The main problem, as outlined by in Potts and Hayden (2009) "Sex and War" is access to affordable birth control. Most African women express a desire to control family size, it is the men who do not.Having not read the book, I cannot respond to what the authors write. It is not clear that the second sentence excerpted above is asserted by the authors or if Dragon Horse is pulling it from somewhere else.
Potts and Hayden's argument is also "Freedom" but not the libertarian kind. Pretty much, they found that the less free a woman is in society, the more a male controlled societal (or religious) structure control their fertility the higher the birth rate tends to be. In societies where women have a higher education level, more legal rights, more say inside their marriage, etc, they have less children.
I am able to see whether or not the World Values Survey lends credence to the claim that African men want more children than African women do, however. A question probing for participants' ideal number of children was not asked in the most recent "fifth wave" survey, so responses are from the fourth wave, spanning 1999-2002. The following table shows the average number of children desired, by gender, for sub-Sahara African countries surveyed by the WVS:
Not much in the way of support for the claim. Differences in the perceived ideal rate of children exist between countries, and, at least in the US, by educational attainment among residents of that country. Gender does not predict fertility in sub-Sahara Africa (or anywhere else to any significant degree, for that matter).
The second point Dragon Horse makes is valid. Specifically, the gender gap in education is a strong predictor of national fertility rates. As measured by the World Economic Forum's 2007 report entitled "The Global Gender Gap", the correlation between fertility and educational parity is an inverse .75 (p=0), far stronger than it is among the other measures of gender equality, including economic participation and opportunity (.22), political empowerment (.22), and health and survival (.01).