Conceptual Understanding in Math – Educational Aspirations

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I’m grading tests this weekend. My third grade group just finished up an assessment on fractions and multiplication. It’s been about a 1-2 month journey full of investigations on this particular topic. Many formative checkpoints were planned along the way and the unit assessment was scheduled for last week.

While reviewing the student work, I’ll sometimes write questions or direct students to a part of the question that would’ve made their answer more complete. There are moments of pride and moments where simple mistakes drive me a bit crazy. You see, there are only eight units with my new resource, so the assessments influence grade reporting quite significantly.

After grading the assessments, I have student analyze their results. They comb through the test and look at how each question aligns with certain skills. They also determine if a missed question was a fixable mistake. I want students to be able to recognize when this occurs and fix them when they can.

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In my experience, 9/10 times students believe that the reason they missed a question was because it was a fixable mistake. That’s not always the case. There’s a certain amount of self-reflection and humility that’s involved in this process. Being able to be a bit more honest and communicating what a simple mistake is and what it isn’t might be in order before the next assessment. So, what steps do students take if a missed questions isn’t a fixable mistake? It’s one step in the right direction to admit that it isn’t fixable, but then what happens next? Do students and teachers have plan for this, or do we move on to the next unit?

If Ss continue to make small mistakes in the details of the problems, is it a minor mistake or a sign of a lack of understanding in what they are doing? #ITeachMath #MTBoS #msmathchat

— Natalie Perez (@Mathymomto5) February 16, 2018

So, was it a simple mistake or something more? This question comes up more often than not while I’m grading student work or reflecting back on a class conversation. Some of the answers are more positive than others. A simple calculation error can vastly impact an answer, but it may be a simple mistake and the student has a solid conceptual understanding of that skill. But, a number model that doesn’t match the problem tells me that the student might not be certain about the operation that needs to be completed. Was the simple mistake putting the wrong operation sign in the number model? I guess you could go down many different paths here.

The question type can influence how well and thorough a student responds. Some questions are quite poor in giving educators quality feedback to help inform instruction. Right off the top of my head, multiple-choice and true/false questions fit that bill. They sure are easy to grade by human or a machine. Hooray! But, they don’t give me quality feedback that I can use immediately.

This also has me wondering about the quality of the assessments that are given. Measuring how proficient a student is on a particular concept doesn’t always have to come from standardized or unit assessment results. Classroom observations and formative checkpoints are beneficial and give teachers insights to what students are thinking. I want to make students’ math thinking visible. Whether that’s using technology or not, making that thinking visible puts the teacher in a better position. From what I observe, some of the best math task questions are open-ended and tend to have a written component where students are asked to explain their thinking. The quality of the question and openness of the answer helps educators dig deeper into how and what students are thinking. I think that’s why teachers are always looking out for better math tasks that help students demonstrate their understanding more accurately.

How do you help students determine if a mistake was simple or something more?

Image by: Renjith

There are “tricks” or “shortcut” techniques that many teachers have in their tool belt when it comes to teaching mathematics. These techniques are often memorized by students to be used later on some type of assignment. I’m not saying that these types of techniques are good or bad, but often students come away with little conceptual mathematical understanding of the concept being taught via the shortcut. Click on the picture below for more information on conceptual understanding.

Conceptual Understanding — © 2007 – 2012 Regents of the University of California

Without that understanding, students are not necessarily being prepared to apply the concepts later in middle school/high school. A very small sample of the short cut techniques I’m referring to are below.

PEMDAS – order of operations

King Henry Died Drinking Chocolate Milk – Metric System

FOIL Method – Algebra

Negative Multiplied by a Negative = Positive

Gallon Guy/Gal = Capacity

Students generally remember the shortcuts and utilize them on assignments/tests. Is that a bad thing? I can almost hear math teachers around the world grumble. If the students truly have a conceptual understanding of the concept then why not use these techniques? Many of these types of shortcuts are used at the late elementary level. When students understand the technique – such as King Henry .. but don’t understand the concept (differences between the units of measurement and in what context they can be applied) then students/teachers run into problems. Students are expected to be able to apply the concepts in multiple situations. Middle school teachers are then are held responsible to deepen the mathematical understanding of the concepts behind the techniques that were briefly utilized at the elementary level. This topic has been on my mind lately, and finally made it’s way into this post based on this post. Elementary teachers, as most teachers do, attempt to use innovative and engaging methods to produce excitement related to learning and school. That motivation is often contagious and beneficial. Whether teachers use these techniques or not (obviously, it’s up to you!) students should understand the concepts before memorizing nifty sayings that don’t really relate to the concept itself. I’m not blaming teachers for using these techniques to engage students, but ensuring that students have a mastery of the concepts should be near the top of the priority list.