In 1998, Tom Carpenter and his colleagues documented grades 1–3 students’ use of invented strategies and standard algorithms. The vast majority of students in the study used some invented strategies. The researchers found that students who used invented strategies before learning standard algorithms showed better understanding of place value and properties of operations than those who learned standard algorithms earlier.
We have seen it over and over: a child is tutored early in memorized arithmetic procedures by a well-meaning parent.
Some children do fine with those traditional algorithms. These are linear thinkers with a facility for memorization. (We would argue that even the best memorizers would benefit from a more investigative, explorative stage of learning, coupled with complex problem-solving).
For about a third of our students, however, this approach does not work. These students retain the procedure for a few days or weeks, and subsequently confuse it with the next algorithms taught. They fall behind, need frequent re-teaching, and lose confidence in their own conceptual abilities. In an earlier century, they would have ended up in agricultural or factory jobs. Today they run the risk of ending up in a long-term underclass, with poor career choices. (See this Article for data.)
Alternatively, if students spend as long as they need to at the concrete/pictorial level, they can then comprehend and internalize the resulting algorithm. They create their own meaning behind that algorithm. They are able to apply it to new, more complex situations and they have an internalized, often visual conceptual understanding to fall back on if they get confused later on. Perhaps most importantly, the acquisition of algorithms through exploration is more fun than drill, more respectful of students’ intelligence than direct teaching, and more likely to instill confidence in learners.
We DO believe in the usefulness of algorithms. They are quick, efficient and expedient. In fact, one of the final goals of mathematics teaching is to help students master its widely accepted abstract representations. We just believe that they are most useful when they are understood.
So… which 5th grade concepts are worth spending that much time on? Our answer: Division, Fractions and Multiplication. Let’s look at division.
TEACHING DIVISION
- CONCRETE
We start with concrete representation; mostly base-10 blocks, but we also play a game called the remainder game. It uses a planting tray from a nursery and plastic spheres, but blocks would work, too.
2. PICTORIAL
Since our students had numerous fraction investigations in 4th grade, we move quickly to pictorial division, but with the blocks available for the few who need them.
For most students, this transfers quickly to semi-mental math:
3. ABSTRACT. Finally, all this work transfers to the long-division algorithm. 
And to decimal division:
The child at right still needs drawings, the child below is doing most of the division mentally.
This problem is only visible as 36 tenths if it is first illustrated with blocks or drawings. 36 tenths divided among 6 groups is 6 tenths each. The rest becomes 12 hundredths, also divided by 6. Voila — division with less pain, less drill, and more number sense.
