When I ask students what they most wish to improve about their playing, the most common answer by far is intonation.
I ask: "What causes poor intonation?"
The usual answers?
"Putting your finger down in the wrong spot."
These aren't bad answers. For the first, I'd say that not understanding the meaning and nuances of intonation is a bigger problem than not listening for them, since you can definitely listen attentively without knowing what you're listening for. (See Chapter 3 and Chapter 6 of Cello Practice, Cello Performance for an explanation of different types of intonation and how the cello can help you determine how best to play in tune.)
The second answer describes what is going on. It's true that putting your finger in the wrong place contributes to poor intonation. But why do we put our fingers down wrongly?
Here's what I think is happening. When a student is playing habitually out of tune, I ask them to play a problematic passage while I observe the left hand from all angles. For intonation problems in the neck position, the main mistake is almost always that the student is squeezing the fingerboard tensely, with the thumb habitually opposing the first finger. The hand gets cramped and tight because the fingers can't move easily from note to note.
Some teachers actually teach beginning students to position the left hand like this, and I think it's a mistake. Others teach that the thumb should oppose the second finger, and while this is a giant improvement over the first way, it can still lead to a rigid, "set" hand position that doesn't extend and shift easily around the neck of the cello.
If you look up the human hand in an anatomy textbook, it'll tell you that the thumb opposes not just the first, not just the second, but all four fingers. This is what enables monkeys to swing between branches and hold bananas. You don't see monkeys holding bananas between thumb and forefinger, because they'd drop the banana. Nor do they squeeze them, because who wants a squashed banana? Instead, they hold them in a balanced manner between thumb and all four fingers.
Which is just the way we ought to be handling the neck of the cello.
In other words, you should let the thumb move softly around the neck of the cello, letting it oppose whichever finger is down as you move from note to note.
This not only helps with intonation, but with shifting, and freeing the left arm in general. It lets you sink your arm's weight into the pads of your fingers so that you use your whole arm to move from finger to finger. It gives your fingers greater independence and frees you to change the speed and amplitude of your vibrato.
So, if you practice this exercise in chromatic tetrachords....
....your thumb should move like this as you move from note to note with your fingers.
Click here to download a free PDF of another useful exercise in thumb opposition.