been swapped out from when you were a 30-year-old symphony. People still call you the Boston Symphony of course (because you still are), but an almost entirely different set of parts is doing duty for “you” under the banner of this name. You are, in a sense, an almost completely different person—one who is, incidentally, made up of at least as many bacterial cells as human ones.

What’s worse (if you think of the above as bad news), the fact of evolution by natural selection tells us that humanity itself is an ens successivum. If you could line up your ancestors—your mother or father, his or her mother or father, and so on—it would be a very short trip down this line before you reached a person with whom you could not communicate at all, save through gestures. Between 30 and 40 people in would be a person who had almost no real knowledge about the physical universe. And there’s a good chance that perhaps the four thousandth person in your row of ancestors would not even be human.

The ‘Successive’ Perspective

Needless to say, seeing people as entia successiva does not come naturally to anyone. Nor should it, ever. We couldn’t go about out our daily lives seeing things this way. But the general invisibility of this ‘successiveness’ is not due to its only being operational at the very macro or very micro levels. It can be seen at the psychological level too. Trouble is, our brains are so good at constructing singular narratives out of even absolute gibberish, we sometimes have to place people in unnatural or extreme situations to get a good look at how much we can delude ourselves.

An Air Force doctor’s experiences investigating the blackouts of pilots in centrifuge training provides a nice example (from here). It’s definitely worth quoting at length:

Over time, he has found striking similarities to the same sorts of things reported by patients who lost consciousness on operating tables, in car crashes, and after returning from other nonbreathing states. The tunnel, the white light, friends and family coming to greet you, memories zooming around—the pilots experienced all of this. In addition, the centrifuge was pretty good at creating out-of-body experiences. Pilots would float over themselves, or hover nearby, looking on as their heads lurched and waggled about . . . the near-death and out-of-body phenomena are both actually the subjective experience of a brain owner watching as his brain tries desperately to figure out what is happening and to orient itself amid its systems going haywire due to oxygen deprivation. Without the ability to map out its borders, the brain often places consciousness outside the head, in a field, swimming in a lake, fighting a dragon—whatever it can connect together as the walls crumble. What the deoxygenated pilots don’t experience is a smeared mess of random images and thoughts. Even as the brain is dying, it refuses to stop generating a narrative . . . Narrative is so important to survival that it is literally the last thing you give up before becoming a sack of meat.

You’ll note, I hope, that not only does the report above disclose how our very mental lives are entia successiva—thoughts and emotions that arise and pass away—but the report assumes this perspective in its own narrative. That’s because the report is written from a scientific point of view. And from that vantage point, people are assumed (correctly) to have parts that “do duty” for them and may even be at odds with each other, as they were with the pilots (a perception part fighting against a powerful narrative-generating part). The unit of analysis in the report is not an entire pilot, but the various mechanisms of her mind. Allowing for these parts allows for functional explanations like the one we see.

An un-scientific analysis, on the other hand, is entirely possible. But it would stop at the pilot. He or she is, after all, an indivisible, permanent entity. There is nothing else “doing duty” for him, so there are really only two choices: his experience was an illusion or it was real. End of analysis. Interpret it as an illusion and you don’t really have much to say; interpret it as real, and you can make a lot of money.

Entia Permanentia

Good scientific research in education will adopt an entia successiva perspective about the people it studies. This does not guarantee that its conclusions are correct. But it makes it more likely that, over time, it will get to the bottom of things.

This is not to say that an alternative perspective is without scientific merit. If we want to know how to improve the performance of the Boston Symphony, we can make some headway with ‘entia permanentia’—seeing the symphony as a whole stable unit rather than a collection of successive parts. We could increase its funding, perhaps try to make sure “it” is treated as well as other symphonies around the world. We could try to change the music, maybe include some movie scores instead of that stuffy old classical music. That would make it more exciting for audiences (and more inclusive), which is certainly one interpretation of “improvement.” But to whatever extent improvement means improving the functioning of the parts of the symphony—the musicians, the director, etc.—we can do nothing, because with entia permanentia these tiny creatures do not exist. Even raising the question about improving the parts would be beyond the scope of our imagination.

Further, seeing students as entia permanentia rather than entia successiva stops us from being appropriately skeptical about both ‘scientific’ and ‘un-scientific’ ideas. Do students learn best when matched to their learning style? What parts of their neurophysiology and psychology could possibly make something like that true? Why would it have evolved, if it did? In what other aspects of our lives might this present itself? Adopting the entia successiva perspective would have slowed the adoption of this myth (even if were not a myth) to a crawl and would have eventually killed it. Instead, entia permanentia, a person-level analysis, holds sway: students benefit from learning-style matching because we see them respond differently to different representations. End of analysis.

A different but similar perspective on this, from a recurring theme in the book Switch:

In a pioneering study of organizational change, described in the book The Critical Path to Corporate Renewal, researchers divided the change efforts they’d studied into three groups: the most successful (the top third), the average (the middle third), and the least successful (the bottom third). They found that, across the spectrum, almost everyone set goals: 89 percent of the top third and 86 percent of the bottom third . . . But the more successful change transformations were more likely to set behavioral goals: 89 percent of the top third versus only 33 percent of the bottom third.

Why do “behavioral” goals work when just “goals” don’t? Behavioral goals are, after all, telling you what to do, forcing you to behave in a certain way. Do you like to be told what to do? Probably not.

But the “you” that responds to behavioral goals isn’t the same “you” whose in-the-moment “likes” are important. You are more than just one solid indivisible self. You are many selves, and the self that can start checking stuff off the to-do list is often pulling the other selves behind it. And when it does, you get to think that “you” are determined, “you” take initiative, “you” have willpower. But in truth, your environment—both immediate and distant, both internal and external—has simply made it possible for that determined self to take the lead. Behavioral goals often create this exact environment.

## Provided vs Generated Examples

research

The results reported in this research (below) about the value of provided examples versus generated examples are a bit surprising. To get a sense of why that’s the case, start with this definition of the concept availability heuristic used in the study—a term from the social psychology literature:

Availability heuristic: the tendency to estimate the likelihood that an event will occur by how easily instances of it come to mind.

All participants first read this definition, along with the definitions of nine other social psychology concepts, in a textbook passage. Participants then completed two blocks of practice trials in one of three groups: (1) subjects in the provided examples group read two different examples, drawn from an undergraduate psychology textbook, of each of the 10 concepts (two practice blocks, so four examples total for each concept), (2) subjects in the generated examples group created their own examples for each concept (four generated examples total for each concept), and (3) subjects in the combination group were provided with an example and then created their own example of each concept (two provided and two generated examples total for each concept).

The researchers—Amanda Zamary and Katharine Rawson at Kent State University in Ohio—made the following predictions, with regard to both student performance and the efficiency of the instructional treatments:

We predicted that long-term learning would be greater following generated examples compared to provided examples. Concerning efficiency, we predicted that less time would be spent studying provided examples compared to generating examples . . . [and] long-term learning would be greater after a combination of provided and generated examples compared to either technique alone. Concerning efficiency, our prediction was that less time would be spent when students study provided examples and generate examples compared to just generating examples.

Achievement Results

All participants completed the same two self-paced tests two days later. The first assessment, an example classification test, asked subjects to classify each of 100 real-world examples into one of the 10 concept definition categories provided. Sixty of these 100 were new (Novel) to the provided-examples group, 80 of the 100 were new to the combination group, and of course all 100 were likely new to the generated-examples group. The second assessment, a definition-cued recall test, asked participants to type in the definition of each of the 10 concepts, given in random order. (The test order was varied among subjects.)

Given that participants in the provided-examples and combination groups had an advantage over participants in the generated-examples group on the classification task (they had seen between 20 and 40 of the examples previously), the researchers helpfully drew out results on just the 60 novel examples.

Subjects who were given only textbook-provided examples of the concepts outperformed other subjects on applying these concepts to classifying real-world examples. This difference was significant. No significant differences were found on the cued-recall test between the provided-examples and generated-examples groups.

Also, Students’ Time Is Valuable

Another measure of interest to the researchers in this study, as mentioned above, was the time used by the participants to read through or create the examples. What the authors say about efficiency is worth quoting, since it does not often seem to be taken as seriously as measures of raw achievement (emphasis mine):

Howe and Singer (1975) note that in practice, the challenge for educators and researchers is not to identify effective learning techniques when time is unlimited. Rather, the problem arises when trying to identify what is most effective when time is fixed. Indeed, long-term learning could easily be achieved if students had an unlimited amount of time and only a limited amount of information to learn (with the caveat that students spend their time employing useful encoding strategies). However, achieving long-term learning is difficult because students have a lot to learn within a limited amount of time (Rawson and Dunlosky 2011). Thus, long-term learning and efficiency are both important to consider when competitively evaluating the effectiveness of learning techniques.

With that in mind, and given the results above, it is noteworthy to learn that the provided-examples group outperformed the generated-examples group on real-world examples after engaging in practice that took less than half as much time. The researchers divided subjects’ novel classification score by the amount of time they spent practicing and determined that the provided-examples group had an average gain of 5.7 points per minute of study, compared to 2.2 points per minute for the generated-examples group and 1.7 points per minute for the combination group.

For learning declarative concepts in a domain and then identifying those concepts in novel real-world situations, provided examples proved to be better than student-generated examples for both long-term learning and for instructional efficiency. The second experiment in the study replicated these findings.

Some Commentary

First, some familiarity with the research literature makes the above results not so surprising. The provided-examples group likely outperformed the other groups because participants in that group practiced with examples generated by experts. Becoming more expert in a domain does not necessarily involve becoming more isolated from other people and their interests. Such expertise is likely positively correlated with better identifying and collating examples within a domain that are conceptually interesting to students and more widely generalizable. I reported on two studies, for example, which showed that greater expertise was associated with a significantly greater number of conceptual explanations, as opposed to “product oriented” (answer-getting) explanations—and these conceptual explanations resulted in the superior performance of students receiving them.

Second, I am sympathetic to the efficiency argument, as laid out here by the study’s authors—that is, I agree that we should focus in education on “trying to identify what is most effective when time is fixed.” Problematically, however, a wide variety of instructional actions can be informed by decisions about what is and isn’t “fixed.” Time is not the only thing that can be fixed in one’s experience. The intuition that students should “own their own learning,” for example, which undergirds the idea in the first place that students should generate their own examples, may rest on the more fundamental notion that students themselves are “fixed” identities that adults must work around rather than try to alter. This notion is itself circumscribed by the research summarized above. So, it is worth having a conversation about what should and should not be considered “fixed” when it comes to learning.

## Sum and Product Loops

It’s something of a truism that mathematical symbolism is difficult. There are some situations, though, where the symbolism is not just difficult, but also annoying and ridiculous. It likely saved a lot of time when people were still mostly writing ideas out by hand, so back then even the annoying and ridiculous could not be righteously pointed at and mocked, but nowadays it is almost certainly more difficult to set some statements in LaTeX than it is to type them—and, if the text is intended to teach students, more difficult to unpack the former than it is to understand the latter.

Examples of symbols that are justly symbolized, even today, are $$\mathtt{\sum}$$ and $$\mathtt{\prod}$$, representing a sum and a product, respectively. More specifically, these symbols represent loops—an addition loop or a multiplication loop.

So, for example, take this expression on the left side of the equals sign, which represents the loop sum on the right of the equals sign: $$\mathtt{\sum_{n=1}^{5}n=1+2+3+4+5}$$. The expression on the left just means (a) start a counter at 1, (b) count up to 5 by 1s, (c) let n = each number you count, then (d) add all the n’s one by one in a loop.

How about this one? $\mathtt{\sum_{n=0}^{4}2n=0+2+4+6+8}$

This one means (a) start a counter at 0, (b) count up to 4 by 1s, (c) let n = each number you count, then (d) add all the 2n’s one by one in a loop.

For products, we just swap out the symbol. Here is the corresponding product for the first loop: $$\mathtt{\prod_{n=1}^{5}n=1\times2\times3\times4\times5}$$. And here’s one for the second loop: $\mathtt{\prod_{n=0}^{4}2n=0\times2\times4\times6\times8}$

Loops and Linear Algebra

You’ll often see the summation loop in linear algebra contexts, because it is an equivalent way to write a dot product, for example. The sum $$\mathtt{\sum_{n=0}^{4}2n=0+2+4+6+8}$$ above can be written as shown below, which looks like more work to write—and is—but when we’re dealing mostly with variables, the savings in writing effort is more evident. $\quad\,\,\,\begin{bmatrix}\mathtt{2}\\\mathtt{2}\\\mathtt{2}\\\mathtt{2}\\\mathtt{2}\end{bmatrix}\cdot \begin{bmatrix}\mathtt{0}\\\mathtt{1}\\\mathtt{2}\\\mathtt{3}\\\mathtt{4}\end{bmatrix}\mathtt{=2\cdot0+2\cdot1+2\cdot2\ldots}$

The loop sum $$\mathtt{\sum_{i}a_{i}x_{i}+b}$$, where $$\mathtt{i}$$ is an index pointing to a component of vector $$\mathtt{a}$$ and vector $$\mathtt{x}$$, can be written more simply as $$\mathtt{a\cdot x+b}$$, as long as the context is clear that $$\mathtt{a}$$ and $$\mathtt{x}$$ are vectors.

## Subtractive Knowledge

I was intrigued by a pedagogical insight offered by the example below, from the introductory class of a course called Computational Linear Algebra. The setup is that the graph (diagram) at the bottom of the image represents a Markov model, and it shows the probabilities of moving from one stage of a disease to another in a year.

So, if a patient is asymptomatic, there is a 7% (0.07) probability of moving from asymptomatic to symptomatic, a 90% chance of staying at asymptomatic (indicated by a curved arrow), and so on. This information is also encoded in the stochastic matrix shown.

Here’s the problem: Given a group of people: 85% are asymptomatic, 10% are symptomatic, 5% have AIDS, and of course 0% are deceased, what percent will be in each health state in a year? Putting yourself in the mind of a student right now, take a moment to try to answer the question and, importantly, reflect on your thinking at this stage, even if that thinking involves having no clue how to proceed.

I hope that, given this setup, you’ll be somewhat surprised to learn the following: If a high school student (or even middle school student) knows a little bit about probability and how to multiply and add, they should be able to answer this question.

Why? Well, if 85 out of every 100 in a group is asymptomatic, and there is a 90% probability of remaining asymptomatic in a year, then (0.85)(0.9) = 76.5% of the group is predicted to be asymptomatic in a year. The symptomatic group has two products that must be added: 10% of the group is symptomatic, and there is a 93% probability of remaining that way, so that’s (0.93)(0.1). But this group also takes on 7% of the 85% that were asymptomatic but moved to symptomatic. So, the total is (0.93)(0.1) + (0.07)(0.85) = 15.25%. The AIDS group percent is the sum of three products, for a total of 6.45%, and the Death group percent is a sum of four products, for a total of 1.8%.

Probability, multiplication, and addition are all you need to know. No doubt, knowing something about matrix-vector multiplication, as we have discussed, (and transposes) can be helpful, but it does not seem to be necessary in this case.

Bamboozled

I think it’s reasonable to suspect that many knowledgeable students—and knowledgeable adults—would be bamboozled by the highfalutin language here into believing that they cannot solve this problem, when in fact they can. If that’s true, then why is that the case?

Knowledge is domain specific, of course, (and very context specific) and that would seem to be the best explanation of students’ hypothesized difficulties. That is, given the cues (both verbal and visual) that this problem involves knowledge of linear algebra, Markov models, and/or stochastic matrices, anyone without that knowledge would naturally assume that they don’t have what is required to solve the problem and give up. And even if they suspected that some simple probability theory, multiplication, and addition were all that they needed, being bombarded by even a handful of foreign mathematical terms would greatly reduce their confidence in this suspicion.

Perhaps, then, the reason we are looking for—the reason students don’t believe they can solve problems when in fact they can—has to do with students’ attitudes, not their knowledge. And situations like these during instruction are enough to convince many that knowledge is overrated. The solution to this psychological reticence is, for many people, to encourage students to be fearless, to have problem-solving orientations and growth mindsets. After all, it’s clear that more knowledge would be helpful, but it’s not necessary in many cases. We’ll teach knowledge, sure, but we can do even better if we spend time encouraging soft skills along the way. Do we want students to give up every time they face a situation in life that they were not explicitly taught to deal with?

The problem with this view is that it construes knowledge as only additive. That is, it is thought, knowledge only works to give its owner things to think about and think with. So, in the above example, students already have all the knowledge things to work with: probability, multiplication, and addition. Anything else would only serve to bring in more things to think about, which would be superfluous.

But this isn’t the only way knowledge works. It can also be subtractive—that is, knowing something can tell you that it is irrelevant to the current problem. Not knowing it means that you can’t know about its relevance (and situations like the above will easily bias you to giving superficial information a high degree of relevance). So, students cannot know with high confidence that matrices are essentially irrelevant to the problem above if they don’t know what matrices are. But even knowing nothing about matrices, knowing that, computationally, linear algebra is fundamentally about multiplying and adding things may be enough. Taking that perspective can allow you to ignore the superficial setup of the problem. But that’s still knowledge.

A better interpretation of students’ difficulties with the above is that, in fact, they do need more knowledge to solve the problem. The knowledge they need is subtractive; it will help them ignore superficial irrelevant details to get at the marrow of the problem.

Knowledge is obviously additive, but it is much more subtly subtractive too, helping to clear away facts that are irrelevant to a given situation. Subtractive knowledge is like the myelinated sheaths around some nerve cells in the brain. It acts as an insulator for thinking—making it faster, more efficient, and, as we have seen, more effective.

## The Gricean Maxims

When we converse with one another, we implicitly obey a principle of cooperation, according to language philosopher Paul Grice’s theory of conversational implicature.

This ‘cooperative principle’ has four maxims, which although stated as commands are intended to be descriptions of specific rules that we follow—and expect others will follow—in conversation:

• quality:    Be truthful.
• quantity:  Don’t say more or less than is required.
• relation:  Be relevant.
• manner:    Be clear and orderly.

I was drawn recently to these maxims (and to Grice’s theory) because they rather closely resemble four principles of instructional explanation that I have been toying with off and on for a long time now: precision, clarity, order, and cohesion.

In fact, there is a fairly snug one-to-one correspondence among our respective principles, a relationship which is encouraging to me precisely because it is coincidental. Here they are in an order corresponding to the above:

• precision:  Instruction should be accurate.
• cohesion: Group related ideas.
• clarity:     Instruction should be understandable and present to its audience.
• order:       Instruction should be sequenced appropriately.

Both sets of principles likely seem dumbfoundingly obvious, but that’s the point. As principles (or maxims), they are footholds on the perimeters of complex ideas—in Grice’s case, the implicit contexts that make up the study of pragmatics; in my case (insert obligatory note that I am not comparing myself with Paul Grice), the explicit “texts” that comprise the content of our teaching and learning.

The All-Consuming Clarity Principle

Frameworks like these can be more than just armchair abstractions; they are helpful scaffolds for thinking about the work we do. Understanding a topic up and down the curriculum, for example, can help us represent it more accurately in instruction. We can think about work in this area as related specifically to the precision principle and, in some sense, as separate from (though connected to) work in other areas, such as topic sequencing (order), explicitly building connections (cohesion), and motivation (clarity).

But principle frameworks can also lift us to some height above this work, where we can find new and useful perspectives. For instance, simply having these principles, plural, in front of us can help us see—I would like to persuade you to see—that “clarity,” or in Grice’s terminology, “relevance,” is the only one we really talk about anymore, and that this is bizarre given that it’s just one aspect of education.

The work of negotiating the accuracy, sequencing, and connectedness of instruction drawn from our shared knowledge has been largely outsourced to publishers and technology startups and Federal agencies, and goes mostly unquestioned by the “delivery agents” in the system, whose role is one of a go-between, tasked with trying to sell a “product” in the classroom to student “customers.”

## Spooky Action at a Distance

I really like this recent post, called Tell Me More, Tell Me More, by math teacher Dani Quinn. The content is an excellent analysis of expert blindness in math teaching. The form, though, is worth seeing as well—it is a traditional educational syllogism, which Quinn helpfully commandeers to arrive at a non-traditional conclusion, that instructional effects have instructional causes, on the right:

There is a problem in how we teach: We typically spoon-feed students procedures for answering questions that will be on some kind of test.

“There is a problem in how we teach: We typically show pupils only the classic forms of a problem or a procedure.”

This is why students can’t generalize to non-routine problems: we got in the way of their thinking and didn’t allow them to take ownership and creatively explore material on their own.“This is why they then can’t generalise: we didn’t show them anything non-standard or, if we did, it was in an exercise when they were floundering on their own with the least support.”

Problematically for education debates, each of these premises and conclusions taken individually are true. That is, they exist. At our (collective) weakest, we do sometimes spoon-feed kids procedures to get them through tests. We do cover only a narrow range of situations—what Engelmann refers to as the problem of stipulation. And we can be, regrettably in either case, systematically unassertive or overbearing.

Solving equations provides a nice example of the instructional effects of both spoon-feeding and stipulation. Remember how to solve equations? Inverse operations. That was the way to do equations. If you have something like $$\mathtt{2x + 5 = 15}$$, the table shows how it goes.

EquationStep
$$\mathtt{2x + 5 \color{red}{- 5} = 15 \color{red}{- 5}}$$Subtract $$\mathtt{5}$$ from both sides of the equation to get $$\mathtt{2x = 10}$$.
$$\mathtt{\color{white}{+ 5 \,\,} 2x \color{red}{\div 2} = 10 \color{red}{\div 2}}$$Divide both sides of the equation by 2.
$$\mathtt{\color{white}{+ 5 \,\,}x = 5}$$You have solved the equation.

Do that a couple dozen times and maybe around 50% of the class freezes when they encounter $$\mathtt{22 = 4x + 6}$$, with the variable on the right side, or, even worse, $$\mathtt{22 = 6 + 4x}$$.

That’s spoon-feeding and stipulation: do it this one way and do it over and over—and, crucially, doing that summarizes most of the instruction around solving equations.

Of course, the lack of prior knowledge exacerbates the negative instructional effects of stipulation and spoon-feeding. But we’ll set that aside for the moment.

The Connection Between Premises and Conclusion

The traditional and alternative arguments above are easily (and often) confused, though, until you include the premise that I have omitted in the middle for each. These help make sense of the conclusions derived in each argument.

There is a problem in how we teach: We typically spoon-feed students procedures for answering questions that will be on some kind of test.

“There is a problem in how we teach: We typically show pupils only the classic forms of a problem or a procedure.”

Students’ success in schooling is determined mostly by internal factors, like creativity, motivation, and self-awareness.

Students’ success in schooling is determined mostly by external factors, like amount of instruction, socioeconomic status, and curricula.

This is why students can’t generalize to non-routine problems: we got in the way of their thinking and didn’t allow them to take ownership and creatively explore material on their own.“This is why they then can’t generalise: we didn’t show them anything non-standard or, if we did, it was in an exercise when they were floundering on their own with the least support.”

In short, the argument on the left tends to diagnose pedagogical illnesses and their concomitant instructional effects as people problems; the alternative sees them as situation problems. The solutions generated by each argument are divergent in just this way: the traditional one looks to pull the levers that mostly benefit personal, internal attributes that contribute to learning; the alternative messes mostly with external inputs.

It’s Not the Spoon-Feeding, It’s What’s on the Spoon

I am and have always been more attracted to the alternative argument than the traditional one. Probably for a very simple reason: my role in education doesn’t involve pulling personal levers. Being close to the problem almost certainly changes your view of it—not necessarily for the better. But, roles aside, it’s also the case that the traditional view is simply more widespread, and informed by the positive version of what is called the Fundamental Attribution Error:

We are frequently blind to the power of situations. In a famous article, Stanford psychologist Lee Ross surveyed dozens of studies in psychology and noted that people have a systematic tendency to ignore the situational forces that shape other people’s behavior. He called this deep-rooted tendency the “Fundamental Attribution Error.” The error lies in our inclination to attribute people’s behavior to the way they are rather than to the situation they are in.

What you get with the traditional view is, to me, a kind of spooky action at a distance—a phrase attributed to Einstein, in remarks about the counterintuitive consequences of quantum physics. Adopting this view forces one to connect positive instructional effects (e.g., thinking flexibly when solving equations) with something internal, ethereal and often poorly defined, like creativity. We might as well attribute success to rabbit’s feet or lucky underwear or horoscopes!

## Intuition and Domain Knowledge

Can you guess what the graphs below show? I’ll give you a couple of hints: (1) each graph measures performance on a different task, (2) one pair of bars in each graph—left or right—represents participants who used their intuition on the task, while the other pair of bars represents folks who used an analytical approach, and (3) one shading represents participants with low domain knowledge while the other represents participants with high domain knowledge (related to the actual task).

It will actually help you to take a moment and go ahead and guess how you would assign those labels, given the little information I have provided. Is the left pair of bars in each graph the “intuitive approach” or the “analytical approach”? Are the darker shaded bars in each graph “high knowledge” participants or “low knowledge” participants?

When Can I Trust My Gut?

A 2012 study by Dane, et. al, published in the journal Organizational Behavior and Human Decision Processes, sets out to address the “scarcity of empirical research spotlighting the circumstances in which intuitive decision making is effective relative to analytical decision making.”

To do this, the researchers conducted two experiments, both employing “non-decomposable” tasks—i.e., tasks that required intuitive decision making. The first task was to rate the difficulty (from 1 to 10) of each of a series of recorded basketball shots. The second task involved deciding whether each of a series of designer handbags was fake or authentic.

Why these tasks? A few snippets from the article can help to answer that question:

Following Dane and Pratt (2007, p. 40), we view intuitions as “affectively-charged judgments that arise through rapid, nonconscious, and holistic associations.” That is, the process of intuition, like nonconscious processing more generally, proceeds rapidly, holistically, and associatively (Betsch, 2008; Betsch & Glöckner, 2010; Sinclair, 2010). [Footnote: “This conceptualization of intuition does not imply that the process giving rise to intuition is without structure or method. Indeed, as with analytical thinking, intuitive thinking may operate based on certain rules and principles (see Kruglanski & Gigerenzer, 2011 for further discussion). In the case of intuition, these rules operate largely automatically and outside conscious awareness.”]

As scholars have posited, analytical decision making involves basing decisions on a process in which individuals consciously attend to and manipulate symbolically encoded rules systematically and sequentially (Alter, Oppenheimer, Epley, & Eyre, 2007).

We viewed [the basketball] task as relatively non-decomposable because, to our knowledge, there is no universally accepted decision rule or procedure available to systematically break down and objectively weight the various elements of what makes a given shot difficult or easy.

We viewed [the handbag] task as relatively non-decomposable for two reasons. First, although there are certain features or clues participants could attend to (e.g., the stitching or the style of the handbags), there is not necessarily a single, definitive procedure available to approach this task . . . Second, because participants were not allowed to touch any of the handbags, they could not physically search for what they might believe to be give-away features of a real or fake handbag (e.g., certain tags or patterns inside the handbag).

Results

Canvas not supported.

As you can see in the graphs at the right (hover for expertise labels), there was a fairly significant difference in both tasks between low- and high-knowledge participants when those participants approached the task using their intuition. In contrast, high- and low-knowledge subjects in the analysis condition in each experiment did not show a significant difference in performance. (The decline in performance of the high-knowledge participants from the Intuition to the Analysis conditions was only significant in the handbag experiment.)

It is important to note that subjects in the analysis conditions (i.e., those who approached each task systematically) were not told what factors to look for in carrying out their analyses. For the basketball task, the researchers simply “instructed these participants to develop a list of factors that would determine the difficulty of a basketball shot and told them to base their decisions on the factors they listed.” For the handbag task, “participants in the analysis condition were given 2 min to list the features they would look for to determine whether a given handbag is real or fake and were told to base their decisions on these factors.”

Also consistent across both experiments was the fact that low-knowledge subjects performed better when approaching the tasks systematically than when using their intuition. For high-knowledge subjects, the results were the opposite. They performed better using their intuition than using a systematic analysis (even though the ‘system’ part of ‘systematic’ here was their own system!).

In addition, while the combined effects of approach and domain knowledge were significant, the approach (intuition or analysis) by itself did not have a significant effect on performance one way or the other in either experiment. Domain knowledge, on the other hand, did have a significant effect by itself in the basketball experiment.

Any Takeaways for K–12?

The clearest takeaway for me is that while knowledge and process are both important, knowledge is more important. Even though each of the tasks was more “intuitive” (non-decomposable) than analytical in nature, and even when the approach taken to the task was “intuitive,” knowledge trumped process. Process had no significant effect by itself. Knowing stuff is good.

Second, the results of this study are very much in line with what is called the ‘expertise reversal effect’:

Low-knowledge learners lack schema-based knowledge in the target domain and so this guidance comes from instructional supports, which help reduce the cognitive load associated with novel tasks. If the instruction fails to provide guidance, low-knowledge learners often resort to inefficient problem-solving strategies that overwhelm working memory and increase cognitive load. Thus, low-knowledge learners benefit more from well-guided instruction than from reduced guidance.

In contrast, higher-knowledge learners enter the situation with schema-based knowledge, which provides internal guidance. If additional instructional guidance is provided it can result in the processing of redundant information and increased cognitive load.

Finally, one wonders just who it is we are thinking about more when we complain, especially in math education, that overly systematized knowledge is ruining the creativity and motivation of our students. Are we primarily hearing the complaints of the 20%—who barely even need school—or those of the children who really need the knowledge we have, who need us to teach them?

Dane, E., Rockmann, K., & Pratt, M. (2012). When should I trust my gut? Linking domain expertise to intuitive decision-making effectiveness Organizational Behavior and Human Decision Processes, 119 (2), 187-194 DOI: 10.1016/j.obhdp.2012.07.009

## Telling vs. No Telling

So, with that in mind, let’s move on to just one of the dichotomies in education, that of “telling” vs. “no telling,” and I hope the reader will forgive my leaving Clarke’s paper behind. I recommend it to you for its international perspective on what we discuss below.

“Reports of My Death Have Been Greatly Exaggerated”

We should start with something that educators know but people outside of education may not: there can be a bit of an incongruity, shall we say, between what teachers want to happen in their classrooms, what they say happens in their classrooms, and what actually happens there. Given how we talk and what we talk about on social media—and even in face-to-face conversations—and the sensationalist tendencies of media reports about education, an outsider could be forgiven, I think, for assuming that teachers have been moving en masse away from the practice of explicit instruction.

There is a large body of research which would suggest that this assumption is almost certainly “greatly exaggerated.”

Typical of this research is a small 2004 study (PDF download) in the U.K. which found that primary classrooms in England remained places full of teacher talk and “low-level” responding by students, despite intentions outlined in the 1998–1999 National Literacy and National Numeracy Strategies. The graph at the right, from the study, shows the categories of discourse observed and a sense of their relative frequencies.

John Goodlad made a similar and more impactful observation in his much larger study of over 1,000 classrooms across the U.S. in the mid-80s (I take this quotation from the 2013 edition of John Hattie’s book Visible Learning, where more of the aforementioned research is cited):

In effect, then, the modal classroom configurations which we observed looked like this: the teacher explaining or lecturing to the total class or a single student, occasionally asking questions requiring factual answers; . . . students listening or appearing to listen to the teacher and occasionally responding to the teacher’s questions; students working individually at their desks on reading or writing assignments.

Thus, despite what more conspiracy-oriented opponents of “no telling” sometimes suggest, the monotonic din of “understanding” and “guide on the side” and “collaboration” we hear today—and have heard for decades—is not the sound of a worldview that has, in practice, taken over education. Rather, it is one of a seemingly quixotic struggle on the part of educators to nudge each other—to open up more space for students to exercise independent and critical thinking. This a finite space, and something has to give way.

Research Overwhelmingly Supports Explicit Instruction

Teacher as
Activator
dTeacher as Facilitatord
Teaching students self-verbalization0.76Inductive Teaching0.33
Teacher clarity0.75Simulation and gaming0.32
Reciprocal teaching0.74Inquiry-based teaching0.31
Feedback0.74Smaller classes0.21
Metacognitive strategies0.67Individualised instruction0.22
Direct instruction0.59Web-based learning0.18
Mastery learning0.57Problem-based learning0.15
Providing worked examples0.57Discovery method (math)0.11

On the other hand, it is manifestly clear from the research literature that, when student achievement is the goal, explicit instruction has generally outperformed its less explicit counterpart.

The table at the left, taken from Hattie’s book referenced above, directly compares the effect sizes of various explicit and indirect instructional protocols, gathered and interpreted across a number of different meta-analyses in the literature.

Results like these are not limited to the K–12 space, nor do they involve only the teaching of lower-level skills or teaching in only in well-structured domains, such as mathematics. These are robust results across many studies and over long periods of time.

And while research supporting less explicit instructional techniques is out there (as obviously Hattie’s results also attest), there is much less of it—and certainly far less than one would expect given the sheer volume of rhetoric in support of such strategies. On this point, it is worth quoting Sigmund Tobias at some length, from his summarizing chapter in the 2009 book Constructivist Instruction: Success or Failure?:

When the AERA 2007 debate was organized, I described myself as an eclectic with respect to whether constructivist instruction was a success or failure, a position I also took in print earlier (Tobias, 1992). The constructivist approach of immersing students in real problems and having them figure out solutions was intuitively appealing. It seemed reasonable that students would feel more motivated to engage in such activities than in those occurring in traditional classrooms. It was, therefore, disappointing to find so little research documenting increased motivation for constructivist activities.

A personal note may be useful here. My Ph.D. was in clinical psychology at the time when projective diagnostic techniques in general, and the Rorschach in particular, were receiving a good deal of criticism. The logic for these techniques was compelling and it seemed reasonable that people’s personality would have a major impact on their interpretation of ambiguous stimuli. Unfortunately, the empirical evidence in support of the validity of projective techniques was largely negative. They are now a minor element in the training of clinical psychologists, except for a few hamlets here or there that still specialize in teaching about projective techniques.

The example of projective techniques seems similar to the issues raised about constructivist instruction. A careful reading and re-reading of all the chapters in this book, and the related literature, has indicated to me that there is stimulating rhetoric for the constructivist position, but relatively little research supporting it. For example, it is encouraging to see that Schwartz et al. (this volume) are conducting research on their hypothesis that constructivist instruction is better for preparing individuals for future learning. Unfortunately, as they acknowledge, there is too little research documenting that hypothesis. As suggested above, such research requires more complex procedures and is more time consuming, for both the researcher and the participants, than procedures advocated by supporters of explicit instruction. However, without supporting research these remain merely a set of interesting hypotheses.

In comparison to constructivists, advocates for explicit instruction seem to justify their recommendations more by references to research than rhetoric. Constructivist approaches have been advocated vigorously for almost two decades now, and it is surprising to find how little research they have stimulated during that time. If constructivist instruction were evaluated by the same criterion that Hilgard (1964) applied to Gestalt psychology, the paucity of research stimulated by that paradigm should be a cause for concern for supporters of constructivist views.

Both the Problem and the Solution

So, it seems that while a “telling” orientation is better supported by research, it is also identified as a barrier, if not the barrier, to progress. And it seems that a lot of our day-to-day struggle with the issue centers around the negative consequences of continued unsuccessful attempts at resolving this paradox.

Yet perhaps we should see that this is not a paradox at all. Of course it is a problem when students learn to rely heavily on explicit instruction to make up their thinking, and it is perfectly appropriate to find ways of punching holes in teacher talk time to reduce the possibility of this dependency. But we could also research ways of tackling this explicitly—differentiating ways in which explicit instruction can solicit student inquiry or creativity and ways in which it promotes rule following, for example.

It is at least worth considering that some of our problems—particularly in mathematics education—have less to do with explicit instruction and more to do with bad explicit instruction. If dealing with instructional problems head on is more effective (even those that are “high level,” such as creativity and critical thinking), then we should be making the sacrifices necessary to give teachers the resources and training required to meet those challenges, explicitly.

## Interleaving

research

Inductive teaching or learning, although it has a special name, happens all the time without our having to pay any attention to technique. It is basically learning through examples. As the authors of the paper we’re discussing here indicate, through inductive learning:

Children . . . learn concepts such as ‘boat’ or ‘fruit’ by being exposed to exemplars of those categories and inducing the commonalities that define the concepts. . . . Such inductive learning is critical in making sense of events, objects, and actions—and, more generally, in structuring and understanding our world.

The paper describes three experiments conducted to further test the benefit of interleaving on inductive learning (“further” because an interleaving effect has been demonstrated in previous studies). Interleaving is one of a handful of powerful learning and practicing strategies mentioned throughout the book Make It Stick: The Science of Successful Learning. In the book, the power of interleaving is highlighted by the following summary of another experiment involving determining volumes:

Two groups of college students were taught how to find the volumes of four obscure geometric solids (wedge, spheroid, spherical cone, and half cone). One group then worked a set of practice problems that were clustered by problem type . . . The other group worked the same practice problems, but the sequence was mixed (interleaved) rather than clustered by type of problem . . . During practice, the students who worked the problems in clusters (that is, massed) averaged 89 percent correct, compared to only 60 percent for those who worked the problems in a mixed sequence. But in the final test a week later, the students who had practiced solving problems clustered by type averaged only 20 percent correct, while the students whose practice was interleaved averaged 63 percent.

The research we look at in this post does not produce such stupendous results, but it is nevertheless an interesting validation of the interleaving effect. Although there are three experiments described, I’ll summarize just the first one.

Discriminative-Contrast Hypothesis

But first, you can try out an experiment like the one reported in the paper. Click start to study pictures of different bird species below. There are 32 pictures, and each one is shown for 4 seconds. After this study period, you will be asked to try to identify 8 birds from pictures that were not shown during the study period, but which belong to one of the species you studied.

Once the study phase is over, click test to start the test and match each picture to a species name. There is no time limit on the test. Simply click next once you have selected each of your answers.

Based on previous research, one would predict that, in general, you would do better in the interleaved condition, where the species are mixed together in the study phase, than you would in the ‘massed,’ or grouped condition, where the pictures are presented in species groups. The question the researchers wanted to home in on in their first experiment was about the mechanism that made interleaved study more effective.

So, their experiment was conducted much like the one above, except with three groups, which all received the interleaved presentation. However, two of the groups were interrupted in their study by trivia questions in different ways. One group—the alternating trivia group—received a trivia question after every picture; the other group—the grouped trivia group—received 8 trivia questions after every group of 8 interleaved pictures. The third group—the contiguous group—received no interruption in their study.

What the researchers discovered is that while the contiguous group performed the best (of course), the grouped trivia group did not perform significantly worse, while the alternating trivia group did perform significantly worse than both the contiguous and grouped trivia groups. This was seen as providing some confirmation for the discriminative-contrast hypothesis:

Interleaved studying might facilitate noticing the differences that separate one category from another. In other words, perhaps interleaving is beneficial because it juxtaposes different categories, which then highlights differences across the categories and supports discrimination learning.

In the grouped trivia condition, participants were still able to take advantage of the interleaving effect because the disruptions (the trivia questions) had less of an effect when grouped in packs of 8. In the alternating trivia condition, however, a trivia question appeared after every picture, frustrating the discrimination mechanism that seems to help make the interleaving effect tick.

Takeaway Goodies (and Questions) for Instruction

The paper makes it clear that interleaving is not a slam dunk for instruction. Massed studying or practice might be more beneficial, for example, when the goal is to understand the similarities among the objects of study rather than the differences. Massed studying may also be preferred when the objects are ‘highly discriminable’ (easy to tell apart).

Yet, many of the misconceptions we deal with in mathematics education in particular can be seen as the result of dealing with objects of ‘low discriminability’ (objects that are hard to tell apart). In many cases, these objects really are hard to tell apart, and in others we simply make them hard through our sequencing. Consider some of the items listed in the NCTM’s wonderful 13 Rules That Expire, which students often misapply:

• When multiplying by ten, just add a zero to the end of the number.
• You cannot take a bigger number from a smaller number.
• Addition and multiplication make numbers bigger.
• You always divide the larger number by the smaller number.

In some sense, these are problematic because they are like the sparrows and finches above when presented only in groups—they are harder to stop because we don’t present them in situations that break the rules, or interleave them. Appending a zero to a number to multiply by 10 does work on counting numbers but not on decimals; addition and multiplication do make counting numbers bigger until they don’t always make fractions bigger; and you cannot take a bigger counting number from a smaller one and get a counting number. For that, you need integers.

Notice any similarities above? Can we please talk about how we keep kids trapped for too long in counting number land? I’ve got this marvelous study to show you which might provide some good reasons to interleave different number systems throughout students’ educations. It’s linked above, and below.


text:
## A mechanics problem

Today, I solved one from the Cracking the AP Physics C Exam 2008 Edition, which I had bought in high school. It is

Questions (c), (d), and (e) are not included. They are to determine the relationship between the magnitude of the acceleration of the block and the linear acceleration of the cylinder, to determine the acceleration of the cylinder, and to determine the acceleration of the block respectively. It is obvious that from (b) that the magnitude of the acceleration of the block is twice the linear acceleration of the cylinder, and the answer to (e) can be derived from that of (d) via this. As for (d), I first solved it with a free body diagram and Newton’s laws. Then, my ultra gifted friend Brian Bi suggested using the Euler-Lagrange equations, which simplifies things greatly, to the extent that he characterized himself as having forgotten how to solve mechanics problems the way I had done (obviously not true, but you get what he’s saying). I was able to work it out this way shortly, getting the same answer.

Note: I left overlooked the $4$ coefficient initially and only filled it in at the end. It is still omitted in some of the intermediate equations.

## First blog post

I plan, hope to be writing out here a miscellany of thoughts and musings pertaining to my intellectual interests, which can be roughly divided into a primary one, which is pure math and other topics of a mathematical nature, and secondary ones, which, of the ones I can think of now, are Chinese language, language (in general), psychometrics, neuroscience, and stories of scientists and of unusual people of one form or another.


text:
# A Field operator eigenstates & Fock states(Hatfield's Sch rep)

1. Dec 30, 2016

### ShayanJ

In chapter 10 of his book "Quantum Field Theory of Point Particles and Strings", Hatfield treats what he calls the Schrodinger representation of QFT. He starts with a free scalar field and introduces field operators $\hat \varphi(\vec x)$ and its eigenstates $\hat \varphi(\vec x)|\phi\rangle=\phi(\vec x)|\phi \rangle$. Then he says that the "coordinate" representation of the field state $|\Psi\rangle$ is given by the wave-functional $\Psi[\phi]=\langle \phi |\Psi\rangle$. I have two questions about this:

1) Is the ground state of the field, one of the eigenstates of $\hat \varphi(\vec x)$, i.e. is $\hat \varphi(\vec x)|0\rangle=0$ correct?( At the bottom of page 224 of his book "A modern introduction to quantum field theory", Maggiore calls $\phi(\vec x)=0$ the vacuum. It seems to me its related to my question. Is he correct or just being sloppy? )

2) What is the relationship between the Fock states that are created by $\hat a_k$ and $\hat a_k^\dagger$ and their superpositions that are created by $\hat \varphi(\vec x)$ and $\hat \varphi^\dagger(\vec x)$ and the eigenstates of $\hat \varphi(\vec x)$? Can we say that the eigenstates of $\hat \varphi(\vec x)$ are some kind of a coherent state because $\hat \varphi(\vec x)$ is an anihilation operator for Fock states?

Thanks

2. Dec 30, 2016

### MathematicalPhysicist

I refer you to the book by Robin Ticciati "QFT for Mathematicians" chapter 2, hopefully after reading this chapter you'll understand your misconceptions. (I don't have time to copy the answer here, I am studyding for Stat Mech... :-)).

3. Dec 30, 2016

### ShayanJ

Thanks. Ticciati's formula for the ground state was surprising for me. But there was no answer to my second question. Or at least I couldn't find it!


text:
3k views

### why is the variance of t-distribution with 1 and 2 degrees of freedom undefined while these distributions can be drawn? [duplicate]

The variance of a t-distribution is given by df/(df-2), hence the t-distribution with 1 and 2 degrees of freedom have no defined variance. Yet these distributions do exist and can be drawn, so one ...
53 views

### Expected value $=\infty$? [duplicate]

If we let $U_1, U_2, U_3,..., U_n$ be uniform (0,1), find $$\mathbb E[\sum_{i=0}^n iU_i^{i-1}]$$which, using the linearity of expectation, gives $$\sum_{i=0}^n \mathbb E[i U_i^{i-1}]$$ Doing this ...
49 views

### Does expectation of a r.v. always converge? [duplicate]

Let X be a random variable. Will its expectation E[X] expressed as a (possibly infinite) sum converge for arbitrary X? My intuition would be yes. Since E[X] is a function of a r.v., E[X] is a r.v. ...
32k views

### What is the difference between finite and infinite variance

What is the difference between finite and infinite variance ? My stats knowledge is rather basic; Wikipedia / Google wasn't much help here.
7k views

### What makes the mean of some distributions undefined?

Many PDFs range from minus to positive infinity, yet some means are defined and some are not. What common trait makes some computable?
3k views

### Why is the Cauchy Distribution so useful?

Could anyone give me some practical examples of the Cauchy Distribution? What makes it so popular?
8k views

24k views

### What is the Mean and Standard Deviation of the division of two random variables? [duplicate]

I have two normally-distributed independent random variables X and Y and I need to calculate its division Z. As far as I understand the mean of Z is $\mu_Z = \frac{\mu_X}{\mu_Y}$, but I don't know ...
807 views

### Bayesian lighthouse location estimation

I am trying to learn Stan in R and as a fun challenge I am trying to estimate the location of a lighthouse based on the observed flashes. But the models I tried do not converge (Rhat != 1) or have ...
3k views

49 views

### A practical example where maximum likelihood correctly estimates an underlying parameter, but where least squares would fail?

Forgive my very limited understanding. I am trying to learn about maximum likelihood estimation, and how it differs from least-squares estimation. From reading a little, I understand that the two are ...


text:
# IOP 2014 Joint HEPP & APP Group Meeting

Apr 7 – 9, 2014
Royal Holloway, University of London
Europe/London timezone
Institute of Physics 2014 Joint High Energy Particle Physics and Astro Particle Physics Groups Annual Meeting

## Using a Regressional Analysis to Improve the b-jet energy correction in $WH\rightarrow\ell\nu bb$ Searches at ATLAS

Apr 8, 2014, 5:45 PM
15m
Room C (Windsor Building)

### Room C

#### Windsor Building

contributed talk The Energy Frontier Programme

### Speaker

Thomas Olivier Ravenscroft (University of Glasgow (GB))

### Description

Presented is a method to improve the b-jet energy resolution with the task of increasing the signal sensitivity in searches decaying to a b, anti-b quark pair in the final state. The machinery has been tested using Monte Carlo simulated samples in the process $WH \rightarrow l\nu bb$. The method trains a function through regression applying a correction factor to bring the reconstructed jet energy closer to that of generator level energy. This narrows the b anti-b jet mass peak of the signal, and therefore increases the signal to background separation

### Primary author

Thomas Olivier Ravenscroft (University of Glasgow (GB))

 Slides


text:
# A person buys 16 litres of milk at ₹ 35 per litre. He adds x litre of water and sells the mixture of milk and water at ₹ 42 per litre and makes a profit of 57.5%. Find the value of x?

1. 5
2. 6
3. 8
4. 4
5. 12

Option 1 : 5

## Detailed Solution

57.5% = 23/40

The cost price of 16 litre of milk = ₹ (16 x 35) = ₹ 560

The selling price of (16 + x) litre of mixture = ₹ (16 + x) 42 = ₹ (672 + 42x)

According to question,

(Note : Assume the price of 1 litre of water is ₹ 0)
The selling price of 16 litre of milk = 560 $$\times \frac{{63}}{{40}}$$ = ₹ 882

⇒ 672 + 42x = 882

⇒ 42x = 882 - 672

⇒ 42x = 210

⇒ x = 5