Hi there! Thank you for this question! I was the urban forester for the City of Corvallis for several years and I dealt with this countless times!

Exposed surface roots are very common with maples as they age. This is partly because they are shallow-rooted species to begin with, and over time, roots naturally grow upward and become visible above the soil. This is especially if you have an area with overhead irrigation near by, the roots just tend to stay very shallow. 

You asked how to address them, and I think my advise would some combination of the following things I did (or didn't do) during my time at the City. 

Most importantly, avoid burying them with soil or mulch. Adding extra soil to cover exposed roots can suffocate them, and piling mulch too deeply can encourage rot. A thin layer of mulch (2–3 inches) is fine, but keep it pulled back from the trunk flare.

Equally important is not cutting or remove large roots. This can destabilize the tree and reduce its ability to take up water and nutrients. Our general recommendations are to not cut any roots 2" or larger without the guidance of a certified arborist. 

In lawn areas, you might consider shifting to a mulched bed or planting groundcovers around the tree instead of trying to mow or seed grass over the roots. This both protects the roots and reduces mower damage.

 If the roots are causing structural issues (lifting sidewalks, for example), or if you’re concerned about the tree’s health, consulting with an ISA Certified Arborist is the safest step. Depending on where you are, I can help you find one or you can visit the ISA's Trees Are Good website, which helps connect people and arborists based on your zipcode: https://www.treesaregood.org/findanarborist 

I'm more than happy to help address any other concerns!
Jennifer 

Oh my! I'm not super happy to hear that an arborist advised you to cut them and I'm glad your gut told you otherwise! I'm always happy to help, if you have any other questions about this or anything else, please let me know!

Jennifer