The version with the article has been losing ground to the version without the article for the past forty years. Corpus data has the version with the article coming in at 37% as common as the version without the article since 1990. So both are still quite common, but the balance has changed.
Let’s start by looking at this ngram chart, in which we compare three versions without the article plotted in cool colors with the corresponding versions with the article plotted in warm colors:
You can see that initially, warm-color versions with the article dwarfed the cool-color versions without the article. In fact, a hundred years ago written sources show the with-article to without-article ratio was 7:1. The article was very gradually dropped over the twentieth century, and in the twenty-first century the cool-color versions without the article are now around twice as common as the one with the article.
Looking at 2012 data, these work out this way:
|in first grade
|in third grade
|in sixth grade
|in second grade
|in fourth grade
|in fifth grade
|in the third grade
|in the first grade
|in the sixth grade
|in the fifth grade
|in the fourth grade
|in the second grade
If you look at the Corpus of Contemporary American English (COCA), which has a lot more spoken sources than Google Books does, you see the same broad trends since 1990. This varies a bit depending on which grade you’re looking at. Here is the ratio and percentage of each of the first six grades with versus without in COCA:
If you look at the COCA results in context, if feels like academic writing might be more apt to use the article and dialogue less likely to do so, but that's not a rigorous conclusion, just a feeling I got through looking at these data. The longer version might be a little wordier or more formal looking back at it, but neither seems wrong. I would rarely use the article myself, but now and then I might.
You do have to be careful of attributive uses of things like first grade or second grade, because now what drives the use of the definite article is completely different. For example:
- in fifth grade students overall
- in the fifth grade students we interviewed
- in the fifth grade classroom
However, attributive uses like these occur much less frequently than non-attributive uses, so they do not throw the calculations off very much at all.