<p>I spent many years with our son on just talking as a form of instruction. Asking him questions and seeing his answers. I spent some time doing this with our daughter too using AOPS as a base. You can travel all over the place with this approach and it provides a small forum to develop the ability to understand common sense and analysis.</p>
<p>It’s pretty clear that there has to be a balance between the concrete and the abstract and perhaps schools go overboard to both extremes.</p>
<p>Our school district uses Everyday Math (hard to pass the state assessments with any other program). Two schools supplement with traditional methods and one doesn’t. The one that doesn’t doesn’t perform as well as the other two schools. The district is trying to fix that (the teachers in the two schools do this on their own in addition to district curricular guidelines).</p>
<p>These days I have discussions with my son on computational complexity which spans the worlds of estimation and the concrete.</p>