The issue of whether or not to count birds is complex. It really depends on what one wants to do, or hopes that others will do, with the information.
When one uses a check or "x" to indicate that you observed a given species, it simply indicates that the species was present at the same time you were there. The absence of a check is not evidence for the absence of the species, although it is a suggestion. But the checkmark gives no hint of how many birds were there. One, a few, many?
On the other hand, making an effort to count the birds contributes additional information, in that at least that many birds were present when you were there. But no one should mistake that as the total number, or an accurate count, of the individuals present. Obtaining reliable quantitative data is one of the most difficult tasks that faces any ornithologist.
Take Christmas counts as an example. Everyone counts the birds that they observe within a 15-mile diameter circle, but no one should interpret those numbers as representing all of the birds that occupied the circle during the count. Christmas counts are designed to maximize the number of species detected, not the actual number of birds within the area.
Detectability of birds is an important issue. Birding has been described as "listening with binoculars". How many birds do you hear before you see them? If you walk through the woods, a park or your neighborhood and just record the species that you hear, your list will be almost as long as the one that includes those that you see. When you make a list of birds that you have detected, do you have some that you actually did not see, but you are quite confident they were present because of their distinctive call? Estimating how many individuals you heard is difficult.
Detectability lessens with distance. The farther away the bird, the less likely you are to detect it. Scientists have developed a way to deal with this issue but it requires accurately measuring the distance from you to the bird observed. The measurement issue has been greatly improved by the development of laser rangefinders. A computer program has been developed over the past 25 years that allows you to compensate for the decrease in detection with increasing distance. The program is called DISTANCE and is free. It is rather neat because it allows you to determine how many birds you missed in your count. Best of all, it allows you to associate the number of birds with the size of the area covered. After all, to say that you saw 100 birds has little meaning if someone does not know how large an area you covered in your observations.
I have used this program with a marine radar, that allows accurate measurement of distance to the bird detected, to estimate bird densities in our coastal waters, as the number of birds per square mile or square kilometer. Jim Stevenson and I are using it to determine the number of birds per acre or hectare at several prairie areas along the coast. I am also using it in Houston to estimate the number of birds in urban areas. I am not recommending it to the casual birder. The method is what we term "data intensive" and requires a lot of work.
I still record sightings as present or absent. That is all that I have asked participants in the Houston Bird Survey to do. If you record the number of birds that you observed, it adds a tiny bit of information that can be useful. Do not worry that you do not have an accurate number. No one expects you to.
Many years ago I participated in the Pacific Ocean Biological Survey Program conducted by the Smithsonian Institution. I quickly realized that we were losing information about birds seen casually while we were doing other chores. I created a bird checklist that had squares for 1-10, 10-100, 100-1000 and 1000+. I filled one out every night for each species. That was a very crude count but it successfully captured information about bird presence and abundance that was otherwise lost because it was not recorded.
You have already gone to a considerable effort to locate the bird and identify it. When you share information about what you saw, and where you saw it, it adds to our information about where and when the bird can be found in what habitat. If you take the time to think about and record how many you have seen, it contributes another tidbit to our collective knowledge about birds.
Bird, and count on! (Don't forget to record)