## More on Math Feedback

Last night I was fortunate to attend ICTM’s chat on feedback.  It was a productive chat and Anne had some great questions cued up for us.  I came away with a few new tools that I need to research.  Chats like these are motivating as the frigid cold of the midwest is ever-present this time of the year and new ideas can spark my planning process.

Teachers know that student feedback is important – it’s everywhere in schools.  It’s on every teacher evaluation tool that I’ve experienced. ASCD describes it as “Basically, feedback is information about how we are doing in our efforts to reach a goal.”  Teachers give feedback all the time – most without even labeling it specifically as feedback.

The chat was still on my mind this morning as my colleagues and I were having a conversation about math units.  After reviewing multiple student papers, I started thinking about feedback in more detail.  Specifically, I started thinking about how feedback takes on different forms and the tools that are used to give that feedback can vary from class to class.  In all cases that I’ve come across, educators want students to actually USE the feedback.

Technology can be used for this although the reliability of the feedback might not match the need.  I’ve also seen cases where the automated feedback is disregarded by students in an effort to score more points.  It depends on what’s needed.  In some cases, a quick verbal prompt might be the feedback that’s needed.  For others, a conversation with a partner can help students identify misconceptions or spur thought.

Let’s take this problem:

This particular students was able to identify the rule and complete everything but the bottom problem.  Being able to anticipate misconceptions can lead to better student feedback. There are a few questions that I might have before approaching the student and giving feedback.

• How can this student divide 14 by 7, but still have trouble with the bottom problem?
• Does this student think of “divide by two” as half of the in?

Or here’s another one:

• I notice that there isn’t any work or model here
• Did the student notice that the denominators weren’t equal?
• What strategy was used here?

Last one:

• Did the student miscount the boxes?
• Is the students missing pieces? (yes, this has happened before)
• How did the student get 6 as the numerator?

In all of these cases a simple mistake is probable.  I’m working with K-6th grade math this year and sometimes rushing leads to simple mistakes.  I try (as much as I can) to limit that option when deciding to give feedback.  In all three of the cases I could ask the student to recheck their work.  Some students will, while others won’t.  I could also write on their paper a statement or question about wondering what strategy they used.  I could also have the students meet with a peer and discuss the problem in more detail.

There are so many ways to communicate feedback and it’s not a simple issue.  Some students are more responsive to written feedback, while other students want to have a conversation with another peer to discuss their strategies.  As students get older the type of feedback also changes.  Many of my upper elementary students prefer a brief comment on a paper or a quick underline, question mark, or specific arrow to help them move towards a goal.  Having a 1:1 feedback conversation with a student is my number one option because then I can see how receptive they are and answer any follow-up questions.  If you don’t have time to do that with every kid (who does?) then you use other options.

There are a ton (I mean a TON) of apps out there that “help” students along their math learning journey. I tend to be a bit caution when deciding to use them in the classroom. Is the feedback appropriate for their needs?  Is the feedback helping them in their efforts to reach a goal?  In some cases it may, but I think it’s worthwhile consider the ways in which feedback is given.

## Math Reasoning and Feedback

Image by:   J. Creationz

Having math reasoning skills is important.  Generally, math reasoning skills are taught and incorporated in early elementary school.  In math, a problem is what a student is asked and expected to answer.  If a student is unable to answer why their answer is correct, I believe that the student might not fully grasp the mathematical concept.  The student might not be utilizing math reasoning skills.

For example, a student that measures area in linear feet might not completely have an understanding that area is measured in square units.  The student could have the correct numerical answer, but include the wrong unit (centimeters compared to square centimeters).

How is mathematical reasoning taught?  I’m going to be taking a proactive step next year to give opportunities for my students to utilize math reasoning.  I’m deciding to use higher level questioning to enable students to think of the process in finding the solution.  The learning process is key.  I’ve found that math instruction isn’t always linear, just as mathematical reasoning isn’t rigid.  By asking students why/how they arrived at a solution is vital in understanding their thinking.

As I’m planning for next school year, I’ve decided to ask students to explain their reasoning more frequently.  By hearing their reasoning, I’m in a better position to give direct feedback.  All math questions have some type of reasoning.  I believe that multiple solution / open-ended questions can be used to display mathematical reasoning. Students need to be able to explain why they responded with a specific answer and what methods/connections were utilized to solve the problem.  Based on the math Common Core, students are expected to reason abstractly and quantitatively.  When students describe their mathematical process, teachers are better able to diagnose and assess a student’s current level of understanding.  Math reasoning isn’t always quantifiable, but it can be documented via journaling and other communication methods.  More importantly, teachers will be able to provide specific feedback to help a student understand concepts more clearly.  I also feel that this questioning process develops self-confidence in students and prepares them to become more responsible for their own learning.  See the chart below.