The 'take away' for all of us from this post is the need, particularly as pension age looms on the horizon, to check what you'll get at 66/67 and whether there are any gaps in your NI history.
In my own case I was a Civil Servant for 35 years until 2013. With an occupational pension as part of my terms I was contracted out of the second state pension and therefore paid reduced NI for all those years. What that means is that part of what I'm guaranteed as pension in relation to them is delivered as part of my Occupational Pension rather than the State Pension.
The State Pension was reformed c2016 with the Second State Pension closing to new entrants/contributions and the option of lower NI for those with Occupational Schemes going.
The headline stuff for the post 2016 New Retirement Pension scheme is that 35 qualifying years will give you the full pension of around £200/week. However, if some of those years were contracted out you won't.
Because I've been in a second career since 2014 I'm still paying NI and, provided I work until I'm 66 in 30 months time, I'll have 35 qualifying years. Otherwise I'd have needed to decide whether to buy in extra years to make up C£170 to £200.
My other half, like the OP's, was a teacher. Initially she was paying into the Teacher's Pension but stopped when she went part time when our kids were young and never got her act together to rejoin. She also spent a lot of time on supply with no Employer's scheme. Because she was NOT paying reduced NI she has got pretty much the full £200 on her prediction and that's in spite of 3+ years not in employment while working on a PhD.
We only picked up on this because somebody on another discussion group I'm in asked about their pension and the qualifying years thing.