There is an approach to teaching children to read Hebrew, which utilizes songs for teaching the letters and nekudos, in combination with other strategies. If a female apecial education teacher has had success teaching children this way, and a boy between 10-12 years old who still cannot read, wants to get reading help from this female teacher, is it a problem of singing these alef-beis/nekudah songs in front of him? They are not real songs meant to make the singer sound beautiful….
If, as you say, the tunes are not very tuneful, but more like the tunes with which pesukim are sung when they are taught to children, there is room to be lenient for the purpose of education. However, if the songs are ordinary songs, it would be prohibited.
Sources: Avnei Yashfei, vol. 2, no. 5 (concerning a woman teaching pesukim to children by singing them; although Avnei Yashfei thought to prohibit the practice, he quotes that Harav Elyashiv shlita ruled that it is permitted, because there is no “attraction” in singing the pesukim. The same would apply to singing aleph-beis songs. If the teaching technique involved regular songs, however, 10-12 is certainly chinuch age (according to the most lenient opinion the age is from nine years old, and most opinions maintain that chinuch begins at a younger age — see a previous post on this issue), and a male teacher/singer would need to be found.