Dispensationalism teaches us that this reference to being a child refers to the Old Testament dispensation, wherein those hearing the word of God were still under the Law of Moses, and that they were released from that when Jesus died. This does not seem to be the meaning here, in my opinion.
Being a child and under tutors and governors has reference to our being, as unbelievers, under the law, constrained by the law for a better moral behavior, and sternly corrected by the law as to our many wrongs. We as unbelievers are under the stringent rule of the law "until the time appointed of the father." In our case it would be the appointed time of our Heavenly Father.
When we were children, unbelievers, we were "in bondage under the elements of the world" (verse 3). This explains to us that, when we were lost in our sins, we were under the influence of only earthly and humanistic elements, worldly elements. Even the law is a worldly element, for all the law can do is bring us to guilt and shame; the law cannot bring freedom, yea, it has no desire to free us from itself: it only holds us to itself in jealous ownership. Thus, salvation cannot come by the law, since it is made up of "elements of the world."
In verses four and five Paul explains to us how that we can, despite the law's ever-insistent commands, be free from the law for salvation. It is because God sent forth his Son, made of a woman, made under the law, to redeem us so that we might receive, by his grace, the adoption of sons.
Now that we are free from the law and sons of God, being born from above (John 3:3), we have the Spirit of his Son (the Holy Spirit) in our hearts by which we call out to our Heavenly Father in a very personal way (6), knowing we are heirs of God and joint-heirs with Jesus Christ (Rom. 8:17).
Now that we are saved by grace through faith (Eph. 2:8-10), we are no more children, but sons, and heirs (7) according to the promise (Gal. 3:29) and according to the hope of eternal life (Titus 3:7).
Verse eight and nine describe that when we were children we "knew not God" and "did service unto them which by nature are no gods." We served the "elements of the world" and not God. But now, after we are known of God, how can we turn again to the "weak and beggarly elements"? How can we desire to be in bondage as a child under the "elements of the world" when we have been made free from them by the death, burial and resurrection of Jesus Christ, that perfect work that redeemed us by his blood through the death of the cross?