This isn't a book, but I recommend (along with looking at the texts suggested here), to read the article by Jaime Escalante.
This is not to say his words are gospel. One of the key issues in math pedagogical research is people thinking there are single proven/research solutions versus competing approaches and a murky topic.
A couple more good articles to read, by James Cargal (On Teaching in the Mathematical Sciences; The Reform Calculus Debate and the Psychology of Learning Mathematics). With the main messages being DON'T be too "hard"...DON'T assume your kids are as smart as you, or headed to R1 math grad school...and DON'T assume that you can give them a more efficient path to wisdom by doing real analysis before calculus, for example.)
Note: Education research tends to be very shoddy. E.g. low sample sizes, confounding variables, teacher/student dropout effects, non representative samples, poor metrics, failures to replicate, even political bias on top of the statistical bias. You are better off skimming such things with a very open/critical mind (hmm...maybe this might work...looks like BS...etc.) versus a "this is research, it was printed, must be right" approach. In particular, you may be BETTER off having anecdotal discussions with teachers (including those with opposing views) than looking for "the answer" in print, when it doesn't definitively exist.
Unfortunately, pedagogical research is also weak on tools for in depth observation such as ethnographic studies, detailed case histories, wide ranging interviews, etc. Probably because they are more work to do...and education schools prefer the shiny appearance of scholarship from the shite statistical-looking studies they churn out, decade after decade.
To make my point further, consider the counter to the "answer" (really an opinion, because that is how this forum especially, but really much of non hard science/math SE functions best) above advocating Thinking Classrooms by Lilejedahl. See for example this criticism of the (facade of) research backing it up:
"We could imagine a version of this book that doesn’t say anything about research at all. “Here are my thoughts on what makes for good teaching,” it would say. “It’s based on my own teaching, observations, and the experiences of the people I’ve worked with. But don’t take my word for it—try it for yourself, and you’ll see that it works wonders.” Would that be OK?
I think the answer is, that would be very OK. That’s just telling people what you think. You have to be allowed to do that.
Now, how differently should we think about this system with the research support that Peter has given us? I’d argue, not much differently. It’s not that there’s no evidence, and it’s not that he’s playing loose with the facts. It’s just that the evidence is weak. It doesn’t support big generalizations. You wouldn’t want to bet the bank on it."
See also, the criticism by Greg Ashman (paywalled, but free 7 day trial):
[Excerpting the main thesis of his pro-DI and anti-IBL view:
"It was Alfred North Whitehead, a mathematician and philosopher, who made the following observation:
'It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.'
Whitehead’s view is now strongly supported by cognitive science. It is widely accepted that working memory — roughly, the thoughts we are conscious we are having — is extremely limited, capable of processing about only four items at any one time. However, these constraints fall away entirely when dealing with organised knowledge held in long-term memory. Educational psychologists call these webs of knowledge that are related to each other by meaning, ‘schemas’. We can effectively activate an entire schema and solve problems without much conscious effort, provided we have a schema in long-term memory to activate."
[None of this is to say Lilejedahl is wrong and his critics are right...go ahead and look at it...just buyer beware of one more buzzworded edfad.]
P.s. If you really want "the research", you are better off starting with a lit search of your own, to include visiting a university library and scanning/skimming the books on the shelf. Then asking more focused questions. Expecting a dispositive lit search from a forum, as if there were a librarian or a grad student here to prepare it for you, is not the right tool for the job. (Not only is it not there, but you are missing the grounding you'd get from some search yourself.) You are better off with focused questions, after some minimal self education, and with accepting and being interested in anecdotes...versus "tha litrachure".