Our physics students have weak math skills. What to do?

I've heard for nearly three decades how weak our students' math skills are.  And such observations are not necessarily incorrect, though they do harken to a mythical, deified past which probably wasn't as great as some want to suggest.  

Fact is, whether or not the pandemic has worsened our students' math ability, even AP Physics students will cancel across a plus sign as in the picture.  They did so in 1999, they did so in 2023, they'll continue to do so even in 2053.

Let's not talk about how we can Make Our Students Great At Math Again.  They never were; and we can't anyway.  Instead, let's deal with our students as they actually are, rather than as we wish they would be.

But how do we do that, when mathematics is the language of physics?

The big deal is, de-emphasize math errors.  

I looked at an SAT math section a few years ago, for the first time since like 1996.  I was annoyed by the second half of the questions.

See, these were all good math questions, questions that good math students should be able to approach.  Each required multiple steps, often three or four or five(!) separate connections between concepts to get to the answer.  Nothing wrong with that - one of the prime differentiators between a middling and high-level math student is their ability to pile connection upon connection without getting lost or frustrated.  

Yet these were numerical multiple choice, or grid-the-numerical-answer questions.  There was nothing to differentiate near-perfect reasoning from guesswork.  

I'm used to conceptual physics and AP Physics 1*, where the multiple choice questions are usually difficult but straightforward; and the free response usually awards credit for each step in the reasoning process.  When there are five-step problems, a student who makes 4/5 of the connections earns 4/5 of the points... and a top score on the exam.  On this SAT, though, it was all or nothing.  Make all the connections just right, or get no credit at all.

*Sometimes called conceptual physics on steroids

Now, I'm well aware that the purpose of the last questions on an SAT math section are for the purpose of differentiating the toppiest-top math students from the mere outstanding math students; I'm also aware that substantial psychometric data exists to demonstrate that these questions attain their purpose.  And I'm acutely aware that it's not in any way practical to design and score a free-response SAT math section.  My concern here is NOT about ETS or the College Board or the existence of the SAT math section.  

My concern is about those teachers, students, and parents who - possibly informed by the all-or-nothing nature of SAT questions - act as though the final numerical answer to a complicated math question represents a judgment from the Almighty Themself of a student's math ability, and of that student's personal worth.  No.  The 15 year old upstart who falls to Serena 7-6 in the third set has, in fact, performed far better than the player who was destroyed 6-0 6-0... even though both folks lost and are out of the tournament.  The Vikings and Bills may have lost four Super Bowls, but they made it to four Super Bowls - as opposed to the perennial sad-sack Lions, who aren't even sure what a Super Bowl is or how a team might be selected to play in it.  

But my students keep getting answers wrong, just as the Vikings keep losing in the postseason.  What can I do?

Separate the math from the physics.  Did the student begin with a fact, relevant equation, or standard approach (like a free body diagram or energy bar chart)?  Did the student use the fact to make a connection, use the equation in the correct context, do the appropriate next step in the standard approach?  If so, then praise them.  And move on.  

If I see a silly math error on a problem set, I usually ignore it.  Maybe I'll put "14/15" as a score on a graded assignment; maybe in class I'll just stamp the paper and move on.  Those who make significant conceptual errors must come back to redo the problem... but math errors don't really matter in this context.

A huge fraction of my student's practice is done in the context of making laboratory predictions.  Here again, I'll ignore minor math errors a lot of the time.  Billy forgot to square the 2 and so made an incorrect prediction that the speed would double, not quadruple?  I don't need to correct Billy.  The universe will correct Billy.  He'll go measure the cart's speed... and the cart's speed won't double like he said it would.  At that point, I'll show Billy that he did everything right but made a math error.  Yet, Billy doesn't feel like he's being shamed or even corrected by a teacher; Billy feels like the teacher is helping him figure out why his prediction didn't match reality.  And Billy's less likely to make that mistake again next time (though the likelihood will never drop to zero).

And finally, all incorrect answers on a test must be corrected, whether the reason for the incorrectness was math- or physics- related.  In this context, students care very much about getting the right answer, because they want to get their correction checked off, and they want to figure out how to improve their next test.  Importantly, students do NOT see their original test!  Thus, often a math error that led to a wrong original answer isn't repeated.  "Why do you think I got this wrong the first time?"  "Don't worry about it!  You likely made a math error.  Not worth thinking about.  You know how to do it just fine.  Move on."  

But when they bring me the correction with that math error repeated, the context is exactly right for me to show them the error.  "Look here.  There's a plus sign in the denominator.  You're not allowed to cancel across a plus sign."  Often the student hits themself in the forehead... goes back to their seat... and does the problem perfectly.  They knew the math concept!  They just didn't apply the math concept when the time was right.

I haven't had a mathematical breakthrough with this student, such that they'll never make such a mistake again!  The message I've sent isn't "I'm teaching you a math skill you should have already known."  No, the message is, "Here's a math concept that you knew, but you reverted to a misconception under pressure.  But your physics was just right!"