Consider the following pasuk in
Vayakhel, 35:23:
| כה וְכָל-אִשָּׁה חַכְמַת-לֵב, בְּיָדֶיהָ טָווּ; וַיָּבִיאוּ מַטְוֶה, אֶת-הַתְּכֵלֶת וְאֶת-הָאַרְגָּמָן, אֶת-תּוֹלַעַת הַשָּׁנִי, וְאֶת-הַשֵּׁשׁ. | 25 And all the women that were wise-hearted did spin with their hands, and brought that which they had spun, the blue, and the purple, the scarlet, and the fine linen. |
There is a slight irregularity in the beginning of the pasuk, in
וְכָל-אִשָּׁה חַכְמַת-לֵב, בְּיָדֶיהָ טָווּ. Maybe it is the number agreement. That is,
וְכָל-אִשָּׁה חַכְמַת-לֵב is either plural (all the women) or singular (each of the women). Then, we see the word
בְּיָדֶיהָ which is singular (her hands, rather than their hands). And finally, we see the word
טָווּ, which is plural (they spun).
There are different ways of resolving this awkwardness. For instance, I suspect that the translation in the
Septuagint is partly motivated by this:
25 καὶ πᾶσα γυνὴ σοφὴ τῇ διανοίᾳ ταῖς χερσὶ νήθειν ἤνεγκαν νενησμένα, τὴν ὑάκινθον καὶ τὴν πορφύραν καὶ τὸ κόκκινον καὶ τὴν βύσσον·
25 And every woman skilled in her heart to spin with her hands, brought spun [articles], the blue, and purple, and scarlet and fine linen.
That is, they take is as
chochmat lev in the sense that
yadeha tavu, her hands spin. Maybe.
I would suggest that we should associate the word
בְּיָדֶיהָ with what precedes rather than with what follows. Thus, it should be
וְכָל-אִשָּׁה חַכְמַת-לֵב, בְּיָדֶיהָ, 'and every woman who was wise-hearted (skilled) in her hands',
טָווּ, did spin. This is against the
trup, but the
trup can be in accordance with the midrash, rather than with the
peshat.
On to the midrash. If we read the pasuk in the traditional way, then it tells us that:
וְכָל-אִשָּׁה חַכְמַת-לֵב --And all the women that were wise-hearted
בְּיָדֶיהָ טָווּ -- did spin with their hands
How can this be? Are there not multiple ways in which someone can be wise-hearted? A woman could be more than just a spinster. She can be an expert cook, an expert tailor, an expert doctor, or an expert scholar! Yet, this pasuk makes the simple assumption that any woman who is skilled / wise-hearted will be an expert spinner of wool.
One answer would be to reparse the pasuk, as I suggested above, so that there is indeed clarification as to the woman's particular skill.
Another answer is that indeed, any skilled wise-hearted woman will know how to spin wool, among her other skills.
And a final answer is that indeed, for women, there is only one skill or
chochma, and that is spinning.
We see these two interpretations suggested by various midrashim.
Let us turn to Torat HaTur, a nice
sefer that cites the Tur's
halachic work every time he cites a
pasuk in the
parasha. Naturally, since the Tur cites gemaras at length in his
halachic work, in order to show the basis of the
halacha in the gemara, what we have is really citations of
gemaras which the Tur brings
lehalacha. This might not be so "useful", but it is a useful way to learn some Tur and some
halacha and tie it in to the
parsha. Also, the author of this work, Rabbi Moshe HaLevi Steinberg, rav of Kiryat Yam and son of Rabbi Yitzchak Shteinberg, has a running commentary in the footnotes on the bottom.
Here is what the
Tur has to say:
וכל אשה חכמת לב בידיה טוו. הטור אבה״ע הל׳ כתובית
סי׳ פ׳. ומה עושה לו, הכל כמנהג המדינה. במקום שדרכן לארוג
אורגת. לרקום, רוקמת. לטוות פשתן או צמר. טווה. ואם לא
היה דרך אנשי העיר לעשות כל המלאכות האלו. אינה כופה
לעשות, אלא בצמר בלבד, שהפשתן מזיק הפה והשפתים. והטווי.
היא מלאכה המיוחדת לנשים, שנאמר וכל אשה חכמת לב בידיה
טוו:
וְכָל-אִשָּׁה חַכְמַת-לֵב, בְּיָדֶיהָ טָווּ -- The Tur in Even HaEzer, Hilchot Ketubot Siman 80: And what [labor] does she do for him [in the marriage]? All is in accordance with the custom of the country. In a place where their way is to weave she weaves. To embroider, she embroiders. To spin linen [flax] or wool, she spins. And if it is not the way of the residents of the city to do each of these labors, he does not compel her to do, except [spinning] with wool alone, for flax damages the mouth and the lips. And spinning is a labor designated for women, for it is stated וְכָל-אִשָּׁה חַכְמַת-לֵב, בְּיָדֶיהָ טָווּ.
Perhaps the idea is that specifically the women did this labor, while men did other labors for the Mishkan.
At any rate, here is the commentary:

"See
Yoma 66b, 'from here [this pasuk] they said that there is no
chochma for a woman except in the spindle. And the
nafka mina in this the Torah Temima wrote in note 16, that we do not give over
divrei Torah to women.
[
Josh: The gemara is:
שאלה אשה חכמה את ר' אליעזר מאחר שמעשה העגל שוין מפני מה אין מיתתן שוה אמר לה אין חכמה לאשה אלא בפלך וכן הוא אומר (שמות לה) וכל אשה חכמת לב בידיה טוו
And so Rabbi Eliezer refused to answer the wise woman's Torah question, based on this idea that there is no
chochma for a woman except in the spindle.
]
And this matter is explicit in the [
bolded words in the parallel]
Yerushalmi Sotah perek 3 halacha 4,
מטרונה שאלה את רבי לעזר מפני מה חט אחת במעשה העגל והן מתים בה שלש מיתות. אמר לה אין חכמתה של אשה אלא בפילכה דכתיב (שמות לה) וכל אשה חכמת לב בידיה טוו. אמר לו הורקנוס בנו בשביל שלא להשיבה דבר אחד מן התורה איבדת ממני שלש מאות כור מעשר בכל שנה. אמר ליה ישרפו דברי תורה ואל ימסרו לנשים. וכשיצתה אמרו לו תלמידיו רבי לזו דחיתה לנו מה אתה משיב.
A [wealthy] matron asked Rabbi Lazer: For what reason was there one sin in the incident of the Golden Calf and they died three deaths? He [Rabbi Lazar] said to her: A woman's wisdom is only in the spindle, as is written (Shemot 35) "And every wise-hearted woman, with her hands spun." Hyrkanus his son said to him: Because you did not answer her one thing from the Torah, you have lost from me 300 kur of
maaser each year!
He [Rabbi Lazer] said to him: Words of Torah should be burnt rather than giving them over to women. And when she had left, his students said to him, "Master, to this one you have pushed off. What will you say to us?"
[
Josh: I am interjecting here once again to comment a bit on this. What were the three deaths? One seemingly being investigated like an accused Sotah, in
Shemot 32:
| כ וַיִּקַּח אֶת-הָעֵגֶל אֲשֶׁר עָשׂוּ, וַיִּשְׂרֹף בָּאֵשׁ, וַיִּטְחַן, עַד אֲשֶׁר-דָּק; וַיִּזֶר עַל-פְּנֵי הַמַּיִם, וַיַּשְׁקְ אֶת-בְּנֵי יִשְׂרָאֵל. | 20 And he took the calf which they had made, and burnt it with fire, and ground it to powder, and strewed it upon the water, and made the children of Israel drink of it. |
at the hand of the Leviim who fought at Moshe's command (same perek)
| כח וַיַּעֲשׂוּ בְנֵי-לֵוִי, כִּדְבַר מֹשֶׁה; וַיִּפֹּל מִן-הָעָם בַּיּוֹם הַהוּא, כִּשְׁלֹשֶׁת אַלְפֵי אִישׁ. | 28 And the sons of Levi did according to the word of Moses; and there fell of the people that day about three thousand men. |
and another, plague (same perek):
| לה וַיִּגֹּף ה, אֶת-הָעָם, עַל אֲשֶׁר עָשׂוּ אֶת-הָעֵגֶל, אֲשֶׁר עָשָׂה אַהֲרֹן. {ס} | 35 And the LORD smote the people, because they made the calf, which Aaron made. |
And thus the resolution of the gemara of the three types of sinners: Whoever sacrificed and burned incense died by the sword; whoever embraced and kissed the calf died by the plague; and whoever rejoiced in his heart died of dropsy (Yoma 66b). Or, these two suggestions from the Yerushalmi:
רבי ברכיה רבי אבא בר כהנא בשם רבי ליעזר כל מי שהיה לו עדים והתראה היה מת בבית דין. עדים ולא התרייה היה נבדק כסוטה. לא עדים ולא התרייה היה מת במגפה. רב ולוי בר סיסי תריהון אמרין זיבה קיטר ניסך היה מת בבית דין. טיפח ריקד שיחק היה נבדק כסוטה שמח בליבו היה מת במגפה.
Perhaps also the distancing Hashem did from the Jewish people from that point, as stated in the
next perek:
| ה וַיֹּאמֶר ה אֶל-מֹשֶׁה, אֱמֹר אֶל-בְּנֵי-יִשְׂרָאֵל אַתֶּם עַם-קְשֵׁה-עֹרֶף--רֶגַע אֶחָד אֶעֱלֶה בְקִרְבְּךָ, וְכִלִּיתִיךָ; וְעַתָּה, הוֹרֵד עֶדְיְךָ מֵעָלֶיךָ, וְאֵדְעָה, מָה אֶעֱשֶׂה-לָּךְ. | 5 And the LORD said unto Moses: 'Say unto the children of Israel: Ye are a stiffnecked people; if I go up into the midst of thee for one moment, I shall consume thee; therefore now put off thy ornaments from thee, that I may know what to do unto thee.' |
It strikes me as quite plausible that this was not really a recommendation against teaching women Torah in general. That the students asked Rabbi Lazer 'to this one you have pushed off, but what will you say to us' suggests that there is a theological difficulty in this. And the woman is called a
matronita.
And indeed, if we investigate a bit, we see that the incident with the Golden Calf was a matter of intense Christian
polemic. For
example:
As early as the immediate post-crucifixion era, Stephen, the first Christian martyr, sharply denounced the Jews (but not Aaron who was held in veneration by the Church) for having made the golden calf, which became the fountain-head of Jewish crimes throughout their history, culminating in the crucifixion of Jesus (Acts 7:41–52). For the Church the golden calf episode served as proof that the divine covenant with Israel had never been consummated, so that the Jewish claim to a special relationship with the Almighty was unacceptable (see Smolar in bibl., p. 91). By worshiping the golden calf, the Jews had revealed their foolish, stubborn, unrepentant, and immoral character (ibid., 100). Augustine also associated the calf cult with the worship of the devil, and the Jews who had drunk the water into which the powder of the golden calf had been cast with the body of the devil (ibid., 100–1). The medieval identification of the Jew with the devil was no doubt influenced by this extreme patristic interpretation (ibid., 101, n. 12).
The question about the three-fold punishment might well have been a pointed Christian polemic, implying that the guilt and punishment for the Golden Calf might well have extended well past those who actually directly participated, to the entirety of the Jewish people. Thus in the 4th century,
Ephram speaks of treating the Israelites like a Sotah. And if so, we can understand Rabbi Lazer's
aggressive response.
]
Resuming the commentary in Torat HaTur, middle of the second paragraph:

"However, our [Bavli] gemara seems to imply not like the Yerushalmi, but in Gevurat Ari [a commentary by R' Aryeh Leib ben Asher Gunzberg] there, he equates the two Talmuds and deduces that the Bavli holds this as well, that we don't give over Torah to women.
And this prohibition, the Rambam rules in
hilchot Talmud Torah perek 1 halacha 13, is only in the oral law, but not in the written Torah. Howbeit, the laws which are relevant to women, they are obligated to learn,
see the Rama siman 246 seif 6. And for this reason we rule in Shulchan Aruch, siman 47 seif 14 that women bless the blessing on Torah, and see in the Rama there.
Behold, in the beginning of Masechet Peah it is stated: these are the things that a man eats of their fruits in this world, etc., and
Talmud Torah is equal to them all. Behold it is evident from here that upon
talmud Torah a person eats the fruits in this world -- that is to say that in regards to
talmud Torah we say that reward for the
mitzvah does exist in this world. And I am in doubt, according to the aforementioned Rama, that women are obligated to learn those laws in which they are obligated, if as well they receive reward for this in this world, like men, or not.
And I saw in the sefer LeOhr Hahalacha in the name of Shu't Beit HaLevi that he distinguishes in the matter of learning Torah between a man and a woman. That is, that women in their learning do not fulfill any positive commandment, but it is only for the sake of keeping the
mitzvot, while by men, there is learning as well as a positive commandment, just like
tefillin and the like. And based on this there is to say that specifically men, who fulfill in their learning a positive commandment receive reward in this world, while women, since they do not fulfill a any
mitzvah in their learning, there is no place to say that they receive reward."
You can see the relevant Torah Temimah
here.