Keilim, Chapter 22, Mishnah 4. This mishnah brings us two wooden seats whose status is debated by Beit Shammai and Beit Hillel, and in each case Shammai the Elder himself adds a ruling of his own that pushes the halachah one step further than his students did.
The first item is a kisei shel kallah, a bridal chair. It was a decorated wooden seat prepared in honor of the bride, on which she would be seated during the wedding festivities. Structurally it resembled any ordinary chair, with legs, a frame and a seat board. What distinguished it were the wooden panels fitted at the back, which allowed her to recline in comfort, along with further panels rising on either side to function as armrests.
The Mishnah asks what happens when those decorative fittings are gone: "Kisei shel kallah shenitlu chipuyav", a bridal chair whose coverings, the slats forming the backrest and the armrests, were removed.
"Beit Shammai metam'in", Beit Shammai rules that it is still susceptible to tum'ah. True, it is no longer elegant enough to be set out at a wedding, but it has not stopped being a chair: a person can still sit on it, and so it remains a keli.
"U'Beit Hillel metaharin", Beit Hillel rules that it is tahor. In their view, this chair was made for one particular purpose, to serve as the seat of a bride at her wedding, and once it can no longer fill that role it is no longer a utensil at all.
Now the Mishnah records the opinion of Shammai himself, not the school that carried his name: "Shammai omer: af malben shel kisei tamei". Shammai endorses the position of his students and then goes a level beyond it. It is not only the chair with its legs and frame that remains susceptible to tum'ah after the fittings are stripped away. Even the bare seat board on its own, the plain wooden panel that a person can sit upon, counts as a keli in its own right and can therefore become tamei.
The Mishnah's second case involves a chair attached to an areivah, one of the large wooden containers used for kneading dough. The chair is fixed to the trough, yet a person can still sit on it exactly as he would sit on any chair. The question is how to classify it: is it a seat, and therefore subject to tum'at midras, the tum'ah that comes to an object designated for sitting when someone tamei sits or leans upon it, or has it become simply a part of the trough, in which case it could only receive tum'ah through direct contact?
"Kisei sheke'va'o ba'areivah", a chair that was fastened into a kneading trough. "Beit Shammai metam'in", Beit Shammai rules that it does receive tum'at midras, since the chair continues to function as a seat. "U'Beit Hillel metaharin", Beit Hillel rules that it is tahor from tum'at midras: the chair is viewed as nothing more than an extension of the trough, it is batel, nullified to it, and consequently it can become tamei only through contact.
Again Shammai the Elder adds his own ruling: "Shammai omer: af he'asui bah", even a seat that is fashioned within the trough itself can contract tum'at midras. His students spoke of a separate chair that was attached to the trough, and Shammai agrees with them, but he extends the halachah: it is not only an attached chair that qualifies. Even if one takes the trough itself and shapes a place to sit inside it, that area is now usable as a chair, and it too becomes susceptible to tum'at midras.
So in both halves of the Mishnah we hear the same pattern. Beit Shammai looks at the object and asks whether a person can still sit on it, and Beit Hillel asks whether it still serves the purpose for which it was made. And in each case Shammai himself confirms his school's approach and stretches it one step further, to the bare seat board of the bridal chair and to the seat carved into the kneading trough.