Some thoughts on effective feedback

Feedback is hard. It is so tempting to fall into the math teacher archetype of just hurling a bunch of grades at students instead of genuinely responding to their hard work. But even though it is time consuming, and I get all inked-up thanks to my left-handedness, feedback is absolutely crucial. As Aguirre, Mayfield-Ingram, & Martin (2013) say in The Impact of Identity in K-8 Mathematics:

Feedback (in math) tends to accentuate what students do not know and cannot do, thus leading them to believe that they are “not smart,” lack ability, or cannot learn.

So a couple of thoughts on how I do feedback and how I want to push myself…

My A^3 (or A Cubed) Feedback Method

Going back to Aguirre and pals, they elaborate that…

Feedback has the power to determine whether children see themselves as mathematically proficient…These approaches include focusing students’ attention on making sense of mathematics, affirming evidence of mathematical progress, and providing students with opportunities to grow mathematically without sacrificing their mathematical confidence.

In trying not to be so pedantic, I whittle down the essence of this idea into something catchy, and I’ve come up with is what I’m calling (PATENT PENDING. MILLION DOLLAR BOOK DEAL COMING) A Cubed Feedback. Essentially…

AFFIRM – Identify something that is good about the student’s thinking or work, as aligned to your vision of content and grading rubric.

ASK – Direct student thinking towards a focus correction area by asking about their response

ADVANCE – Give them a “foot in the door” to actually resolving the question you’ve asked.

So for instance, when one of my teacher’s did the Math Assessment Project’s Gold Rush task, he got this response:


First off, there is a TON that I could affirm about this work. The student really has a lot of great thinking that he has shared here. There are also some things missing that I’m curious about – why are there only squares considered here? The purpose of the task is to see what shape is the most effective at maximizing the area. While the square IS the solution here, I don’t see any evidence that this student considered other configurations. I could go on and on, and notice how much time I’m spending here (because there’s so much cool stuff I want to address!), but let me whittle down some feedback with A^3…

I really like how clearly you explained your strategy, and how you drew examples to demonstrate that strategy (AFFIRM)! I’m wondering, how did you end up with dimensions of 25 m. by 25 m. for the first question (ASK)? Why wouldn’t it be more effective for the gold prospector to use different dimensions with their 100 meters of rope (ADVANCE)?

Is that the only direction I could have taken with this feedback? No, but it is affirming, actionable, and concise (3 sentences). A^3 Feedback allows me to get through 120-150 responses in a timely fashion, while also giving the student a way to “grow mathematically without sacrificing their mathematical competence,” as Aguirre and company would say.

I’m currently thinking about Revision Symbols in feedback…




Revision symbols can be a great way for ELA or social studies teachers to give consistent, common feedback to students without having to write it out in complete sentences over and over and over again. Now, I also see the risk of such symbols as sliding us back into bad math feedback – if these symbols are the entirety of what we throw at students it can emphasize that we’re not really looking deeply at their work. These would still have to be balanced with written feedback like what I’ve provided above, but they could allow us to be a bit more comprehensive in our feedback while still saving time AND still focusing students on the big ideas of mathematical thinking. I’ve put a few together in a Google doc which you can access by clicking the picture below (or you can just look at the picture I guess. They’re the same for now.)


Now some of these are boring. Some of them could probably be improved. But it is a start. These would have to be clearly communicated to students before they receive their feedback, but – if done systematically and in tandem with more specific written feedback – these could double down on the quantity and quality of feedback that I’m giving without wasting a full pack of pens.

What are your thoughts? Are these ideas about feedback on the right track? Do anyone else use a different approach that I can steal or build off of or add to this? Let me know!

Mathematically yours,



One thought on “Some thoughts on effective feedback

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