user21840: I saw some grandmasters looking up or sideways during calculations. Why don't they just look at the board and visualize? Wouldn't that be easier?

user11749: Expanding on the great comments of @David and @Timothy Chow, GM Nikolai Krogius talks about the role of the residual image in his book Psychology in Chess. This is an image that stays  and blurs the calculations. The residual image is the transfer of judgment from a past position to a new situation. The past then acts on the present. 
To combat this residual image and to improve concentration, Krogius recommends practicing blindfold chess or analyzing without using a chess board. By not looking at the board, the player must constantly check the position of the pieces, thus erasing the residual image.

Here is an example of the residual image. At the beginning of the position the Bishop in b3 controls the square g8, but not ... at the end.
[FEN "r3k3/1p5p/p2R4/4p1Q1/4P1bP/PBq5/6P1/5K2 w - - 0 1"]
[title "Ilyin-Genevsky - Nenarokov, Moscow 1922"]

1.Bf7+! Kf8 2.Qh6+?? (2.Rf6!+-) Kxf7 3.Rf6+? (3.Qxh7+ Ke8 4.Qg6+ Ke7 5.Qf6+=) Kg8! {and now Black is better!}


Check also this article on ChessBase India, which asks your question

*We often see top chess players not looking at the board while
  calculating. It seems that their attention is somewhere else, yet
  after a few minutes they make the move without having even glanced at
  the position. What do you think they were doing for the last few
  minutes? Well, they were thinking in their head. They did not need to
  see the chess board to calculate the variations. They could do it in
  their head. So how do these super GMs calculate without a chess board?
  How are they able to make complex decisions in their mind, which
  normal players cannot fathom.

They welcome GM Vidit Gujrathi on how to calculate better in blindfold. 


21840: <p>I saw some grandmasters looking up or sideways during calculations. Why don't they just look at the board and visualize? Wouldn't that be easier?</p>
 11749: <p>Expanding on the great comments of @David and @Timothy Chow, GM Nikolai Krogius talks about the role of the <strong>residual image</strong> in his book <em>Psychology in Chess</em>. This is an image that stays  and blurs the calculations. The residual image is the transfer of judgment from a past position to a new situation. The past then acts on the present. </p>

<p>To combat this residual image and to improve concentration, Krogius recommends practicing <strong>blindfold chess or analyzing without using a chess board</strong>. By not looking at the board, the player must constantly check the position of the pieces, thus erasing the residual image.</p>

<hr>

<p>Here is an example of the residual image. At the beginning of the position the Bishop in b3 controls the square g8, but not ... at the end.</p>

<pre><code>[FEN "r3k3/1p5p/p2R4/4p1Q1/4P1bP/PBq5/6P1/5K2 w - - 0 1"]
[title "Ilyin-Genevsky - Nenarokov, Moscow 1922"]

1.Bf7+! Kf8 2.Qh6+?? (2.Rf6!+-) Kxf7 3.Rf6+? (3.Qxh7+ Ke8 4.Qg6+ Ke7 5.Qf6+=) Kg8! {and now Black is better!}
</code></pre>

<hr>

<p>Check also this article on <a href="https://chessbase.in/news/Vidit-Gujrathi-solving-studies" rel="nofollow noreferrer">ChessBase India</a>, which asks your question</p>

<blockquote>
  <p><strong>*We often see top chess players not looking at the board while
  calculating</strong>. It seems that their attention is somewhere else, yet
  after a few minutes they make the move without having even glanced at
  the position. What do you think they were doing for the last few
  minutes? Well, they were thinking in their head. They did not need to
  see the chess board to calculate the variations. They could do it in
  their head. So how do these super GMs calculate without a chess board?
  How are they able to make complex decisions in their mind, which
  normal players cannot fathom.</p>
</blockquote>

<p>They welcome GM Vidit Gujrathi on how to calculate better in blindfold. </p>

<p>
