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 Math Notes >> Math Cognition

The perils of too much information [Math Cognition
Posted on January 5, 2007 @ 02:43:32 PM by Paul Meagher

Reading the bestselling book "Blink" (2005) by Malcom Gladwell. He has a section in which he discusses the Goldman algorithm used in ER settings for diagnosing an impending heart attack. The Goldman algorithm is interesting because 1) it is demonstrably more successful in predicting an impending heart attack then relying upon clinical judgement to do so, 2) it takes into account less information than a doctor might prefer to use in trying to diagnose a possible heart attack (it only takes into account ECG readings, blood pressure readings, whether fluid is present in the lungs, and whether the angina pain is stable or not), and 3) it implies a fairly radical insight that more information about a patient is not necessarily a good thing especially in medical decision making in ER settings:

Take, for instance, the hypothetical case of a man who comes into the ER complaining of intermittent left-side chest pain that occasionally comes when he walks up the stairs and that lasts from five minutes to three hours. His chest exam, heart exam, and ECG are normal, and his systolic blood pressure is 165, meaning it doesn't qualify as an urgent factor. But he's in his sixties. He's a hard-charging exectuive. He's under constant pressure. He smokes. He doesn't exercise. He's had high blood pressure for years. He's overweight. He had heart surgery two years ago. He's sweating. It certainly seems like he ought to be admitted to the coronary care unit right away. But the algorithm says he shouldn't be. All those extra factors certainly matter in the long term. The patient's condition and diet and lifestyle put him at serious risk of developing heart disease over the next few years. It may even be that those factors play a very subtle and complex role in increasing the odds of something happening to him in the next seventy-two hours. What Goldman's algorithm indicates, though, is that the role of these other factors is so small in determining what is happening to the man right now that an accurate diagnosis can be made without them. In fact [...] that extra information is more than useless. It's harmful. It confuses the issues. What screws up doctors when they are trying to predict heart attacks is that they take too much information into account. p. 137

There are lots of other reasons to collect a full history on patients admitted to ER for chest pains but arriving at a diagnosis of whether to route them to a costly coronary care unit is not one of them. Another reason why it is often not a good idea to collect too much patient information is because it tends to increase the confidence of the doctor in their diagnosis when there is in fact no grounds for such increased confidence (i.e., they are attending to information that, for the most part, is irrelevant and possibly misleading for the present purposes but which falsely increases their confidence in their diagnosis because they think they are getting the "full picture").

There are important lessons here to be considered when designing web applications to support decision making: giving users more information to incorporate into their decision making is not a good idea if that information is causally impotent or irrelevant for the decisions the system is designed to support. In fact, the mere presence of such information might negitively impact decision making by falsely implying the relevance of such information to the decision at hand.

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Effects of music training on math performance [Math Cognition
Posted on November 10, 2006 @ 03:45:26 AM by Paul Meagher

There is quite a bit of research showing a positive correlation between music training and math performance in early grades. Some of this research suggests that music training helps develop spatio-temporal reasoning in kids. Today as I watched my 9 year-old daugter transcribe some music notation I was very impressed with the complexity of the task of transcribing music notation and the thought occurred to me that perhaps the benefit of music training has more to do with learning to master complex notational systems than higher levels of activation in putative spatio-temporal reasoning areas of the brain. These are not mutually exclusive explanations of the linkage, however, I think that the notational mastery hypothesis needs to be pursued more vigorously as an important contributing factor in the linkage.

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Quadratic scoring rule [Math Cognition
Posted on November 7, 2006 @ 02:56:54 PM by Paul Meagher

David V. Lindley, in Making Decisions (1985), argues for the usefulness of teaching people to use the quadratic scoring rule as a way to learn to better estimate event probabilities.

The idea behind the quadratic scoring rule is to have people estimate the probabability of various events being true or false (e.g., The Golden Bough was written by Henry James?) and then score their probability estimates.

The particular score proposed is (1-p)2 if the event is true and p2 if it is false. The score is to be thought of as a penalty score, so that the smaller the score the better you have done. It is usual to multiply the values by 100 and to ignore values after the decimal point, if any.

So, for example, if an event was true and you estimated that it was true with a p value of .90 then your score would be (1 - 0.9)2 = .12 = .01. Now multiply this by 100 and your penalty score is only 1. Contrast this with a p value estimate of 0.2 which produces a penalty score of 64. If you thought the event was false (p=0) then you would receive the maximum penalty score of 100. Note that it is a nice feature of this penalty score that the range is between 0 and 100.

Exercise

Implement the quadratic scoring rule as a PHP script that generates the following table:

  Score
Probability True Event False Event
0.0 100 0
0.1 81 1
0.2 64 4
0.3 49 9
0.4 36 16
0.5 25 25
0.6 16 36
0.7 9 49
0.8 4 64
0.9 1 81
1.0 0 100

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Interactive training on arithmetic facts using PHP [Math Cognition
Posted on October 18, 2006 @ 11:57:12 AM by Paul Meagher

At the end of the last school semester I developed the PHP script below to help my daugher practice her arithmetic facts. The impetus was end of year speed tests for arithmetic retrieval. All kids at the end of grade 3 should to be able to generate an answer to basic arithmetic questions in under 3000 msec. In many schools, failure to achieve the retrieval criterion of 3000 msec per fact can be used to prevent kids from getting into the next grade so it is an critical skill to master at an early age.

<?php
/**
* @author Paul Meagher
* @updated May 18, 2005
*/
session_name("practice");
session_start();

$min_rand 0;
$max_rand 18;

$first_number  rand($min_rand$max_rand);
$second_number rand($min_rand$max_rand);

if (
$_GET['op'] == "clear"$_SESSION = array();

if (!isset(
$_SESSION['num_right'])) $_SESSION['num_right'] = 0;
if (!isset(
$_SESSION['num_wrong'])) $_SESSION['num_wrong'] = 0
if (!isset(
$_SESSION['num_probs'])) $_SESSION['num_probs'] = 0

if (
$_POST['form_submitted'] == "yes") {
  
$fn $_POST['fn'];
  
$sn $_POST['sn'];
  
$ca $fn $sn;
  
$_SESSION['num_probs']++;      
  
$np $_SESSION['num_probs'];  
  
$sa $_POST['submitted_answer'];    
  if (
$ca == $sa) {    
    echo 
"<p>Hooray, you answered problem # $np correctly: $fn + $sn = $ca</p>";
    
$_SESSION['num_right']++;
  } else { 
    echo 
"<p><font color='red'><b>Darn, you answered problem # $np incorrectly: $fn + $sn = $ca, <i>not</i> $sa</b></font> </p>";
    
$_SESSION['num_wrong']++;    
  }
}
?>

<table>
  <form method='post' action='<?php echo $_SERVER['PHP_SELF'?>'>
  <input type='hidden' name='form_submitted' value='yes'>   
  <input type='hidden' name='fn' value='<?php echo $first_number ?>'> 
  <input type='hidden' name='sn' value='<?php echo $second_number ?>'>   
  <tr>
    <td><?php echo $first_number ?></td>
    <td> + </td>        
    <td><?php echo $second_number ?></td>    
    <td> = </td>            
    <td><input type='text' name='submitted_answer' value='' size='2'></td>
    <td><input type='submit' value='Check Answer'></td>    
  </tr>
  </form>
</table> 

<br />

Number of right answers: <?php echo $_SESSION['num_right'?><br />
Number of wrong answers: <?php echo $_SESSION['num_wrong'?><br />

<br />

<a href='<?php echo $_SERVER['PHP_SELF']."?op=clear"?>'>Start over</a>.

This year I started using it with my son who is two grades behind in the same school. I was pleasantly suprised at how quickly both of them took to using this script. My 7 year old son in particular related very naturally to this form of interactive learning, probably because he consumes a steady diet of computers, internet, and video games (e.g., lego star wars has kept his interest for a good while now).

I have started to experiment with adding some more features to my basic practrice script. For example, I now have a graphic that displays when they get an answer right or wrong. The wrong answer is a humorous "I failed you yoda" graphic.

What I am doing here is something that any person calling themselves a PHP developer can do with their son or daughter (i.e., develop scripts to help them practice their arithmetic facts). For those of you with sons or daughters who are learning their arthmetic facts inefficiently, perhaps what they need is to be given a more interactive approach not necessarily by a human to learn how to quickly retrieve arithmetic facts.

Now, dear reader, go practice your addition facts.

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The bat and ball problem [Math Cognition
Posted on August 14, 2006 @ 01:22:21 PM by Paul Meagher

Try to solve this problem:

A bat and a ball cost $1.10 in total. The bat costs $1 more that the ball. How much does the ball cost?

I have added the answer to the comments for this posting.

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