Even for Europe this map is pretty bad. All of Western Europe was practicing agriculture by 3000 BC, and most of the green part by 5000 BC.

As is to be expected of a World History textbook, the Peru part is useless.

Timeline Errors: It is correct to separate the development between the coastal and eastern mountain highlands. However, Tom Dillehay's research puts the earliest horticulture in the coastal valleys around at least 6000 BC. By 4000 BC, there's evidence of complex, gravity-assisted canal systems. In the highlands, Linda Perry, et al., have identified corn samples from 2000 BC. At that point it was still a somewhat "exotic" crop, so we can assume agriculture in general occurred earlier.

Location Errors: The map shows only a tiny portion of people that developed farming. On the coast, it should extend much further down into Chile, and the highlands should extend a bit past Lake Titicaca, the blue sploch right above "DESERT." It also shows no farming in the Amazon; Clark Erickson's research in Bolivia shows Amazonian canal and patty systems that may go back as early as 1500 BC, but are at least around by 500 BC.

Cotton should also be shown in the Andes. Evidence suggests the trading of Andean cotton for fish and other sea food. The cotton was used for making fishing nets as well as clothing. edit: Not a historian but I am currently reading the book 1491: New Revelations of the Americas Before Columbus.

There should be a yellow/purple fringe on Mexico's agriculture extending up into Utah/California/Arizona/New Mexico, where Maize was definitely present by 2000BC. Cotton should be listed near Teotihuacan valley in the earliest shade and in the Sonoran Desert by the latest.

The western Arabian coast should have some agriculture as well, not to mention New Guinea (which is strangely omitted from this chart, but apparently not marked).

The dates are wrong for Mesoamerica. While not the main part of the diet since people at the time practiced mix strategies, people in Mesoamerica were already doing maize agriculture by 4350 BC.

This map is pretty bad. You are right about Ireland. The Céide fields in Ireland are evidence of agricultural crops for more than 3500BCE.

This map also dates the area around Papua New Guinea with agriculture by 7000BCE.

it seemed at least a little fishy to me since it doesn't include much going on in North America and I thought there was a bit more going on around the Mississippi

There's evidence for agriculture in the middle Mississippi by at least 5000 BP, in the form of independent domestication of squash (a different subspecies than that domesticated in Mesoamerica). By 3000 BP, the various other components the Eastern Agricultural Complex had come together throughout most of the Eastern Woodlands (particularly in association with the various Hopewell-affiliated cultures). The EAC consisted of squash, sunflower, marsh elder, goosefoot (a relative of the more famous quinoa from South America), little barley, and a few others. However, chances are you're thinking of maize-based agriculture which was imported from Mesoamerica via the Southwest. Maize starts showing east of the Mississippi around 2000 BP, but doesn't become intensively utilized until around 1400 BP, in the vicinity were Cahokia would rise to prominence a few centuries later (at the time the place that would be Cahokia was just one large town among many in the area). Around 1000 BP, maize-based agriculture had largely spread eastward as far as the Appalachians and northwards to the Great Lakes (which was as far north as it could reliable be grown). A century or so later, it was common place east of the Appalachians as well. The plants of the EAC hadn't been completely abandoned though. Squash and sunflowers remained important crops, and many of the those, especially chenopodium, became common garden herbs.

Also missing from this map is the domestication of cassava in the Amazon many thousands of years of ago (preservation of plant remains is spotty at best in the region, but the domesticated cassava had already spread to Mesoamerica from Brazil by 6000 BP, while sweet potatoes came south from the Yucatan).

Another agricultural epicenter missing from the map occurred in the New Guinea highlands around 9000 BP. At the Kuk Swamp archaeological site, the people of the constructed a huge network of drainage canals to make the land suitable for the cultivation of taro, and later by 6500 BP, for bananas and sugar too. Unfortunately, New Guinea doesn't even show up on the map.