Measurement, by Paul Lockhart, may be helpful to start with. It carefully and clearly derives mathematical principles from basic ideas, in a very approachable way.

Here's how it describes its content:

Part One: Size and Shape

In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves.

Part Two: Time and Space

Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.