Students Need Your Conceptual Understanding!

What IS conceptual understanding?

Amy established common ground by sharing these definitions:

Procedural Fluency: can reliably follow a learned procedure to a mathematical goal

Conceptual Understanding:

• • an understanding of the underlying concepts;
  • the concept fits into your schema of mathematics;
  • the topic can be approached from multiple directions

Example:

I can know the formula P = s₁ + s₂ + s₃ + … and make an accurate calculation (procedural fluency)

or

I can know that perimeter is the distance around an object or geometric figure and use that knowledge to solve a problem – even if I don’t remember the formula (conceptual understanding)

Why is it important for teachers to build our own conceptual understanding?

As Amy shared, “The more we understand, the more that we can say yes to students’ good ideas.” (emphasis added)

Take, for example, the following student work from Dylan:

Question to think about:

What questions could you ask to uncover Dylan’s thinking?

Or consider this student work from Savannah:

Question to think about:

How could you and Savannah unweave and look closely at all of the math happening here in a way that would advance her understanding of fractions?

In my opinion, it can be all too easy to label student solutions as incorrect if they don’t match our own understanding of solving the problem. However, amidst Dylan’s many different (and arguably unorganized) calculations, we might miss that Dylan changed the denominators to work with smaller numbers, all while still maintaining equivalency. Fantastic! Or we might miss that he saw a scaling of 2 ½: multiply 8 by 2 and then add half of 8 to get 20, then repeat that relationship by multiplying 7 by 2 and adding half of 7 to find the missing value, 17.5. (Note his work at the bottom of his page.)

And if we aren’t able to flexibly think about fraction concepts, we might miss that Savannah sees the need for a common denominator but maybe needs a deeper understanding of when and why cross-multiplication works. There’s also a “doubling” relationship that we could draw out: 14/8 is double 7/8, and 8/8 is double 1/2. Savannah essentially doubled each quantity before subtracting. Did she mean to? I’m not sure. But if she did, this is a perfectly legitimate solution pathway as long as she compensates by halving her answer from 6/8 to 3/8 since she originally started with half as much of both quantities.

Full transparency: I saw the “times 2” relationship and at first could not wrap my head around why the answer did not come out correctly! I mean, both terms were multiplied by the same amount, so it should work, right?! It wasn’t until I drew a picture of 7/8 and 1/2, realized the answer was 3/8, repeated that process with ANOTHER 7/8 and 1/2 (to double the values), that I realized I would be left with two 3/8, or 6/8…double what I should have! Wrestling with the equations on Savannah’s paper gave me the opportunity to strengthen my own ability to think flexibly not only of equivalent fractions but of equivalent expressions. As a teacher, moving beyond commenting on the procedure and instead looking to understand student thinking can make room for dynamic conversations that sharpen everyone’s math knowledge.

How do I uncover the thinking my student used if I can’t make sense of what’s on their paper?

Here are three suggestions from Amy that we can use as starting points:

1. Listen to our students. Who is doing the talking? Are students encouraged to explain in their own words, even if they aren’t using the exact vocabulary or phrasing that we would? Are we listening for how they are visualizing and working through the problem for us to then use their own words and processes as part of how we either confirm, clarify or advance their thinking?
2. Stay curious about math concepts and various solution methods. Take the time to explore your own questions (ex: sketch out Savannah’s problem of ⅞ – ½ twice to see what’s leftover). Ask students questions about what you see on their paper (ex: How did you (Dylan) identify the 2 ½ relationship? How did you connect that to finding the missing value?).
3. Accept the chaos that might ensue from exploring ideas alongside our students. It’s okay to not know. It’s okay to try things to see what works, what doesn’t and follow where the conversation leads.

All three of these suggestions honor student thinking and help us use their current understanding to advance deeper learning of concepts.

Amy ended by sharing a line from the Adult Numeracy Network’s Professional Development Principles: “Sound professional development in adult education mathematics should be designed to begin with teachers as mathematics learners and thinkers.” Exploring the math ourselves allows us to understand ideas from multiple angles…and say yes to students’ good ideas!

To hear the conversation firsthand and to learn from your colleagues’ many ways of thinking about the same problems, watch the recording of this workshop on the MN Adult Education Professional Development YouTube Channel.