user11518: Let's assume modern scientific discoveries contradict or alter the Buddha's explanations. How are we supposed to approach this new information as it relates to the Buddha's teachings?

user8157: The Buddha taught his Dhamma is visible here-&-now therefore science cannot contradict it. 
The Thai monk Ajahn Buddhadasa explained about the Pali word "jati" in Dependent Origination: 

The word "birth" ("jati") refers to the arising of the mistaken idea "I," "myself". It does not refer to physical birth, as generally supposed.
  The mistaken assumption that this word "birth" refers to physical
  birth is a major obstacle to comprehending the Buddha's teaching.
Another Kind of Birth

In other words, the Buddha did not teach about "reincarnation" or "physical rebirth". However, if modern science discovers reincarnation is true, it will then contradict the teachings of the Buddha.

user254: I tried to understand (classical) Buddhist doctrine even though I have modern preconceptions (from a "modern scientific" education).
Generally IMO "modern science" and "Buddhism" are different topics: e.g. modern science provides models for the phenomena you observe through a microscope or telescope, which IMO is not at all what Buddhism is trying to teach.
IMO the fact that Buddhist doctrine is translated using modern words (including e.g. "elements"), which are also associated with modern doctrines (e.g. "the periodic table"), is a potential source of confusion (in the student who uses these words), but shouldn't be understood as proof that the topics (Buddhism and science) are the same topic (e.g. that they're both attempting to solve the same problem or describe the same phenomenal observations).
Thus IMO a lot of problems are avoided by not accepting the premise which you granted (i.e. that we should "assume modern scientific discoveries contradict or alter the Buddha's explanations").
For example, let's imagine that you and I were having an argument: where you were saying one thing and I was saying another. I think I should assume that you're trying to say something, that "means something" (or "is true") to you. The question is, what's Buddhism trying to say, what's it getting at, and why, what does it see that I don't, where's it coming from, what can I learn from it?
Some other problems can be avoided by understanding some of the doctrine as metaphorical: expressed using the language (including the "scientific" models) of the time.
Some people say that some small parts of it (e.g. a sutta which includes a cosmological model, and a description of the evolution of the human species) is a parody of contemporary models. That's a theory. Personally if something doesn't make sense to me then I'm inclined to leave it and look for something which does make sense (or makes more sense). Within that sutta, for example, as in many other Buddhist stories, I'd ask "what's the moral of the story, what lesson is it trying to teach?"
I am (personal history) accustomed to literature that uses or consists of parables, which uses talking animals as characters for example to teach the reader something about life. I don't want to say that Buddhism is a fable, or shouldn't be taken literally, because I think it's often carefully expressed and well-expounded (and purposeful). But there may be something of a mix in the canon, see for example ...

What is the difference between relative and absolute truth in Buddhist philosophy?
What are 'suttas of indirect meaning' in the Pali canon?

... which doesn't even get into the topic of "metaphor" etc., but in summary perhaps you shouldn't be too rigid when you read it (also when you begin to understand how difficult it may be to translate the written canon accurately).
I personally know little (probably less than you do) about modern psychology (so I can't comment on whatever disparities you may see between Buddhism and that science), although the little I've read of Freud and successors seems to me kind of puerile (especially compared with Buddhism). I've had some (limited) personal experience with clinical psychiatry and I don't think I'm overly impressed by that as a "science" either (it's not as if it has made Buddhism obsolete).
I suppose this has been an egocentric and wordy answer on the whole, but maybe some bit of it could help you to answer your own question.

user11788: It would depend on the information, but when in doubt, I go back to the basics of the Four Noble Truths, and the Buddha's last words of "Be a light unto yourself." and would see where the new information fit in with those. Without an idea of how a Teaching might be disproved, it is difficult to speculate. 
If ultimately, it was somehow proven, and I feel this is unlikely, the Teaching is incorrect, then another path would need to be considered.

user12365: Dhammika all over the world, those who have found certain faith, reflect daily by recitation the sublimness of the holly Dhamma.
[Yo so svākkhāto] bhagavatā dhammo,
The Dhamma well-expounded by the Blessed One,
Sandiṭṭhiko akāliko ehipassiko,
to be seen here & now, timeless, inviting all to come & see,
Opanayiko paccattaṃ veditabbo viññūhi:
leading inward, to be seen by the wise for themselves:
Tam-ahaṃ dhammaṃ abhipūjayāmi,
Tam-ahaṃ dhammaṃ sirasā namāmi.
I worship most highly that Dhamma,
To that Dhamma I bow my head down.

akāliko timeless, independent of certain times, views...

That derives from the Canon, the refuge in the Dhamma and the reflective meditation on Dhamma.

"There is the case where the disciple of the noble ones recollects the Dhamma, thus: 'The Dhamma is well-expounded by the Blessed One, to be seen here & now, timeless, inviting verification, pertinent, to be realized by the wise for themselves.' As he is recollecting the Dhamma, his mind is calmed, and joy arises; the defilements of his mind are abandoned, just as when the body is cleansed through the proper technique. And how is the body cleansed through the proper technique? Through the use of scouring balls & bath powder & the appropriate human effort. This is how the body is cleansed through the proper technique. In the same way, the defiled mind is cleansed through the proper technique. And how is the defiled mind cleansed through the proper technique? There is the case where the disciple of the noble ones recollects the Dhamma... As he is recollecting the Dhamma, his mind is cleansed, and joy arises; the defilements of his mind are abandoned. He is thus called a disciple of the noble ones undertaking the Dhamma-Uposatha. He lives with Dhamma. It is owing to Dhamma that his mind is calmed, that joy arises, and that whatever defilements there are in his mind are abandoned. This is how the mind is cleansed through the proper technique.Muluposatha Sutta: The Roots of the Uposatha

That's the cleaning of the Noble Ones. If wihing to see it, know it, arive at Dhamma, the cleaning from defilements is required to see the truth in the jungle of cheatings. For it's impossible to judge the Dhamma with the means of wordily views and with defilements, having a stand like "me, "we", "they"...
How best approach "new daily opinions ideas" changing every day? Seeing that meaningless, not lasting even one day, let go of them, knowing that normal humans are not able to find the right way outside of the timesless eightfold path. 
There are many who, yet not able to conduct virtuose, attached to countless things, no liberation at all gained, think to be smarter then the Awakened One. Let them... orjoin if finding worthy refuge and real benefit from it. Aging, sickness death and those few going beyond will last nevertheless.
Kauva Ātman: if searching for Brahmanic or Mohayaniv approches, this modified ideas you could also find here for example: [English] Sarvastivada School of Realism, by Venerable Vira Avalokita. Imortand again: this are secular approches, defusing the Dhamma of the Buddha and certain counterfies of the well expounded sublime Dhamma.
[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]

11518: <p>Let's assume modern scientific discoveries contradict or alter the Buddha's explanations. How are we supposed to approach this new information as it relates to the Buddha's teachings?</p>
 8157: <p>The Buddha taught his Dhamma is visible here-&amp;-now therefore science cannot contradict it. </p>

<p>The Thai monk Ajahn Buddhadasa explained about the Pali word "<strong>jati</strong>" in Dependent Origination: </p>

<blockquote>
  <p><em>The word "birth" ("<strong>jati</strong>") refers to the arising of the mistaken idea "I," "myself". It does not refer to physical birth, as generally supposed.
  The mistaken assumption that this word "birth" refers to physical
  birth is a major obstacle to comprehending the Buddha's teaching.</em></p>
  
  <p><a href="http://www.dhammatalks.net/Books3/Bhikkhu_Buddhadasa_Another_Kind_of_Birth.htm" rel="nofollow noreferrer"><em>Another Kind of Birth</em></a></p>
</blockquote>

<p>In other words, the Buddha did not teach about "reincarnation" or "physical rebirth". However, if modern science discovers reincarnation is true, it will then contradict the teachings of the Buddha.</p>
 254: <p>I tried to understand (classical) Buddhist doctrine even though I have modern preconceptions (from a "modern scientific" education).</p>

<p>Generally IMO "modern science" and "Buddhism" are different topics: e.g. modern science provides models for the phenomena you observe through a microscope or telescope, which IMO is <em>not at all</em> what Buddhism is trying to teach.</p>

<p>IMO the fact that Buddhist doctrine is translated using modern words (including e.g. "elements"), which are also associated with modern doctrines (e.g. "the periodic table"), is a potential source of confusion (in the student who uses these words), but shouldn't be understood as proof that the topics (Buddhism and science) are the same topic (e.g. that they're both attempting to solve the same problem or describe the same phenomenal observations).</p>

<p>Thus IMO a lot of problems are avoided by not accepting the premise which you granted (i.e. that we should "assume modern scientific discoveries contradict or alter the Buddha's explanations").</p>

<p>For example, let's imagine that you and I were having an argument: where you were saying one thing and I was saying another. I think I should assume that you're trying to say <em>something</em>, that "means something" (or "is true") to you. The question is, what's Buddhism trying to say, what's it getting at, and why, what does it see that I don't, where's it coming from, what can I learn from it?</p>

<p>Some other problems can be avoided by understanding some of the doctrine as metaphorical: expressed using the language (including the "scientific" models) of the time.</p>

<p>Some people say that some small parts of it (e.g. a sutta which includes a cosmological model, and a description of the evolution of the human species) is a parody of contemporary models. That's a theory. Personally if something doesn't make sense to me then I'm inclined to leave it and look for something which <em>does</em> make sense (or makes <em>more</em> sense). Within that sutta, for example, as in many other Buddhist stories, I'd ask "what's the <em>moral</em> of the story, what <em>lesson</em> is it trying to teach?"</p>

<p>I am (personal history) accustomed to literature that uses or consists of parables, which uses talking animals as characters for example to teach the reader something about life. I don't want to say that Buddhism is a fable, or shouldn't be taken literally, because I think it's often carefully expressed and well-expounded (and purposeful). But there may be something of a mix in the canon, see for example ...</p>

<ul>
<li><a href="https://buddhism.stackexchange.com/q/1259/254">What is the difference between relative and absolute truth in Buddhist philosophy?</a></li>
<li><a href="https://buddhism.stackexchange.com/q/11627/254">What are &#39;suttas of indirect meaning&#39; in the Pali canon?</a></li>
</ul>

<p>... which doesn't even get into the topic of "metaphor" etc., but in summary perhaps you shouldn't be too rigid when you read it (also when you begin to understand how difficult it may be to translate the written canon accurately).</p>

<p>I personally know little (probably less than you do) about modern psychology (so I can't comment on whatever disparities you may see between Buddhism and that science), although the little I've read of Freud and successors seems to me kind of puerile (especially compared with Buddhism). I've had some (limited) personal experience with clinical psychiatry and I don't think I'm overly impressed by that as a "science" either (it's not as if <em>it</em> has made Buddhism obsolete).</p>

<p>I suppose this has been an egocentric and wordy answer on the whole, but maybe some bit of it could help you to answer your own question.</p>
 11788: <p>It would depend on the information, but when in doubt, I go back to the basics of the Four Noble Truths, and the Buddha's last words of "Be a light unto yourself." and would see where the new information fit in with those. Without an idea of how a Teaching might be disproved, it is difficult to speculate. </p>

<p>If ultimately, it was somehow proven, and I feel this is unlikely, the Teaching is incorrect, then another path would need to be considered.</p>
 12365: <p>Dhammika all over the world, those who have found certain faith, reflect daily by recitation the sublimness of the holly Dhamma.</p>

<p><code>[Yo so svākkhāto] bhagavatā dhammo,
The Dhamma well-expounded by the Blessed One,</code></p>

<p><code>Sandiṭṭhiko akāliko ehipassiko,
to be seen here &amp; now, timeless, inviting all to come &amp; see,</code></p>

<p><code>Opanayiko paccattaṃ veditabbo viññūhi:
leading inward, to be seen by the wise for themselves:</code></p>

<p><code>Tam-ahaṃ dhammaṃ abhipūjayāmi,
Tam-ahaṃ dhammaṃ sirasā namāmi.
I worship most highly that Dhamma,
To that Dhamma I bow my head down.</code></p>

<ul>
<li><strong>akāliko</strong> timeless, independent of certain times, views...</li>
</ul>

<p>That derives from the Canon, the refuge in the Dhamma and the reflective meditation on Dhamma.</p>

<blockquote>
  <p>"There is the case where the disciple of the noble ones recollects the Dhamma, thus: 'The Dhamma is well-expounded by the Blessed One, to be seen here &amp; now, timeless, inviting verification, pertinent, to be realized by the wise for themselves.' As he is recollecting the Dhamma, his mind is calmed, and joy arises; the defilements of his mind are abandoned, just as when the body is cleansed through the proper technique. And how is the body cleansed through the proper technique? Through the use of scouring balls &amp; bath powder &amp; the appropriate human effort. This is how the body is cleansed through the proper technique. In the same way, the defiled mind is cleansed through the proper technique. And how is the defiled mind cleansed through the proper technique? There is the case where the disciple of the noble ones recollects the Dhamma... As he is recollecting the Dhamma, his mind is cleansed, and joy arises; the defilements of his mind are abandoned. He is thus called a disciple of the noble ones undertaking the Dhamma-Uposatha. He lives with Dhamma. It is owing to Dhamma that his mind is calmed, that joy arises, and that whatever defilements there are in his mind are abandoned. This is how the mind is cleansed through the proper technique.<a href="http://zugangzureinsicht.org/html/tipitaka/an/an03/an03.070.than_en.html" rel="nofollow noreferrer">Muluposatha Sutta: The Roots of the Uposatha</a></p>
</blockquote>

<p>That's the cleaning of the Noble Ones. If wihing to see it, know it, arive at Dhamma, the cleaning from defilements is required to see the truth in the jungle of cheatings. For it's impossible to judge the Dhamma with the means of wordily views and with defilements, having a stand like "me, "we", "they"...</p>

<p>How best approach "new daily opinions ideas" changing every day? Seeing that meaningless, not lasting even one day, let go of them, knowing that normal humans are not able to find the right way outside of the timesless eightfold path. </p>

<p>There are many who, yet not able to conduct virtuose, attached to countless things, no liberation at all gained, think to be smarter then the Awakened One. Let them... orjoin if finding worthy refuge and real benefit from it. Aging, sickness death and those few going beyond will last nevertheless.</p>

<p>Kauva Ātman: if searching for Brahmanic or Mohayaniv approches, this modified ideas you could also find here for example: <a href="http://sangham.net/index.php/topic,1087.msg4381.html#msg4381" rel="nofollow noreferrer">[English] Sarvastivada School of Realism, by Venerable Vira Avalokita</a>. Imortand again: this are secular approches, defusing the Dhamma of the Buddha and certain counterfies of the well expounded sublime Dhamma.</p>

<p><em>[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]</em></p>
