I think when it is said that Singapore is a mental program what is meant by that is that it teaches dc to think about the concepts of how the process is done and often times be able to solve the problems in their mind without pencil and paper. As an example, instead of just teaching dc to write out a math problem that involves regrouping, it teaches them how to think about it in groups (tens, ones, etc.) so that they can do it without writing it down. When I was in school, I was taught just to follow the steps of adding the first column, carry to the second column, etc. I wasn't taught why! Singapore teaches the why and when you know the why you can often do the problem without having to go through the trouble of writing down the steps. Now my ds is not typical, but on his standardized testing this year, the tester told me that he didn't even use his scrap paper to do any of his math problems.
He did all of the problems in his head and tested many levels above his grade level.
I don't think you can find a program that better teaches how to think conceptually about math. It still follows the method of teaching first with concrete examples (manipulatives), then moving to pictoral examples, and finally the abstract concepts. In both level 1 and 2, you'll be using the hands on (concrete) examples that Carrie has provided in the math exploratations. When you move into level 3, you'll be moving into the textbooks which use pictures to explain the process. By this time most dc can understand the concepts through the picture examples. I find the pictures very clear and helpful. It would always be possible at this level to still include manipulatives if a dc needed them. We're just now in level 4 and there are still great pictoral examples in the textbook. I'm guessing that these will continue in all of primary math.
I have a high regard for Singapore math now. When I first came to HOD, I was using another math program, Miquon along with some other living math concepts. I was hesitant to switch to Singapore. The thought process behind Singapore and MIquon are similar though and Singapore is laid out much simpler. Many people use the two programs together so I thought I would do the same. As time passed, I didn't find the need to continue with Miquon and just continued with Singapore. My dc are thriving with it. My ds is mathematically gifted and he soars with Singapore. Math is not my dd's strength yet she has excelled with it also. Both of them are testing
well above grade level in mathematics on standardized tests. The tester this year asked my ds what math program he uses and when he told her Singapore, she said, "I knew it!" It's her program of choice for teaching higher level reasoning skills. I tell you this to just encourage you to give it a chance. I think it can work well for many dc, and it is such a solid program especially with the wonderful concrete examples and activities that Carrie has written to go with the program.
In other words, you can do Singapore in a non-mental way if you aren't doing it the way the math program was intended and your child will still learn math. I think it was when I got to Sing 3 that I realized what I'd done and we adjusted for the way my dd had been taught (by me using Sing). Oddly, she now (in 4A) prefers to solve 2 digit multiplication in her head (because it's faster).