As the rest of my household heads back to school, I am getting down to the business of planning for my new semester. One instructional problem I am trying to solve is that of pushing my future elementary teachers to understand decimals more deeply. Whole number place value I have nailed down. But decimal place value is another story. The rules are so deeply embedded in them that it is very difficult to get them to question these rules, or to seek a better understanding of them.
I have collected substantial data over recent semesters (not research quality data, but consistent formative and summative assessment results) to demonstrate that my students operate on decimals using a combination of known rules and whole-number place value principles. They can switch back and forth between these fluidly and generate right answers to nearly any decimal task.
Which leaves me seeking questions that they cannot answer without digging more deeply.
I have a candidate question in mind: Why can we put zeroes after a decimal number without changing the value, but not after a whole number without changing its value?
It occurred to me last evening that I knew what my answer to this was, but that I lacked a wide repertoire of correct explanations. So I asked Twitter. Lots of good stuff followed. Read the full conversation on Storify.
Overthinking my teaching 
This seems related to the question of why there is a 1s place, 10s place, 100s place, etc. to the left of the decimal, but no 1ths place to the right of the decimal. Why isn’t it 1ths, 10ths, 100ths, etc. as we go to the right? I wonder if it would be easier to explain why 025 and .250 are non-changy ways to “append” 0s, while 250 and .025 are changy if we instead had a decimal underline or something that went under the term we would consider the unit term. Then it might be clearer that we were moving non-zero values away from the unit in one case, and not in the other case.
Aaaaaand now I see that Max beat my Twitter comment by four minutes. Priority has officially been established. Plus, I like “non-changy.” The other benefit of this method is that it makes James’s “distance from the decimal point” idea more natural, more properly analogous to the absolute value analogy (I suspect) he’s trying to make.