Not on TD, which bars employment, and not on TN as a schoolteacher, because the Teacher listing covers only college, seminary and university positions. The spouse needs an employer willing to sponsor a different category.
The spouse’s own profession decides the spouse’s own route
TD status for a spouse and unmarried children under 21 permits study and residence but no employment, so a teaching spouse cannot simply take a job at a local school. The TN list does include Teacher, but the listing is restricted to college, seminary and university teaching, so a primary or secondary school position does not fit. A spouse with a university teaching offer could pursue her own TN with the required degree; a school teacher would need an employer prepared to sponsor a category that permits that work, which has its own requirements and timing.
The children may attend school in TD status. Because the family’s income may depend on the answer, this question should be settled before the writer accepts the U.S. start date, not after the household has moved.
Hypothetical example: an accountant’s partner expects to keep a remote Canadian payroll job after entering as a TD dependant. The work location, not the payroll address, is the problem: TD does not authorize that work while the partner is in the United States. The household should price a period without that income before it signs a lease.
If the partner has a separate work option, its employer and timing need their own review rather than an assumption that the principal’s status covers it.