A helpful fact that few physicists will disagree with is that the wavefunction of the electron certainly goes through both slits. Also, personally I think that describing the whole wavefunction of the electron sufficiently describes that electron.
The origional and simplest model of QM models the electron as follows:
When the electron is not being measured, a "wavefunction" (which moves similarly to a wave) propagates in space. This wave can interfere with itself just like how a wave of water can split into smaller waves and interfere with itself. When the electron is measured, the probability of finding the electron is proportional with how high that wave is at the point in space that wave is located at.
Because the wave is associated with probability of seeing the electron at a specific spot, it's not as if the wave-itself can be seen. Furthermore, because the electron is not seen when it is traveling (since measuring it would stop this wave from existing), we cannot make statements easily about what is happening when it is traveling.
Ultimately though, there is a lot of good conceptual intuition that the electron is going through "Both" slits, simply because it behaves in a way that is different than if it had simply went through either one or the other.