locust: what I have to say is based in lots of years of experience, but it won't do you a bit of good for this year! I do the soil preparation in October/November, adding compost and such rock powders as might be called for. Then I cover bare soil (that which nature abhors, along with perfect vacuums) with a natural mulch of shredded leaves or straw, or a green manure crop (we tend to use cereal rye and clover -- it's much colder here than where you are in winter). I like the idea of green manure crops for a school, definitely a teachable moment. And I never use a rototiller because they are death to earthworms and fungal hyphae and create a hardpan at the rotor tip with annual repetition. A class all armed with spading forks should be able to do the job. Furthermore, Fall is a good time to add shredded leaves to your soil; the clay will eventually turn into rich clay loam if you do. Hope this helps -- for next year. Regards, Peter.

"A class all armed with spading forks" creates a picture in my mind (LOL). Have an ambulance standing by.
Actually kids are great in the garden and it is good experience for them but I think adult/child ratio should be closer to 1:1.

If the students at that school are at all involved turn the whole project into a science lesson. Start with these simple tests any school child could do;
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top. Your primary concern is the OM level and yours may be pretty good.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer you soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

Then work at getting more organic matter into the soil since organic matter will seperate the clay particles so air, water, plant nutrients, and plant roots can move through the soil better and easier. And that can be demonstrated with another science project by setting up two aquariums, one with the "natural" soil and one with well amended soil.

locust, I'm also in the Bay Area and have been working on my soil for the past month or so. I just got some new community garden space, and the previous person took her soil from the raised beds (fine with me, I have 4 compost bins and needed a place to put all the compost), so I have been adding a couple inches of coarsely sifted homemade compost, an inch or so of used coffee grounds (because they're free, they add more organic matter, and the worms like them), and an inch or two of mulch (free from tree trimmers) on top.

I have reconfigured the garden beds, and some of them needed more soil, so I fine-sifted the path mulch and got some more good compost/organic matter to add. I also added some soil from the paths. Even though it has been raining (seems like) every other day, it was easy to move soil from path to bed with the digging fork (I would not use a shovel in such wet soil, though).

I always mulch my tomato gardens and the tomatoes have done very well, and in fact, a couple sheets of newspaper and half an inch of wood-chip mulch (plus the shade from tomato plants) got rid of bindweed in the tomato garden in one season. It's still too early to plant tomatoes (though everyone I know, including me, could not resist planting a few already, with lots of protection), and I think sheet mulching is much preferable to leaving the soil bare.
I keep my soil covered with mulch at all times to protect the earthworms. Even a couple days without rain and with hot sun makes the soil crack, and when it rains hard the worms need the mulch as a refuge. I do not till; tilling destroys the soil structure.

So for your soil, I would add compost and mulch on top of the existing soil now, and plant into that when you get the plants. If weeds are a problem (and they soon will be, if your soil is uncovered!), pull out or at least knock down the weeds, then add a couple sheets of newspaper underneath the compost and mulch, or just under the mulch. Around here, the birds will eat young seedlings until about May or so, and we sometimes still get cold night temperatures into April and May, so the plants need to be protected with cloches, water walls, bird netting, or whatever you have.

After I add extra soil, I smooth out the surface with a rake and break up any large clods of clay soil, and then I lightly rake the compost and coffee grounds to make sure the surface is more or less evenly covered. I lightly forked the compost into the raked layer of added soil in one bed, but mostly I just add the compost etc. on top. I let the worms do the work of incorporating the organic matter into the soil.

A great alternative to wood-chip mulch is straw bales. I am going to be using straw mulch on my strawberry bed this year. I have worked on a permaculture garden that was essentially made from large amounts of free coffee grounds added regularly to a dry hillside, topped by about 6-8 inches of straw mulch. We planted big seeds (beans, squash, etc.) by moving the mulch aside a little, and planting at the appropriate soil depth.

Peter has given you great advice. It is what I was going to recommend also. If you plant this year, look for and use plenty of good organic mulch. Then in the fall, just broadcast your covercrop seed over the plot, cover with compost, and let it go till spring. Good luck.