Shape- I would like the logo to be adaptable for both a square and an elongated shape. The emblem and wordmark will ideally be able to be "arranged" in two such configurations.
Colors- I would like a primary logo to include red and gray or potentially just red. Additionally, I would like it to translate well in both all black and all white.
Feeling- The words listed above in regard to the feelings I would like the website to evoke overall should be reflected in the logo. I hope it will have an "illustrated look" that suggests a softness and "cutesy" playfulness to appeal to mothers. However, it should also project authority and credibility that matches the tone of the valuable content.
Font-I don't know that I can provide any direction in choosing a font as I find it difficult to describe fonts. I guess I'll know it when I see it. Hopefully, the description I've offered regarding the tone I'm going for overall will direct you as you select a font for the wordmark,
Imagery- I do not want the logo to simply be some sort of a stylized book. The imagery should imply more engagement-- ideally a child reading a book.
For your consideration, attached please find two concepts that I mocked up featuring two basic images I like.
One shows a child sitting on a stack of books while reading. This was created using a clipart drawing. I would like to see an original version of this concept that is more refined. I wonder if it can be simplified as I know detailed images don't often read well as logos, especially when used for a small application.
Another concept shows a child reading while leaning in, engrossed in a book. This is taken from a photograph I find to be very powerful and fitting. I wonder if it can be illustrated and retain all of its strength.
Again, I would potentially like to see refined/better takes of these concepts but I am definitely also open to other good ideas that fit in line with everything I have laid out above.
Here are some keywords that might provide inspiration:
math, core curriculum, literature, books, reading, addition, subtraction, division, multiplication, arithmetic, fractions,
support, learning, tutor, exploration, games, play, fun, toys, activities, kids, parents, teachers, learning at home
And, attached, please see some very basic mockups I created of the website front page that shows some context for how the logo would be applied (although it should be versatile enough to look good elsewhere on the web and social media as well as in print).