The question is not whether women may teach publicly in the church, for all orthodox Christians agree that Scripture prohibits women from holding office in the Church and from doing duties reserved for church officers (1 Cor. 14:34-35; 1 Tim. 2:11-15). Rather, the question is if women may otherwise exercise public authority over men. Secondly, we distinguish between private teaching, admonition, and exhortation of one another and public capacities and acts of authority.
Without getting into specific examples (which certainly need to be discussed after all parties agree with the thesis of this article), we will address this question focusing primarily on the two passages previously cited.
It may be a legitimate question to ask what constitutes an exercise of authority, but it would be problematic to restrict the public exercises of authority by women to merely the ecclesiastical sphere and to deny that Scripture roots this prohibition in the created order.
“It is not permitted unto them to speak” — Why not?
34 Let your women keep silence in the churches: for it is not permitted unto them to speak; but they are commanded to be under obedience as also saith the law.
35 And if they will learn any thing, let them ask their husbands at home: for it is a shame for women to speak in the church.
(1 Corinthians 14:34-35)
Matthew Poole notes that this prohibition does not apply to extraordinary occasions where God saw fit to make a woman a prophetess, such as Anna (Luke 2:38), the four daughters in Acts 21:9, Miriam (Exodus 15:20), Huldah (2 Chronicles 34:22), etc. but these are exceptions and not the norm. Poole notes that the Apostle’s reference to the Law refers to Genesis 3:16, which is important for our consideration because it has to do with the created order, not something specific to old covenant Israel, or to the sphere of the New Testament Church.
“The law to which the apostle here refers, is thought to be that, Genesis 3:16, where the woman is commanded to be subject to her husband, and it is said, that he should rule over her…”
John Calvin comments on this passage noting that the reasoning for prohibiting women from speaking in an official, public, or authoritative capacity is grounded in the Law, nature, and common sense, such that it was even a moral principle held by people without access to special revelation; it is not grounded in super-added positive law specific only to the Church.
“If the woman is under subjection, she is, consequently, prohibited from authority to teach in public. And unquestionably, among all nations and peoples wherever even natural propriety has been maintained, women have in all ages been excluded from the public management of affairs. It is the dictate of common sense, that female government is improper and unseemly. Nay more, while originally they had permission given to them at Rome to plead before a court, the effrontery of Caia Afrania led to their being interdicted, even from this. Paul’s reasoning, however, is simple — that authority to teach is not suitable to the station that a woman occupies, because, if she teaches, she presides over all the men, while it becomes her to be under subjection.”
John Calvin, commentary on 1 Corinthians 14:34
“I suffer not a woman to teach” — Why not?
11 Let the woman learn in silence with all subjection.
12 But I suffer not a woman to teach, nor to usurp authority over the man, but to be in silence.
13 For Adam was first formed, then Eve.
14 And Adam was not deceived, but the woman being deceived was in the transgression.
15 Notwithstanding she shall be saved in childbearing, if they continue in faith and charity and holiness with sobriety.
(1 Timothy 2:11-15).
The immediate context is about prohibiting women from teaching publicly in the Church, but does the reasoning behind the prohibition apply only to the Church, or does it have a broader application? The prohibition in 1 Timothy 2:12 cannot be solely an ecclesiastical ordinance because the Apostle Paul grounds it in creation, not in anything specific to the Church. He is applying a general principle to a specific situation, therefore anywhere that same general principle is true, it applies.
John Gill rightly understands there to be natural equity in this passage that applies not only to church officers, but to other spheres of public life as well:
‘But I suffer not a woman to teach,’
They may teach in private, in their own houses and families; they are to be “teachers of good things” (Titus 2:3). They are to bring up their children “in the nurture and admonition of the Lord” (Eph. 6:4); nor is the law or doctrine of a mother to be forsaken, any more than the instruction of a father (see Prov. 1:8; 31:1-4). Timothy, no doubt, received much advantage, from the private teachings and instructions of his mother Eunice, and grandmother Lois; but then women are not to teach in the church; for that is an act of power and authority, and supposes the persons that teach to be of a superior degree, and in a superior office, and to have superior abilities to those who are taught by them:
‘nor to usurp authority over the man;’
as not in civil and political things, or in things relating to civil government; and in things domestic, or the affairs of the family; so not in things ecclesiastical, or what relate to the church and government of it; for one part of rule is to feed the church with knowledge and understanding; and for a woman to take upon her to do this, is to usurp an authority over the man: this therefore she ought not to do.
John Gill, commentary on 1 Timothy 2:12.
The Apostle Paul is speaking in the context of public worship, but the violation of the law of which he speaks is more general, which is why he reasons from creation. The Apostle is making an argument from the greater to the lesser. He uses a general moral principle rooted in the created order to explain why a particular instance of that same principle holds for the application he’s making. The fact that God created Adam first and then Eve was to teach us that the female gender is intended to be a helpmate to the male, not a ruler. He also reasons from the Fall. “Last in being, she was first in sin. The subtle serpent knew she was ‘the weaker vessel.’ He therefore tempted her.”  Therefore Paul does not allow a woman to teach or exercise authority, which would, as an application of the law, forbid her from such an exercise of authority in the churches. However, it also would have many other areas of application. This is why Isaiah declares that women ruling over men is a judgment of God upon a people (Isa. 3:12), and why Paul instructs that women are to “bear children, guide the house” (1 Tim. 5:14), and to be “keepers at home” (Titus 2:5).
Moreover, in this passage, the Apostle redirects the focus of the role of women from public and formal authority to “childbearing“, the sphere of the home, according to God’s design. There is certainly more to unpack in verse 15 , but this point suffices for our purpose here.
“It is implied that the very curse will be a condition favourable to her salvation, by her faithfully performing her part in doing and suffering what God has assigned to her, namely, child-bearing and home duties, her sphere, as distinguished from public teaching, which is not hers, but man’s (1 Tim. 2:11-12). In this home sphere, not ordinarily in public service for the kingdom of God, she will be saved on the same terms as all others, namely, by living faith.”
Jamieson Fausset Brown commentary on 1 Timothy 2:15, vol. 3, p. 486.
Likewise, J.G. Vos and Phil Martin, in their judicious pamphlet “Are Women Elders Scriptural?” (1939), note that, “The common interpretation [of 1 Tim. 2:15] is that by keeping house and being a joyful mother of children (Psalm 113:9) woman will find her true sphere and exalted mission in life, rather than in occupying positions of leadership and authority over men.” (point 14). Again, this is an application of a broad principle—established in nature and God’s design for the sexes—to a specific circumstance.
Again, it may be a legitimate question to ask what constitutes an exercise of authority, but it would be problematic to restrict the public exercises of authority by women to merely the ecclesiastical sphere and to deny that Scripture roots this prohibition in the created order.
Private instruction, admonition, or exhortation is not the same as exercising authority over men in the church, state, home, or any other sphere. That is why Reformed commentators draw a distinction between public positions of authority and private discourse.
For instance, John Gill comments on Acts 18:26 that Aquila and Priscilla did not correct Apollos publicly, rather they took him aside privately for the following three reasons: 1) For the sake of the Gospel “that they might not lay any stumblingblocks in the way of that, and of young converts, and give an occasion to the adversary to make advantages.” 2) For the sake of Apollos himself, “that they might not put him to the blush, and discourage him.” And 3) Because Priscilla was a woman, “it not being proper, especially for Priscilla, to speak in public, nor was it allowed in the Jewish synagogues for a woman to speak there.” He goes on, “and from hence it may be observed, that women of grace, knowledge, and experience, though they are not allowed to teach in public, yet they may, and ought to communicate in private, what they know of divine things, for the use of others.“
On the same passage John Calvin notes, “we must remember that Priscilla did execute this function of teaching at home in her own house, that she might not overthrow the order prescribed by God and nature.” Likewise Matthew Poole distinguishes between public positions of authority and private instruction:
“If we allow Priscilla to have contributed towards the instruction of Apollos, as doubtless we may, it is certain it was only in private discourse; which being joined with a meek and humble behaviour, might be very effectual for the conversion of souls (1 Peter 3:1-2). Thus Timothy was indebted for his knowledge in the things of God to his mother and grandmother (2 Timothy 1:5). But otherwise it is not lawful for a woman to teach (1 Timothy 2:11-12).”
Matthew Poole, commentary on Acts 18:26
With this principle properly understood, from the light of nature and from Scripture, Christians have a common basis to discuss specific roles and functions and what constitutes a public exercise of authority. This principle has been understood in the church for centuries, as seen from the preceding quotes. Although this is not a popular idea in modern society, we understand that this applies to both civil and ecclesiastical spheres. By fulfilling their proper role in society, women fulfill their duty to God and benefit their families and society as a whole. By acting outside of their God given role, women actually harm themselves and society when they exercise authority over men not only in the church, but in public as well.
 Jamieson Fausset Brown commentary on 1 Timothy 2:13, vol. 3, p. 486.
 See Stephen Charnock’s exegesis of this verse here: Discourse For The Comfort of Childbearing Women