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What Exactly Is "Understanding?" And How Do We Assess It?

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What Exactly Is "Understanding?" And How Do We Assess It?

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What Exactly Is “Understanding?” And How Do We Assess It?

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Blogger Terry Heick looks at the nebulous term “understanding” and how its many interpretations can make student assessment so difficult.

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Building Conceptual Understanding through Concrete, Real-Life Example

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Building Conceptual Understanding through Concrete, Real-Life Example

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Building Conceptual Understanding through Concrete, Real-Life Examples - Everyday Mathematics

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If children are introduced to abstract concepts before they have a solid basis for understanding those concepts, they tend to resort to memorization and rote learning, which is not a solid foundation for further learning. Everyday Mathematics focuses on first developing student's understanding of concepts through:

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Building Conceptual Understanding in Mathematics

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© Balka, Hull, and Harbin Miles

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As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students' conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.

As my teaching methods have evolved, especially since embedding the math practices in every lesson, I've pushed myself to use inquiry-based tasks and hands-on manipulatives with my students as a way to help them develop their conceptual understanding. Yet there are still topics (such as solving inequalities with a negative coefficient) where I feel unprepared to teach in this way.

I recently got a sneak peak of Leah Alcala's new video, Concept First, Notation Last. While Leah is probably best known on Teaching Channel for her My Favorite No and Highlighting Mistakes strategy videos, this new example of practice is a chance for us to see how Leah helps her students develop conceptual understanding in math. After watching, I felt as if some of my own questions on the topic had been answered.

Here are my 5 takeaways, which are now reminders to myself and tips for you:

1. Belief
Leah Alcala has the utmost belief that her students will be able to access and attempt the task at hand. In the video, you'll hear her make the statement, "Everyone likes to learn." It's clear that her beliefs move students forward. Take inventory of yourself. Do you believe that ALL students in your classroom can access mathematical ideas without being told what to do and how to do it? To continue to deepen your beliefs, I recommend reading Jo Boaler's research on how all students are capable of developing mathematical ideas.

2. Sense Making
When my teaching started to evolve into one that focused more on conceptual understanding rather than rote memorization, I started asking one question of all my students: "How/Why does your solution make sense to you?" I found that by developing justification as a norm, I could understand exactly what students believed, why they believed it, and respond with more effective questions. Moreover, in order to have those conversations, the task they were responding to had to be a rich, inquiry-based task. That meant I was no longer teaching tricks or math shortcuts. Students had to be able to construct viable arguments and critique the reasoning of others. In the video, you'll see Leah posing carefully developed tasks alongside constant questioning, as opposed to teaching mathematical tricks. This allows her students to use sense making to come to a conclusion on their own.

3. Scaffolding
From the outset of this lesson, Leah keeps the numbers easy to manipulate while the mathematics remains demanding. If the numbers become too difficult at the outset, students can get overwhelmed and will often shut down. As Leah mentions, easier numbers help give an access point to all students. Over time, you can increase to more challenging numbers while maintaining challenging thinking as well.

4. Time
Conceptual ideas are not built in one day or even two. They are developed after repeated exposure to a particular mathematical idea in various contexts. Students have to struggle and resolve that struggle to internalize a concept. This process takes time. Leah mentions that the one day her class is being filmed will, by itself, not be sufficient for complete learning to occur. In my classroom, this means a learning objective may remain consistent for a week or two as we are developing the idea.

5. Multiple Representations
Leah allows her students to engage with the mathematical idea of solving inequalities through graphs, lists, and/or mathematical notation. When developing conceptual understanding, it's imperative to give students freedom of choice in how they might potentially respond. Narrowing to one representation too often causes students to try to find the one correct path towards a solution, rather than thinking expansively and for themselves.

[video_embed src=" https://www.teachingchannel.org/videos/teaching-inequalities/embed.js" width="480"]

Since I've begun to focus on the development of conceptual understanding, I've run into some challenges. Students and parents alike want me to teach short cuts and algorithms. Yet, when I hear students talking and thinking mathematically, I'm certain that this struggle will prepare them to be risk-takers, not only in my classroom but in their daily lives, too. My hope is that this will lead to self confidence and an internal belief that when faced with any problem, they have ideas worth exploring.

I encourage you to watch this video and think about how developing conceptual understanding is both doable and essential in your classroom. I would love to read your responses in the comments section below on how you were able to take these ideas and apply them with your students!
Crystal Morey is a middle school math teacher in Enumclaw, WA. She has served on the county's Common Core transition team, and is a teacher leader on a statewide project with Washington State's Office of the Superintendent of Public Instruction and the Mathematics Education Collaborative.

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5 Tips To Help Develop Conceptual Understanding In Math

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As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students' conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.

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Building Conceptual Understanding
and Fluency Through Games

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Building Conceptual Understanding
and Fluency Through Games

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FOR THE COMMON CORE STATE STANDARDS IN MATHEMATICS

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This post is an excerpt from a post entitled " Don't lock me into 8 key concepts" by Shannon O'Dwyer. The full post explains concepts in a nutshell and why concept-driven teaching is both easy and hard. This excerpt provides an insight into how teachers can make it happen. Concepts in a Nutshell
Concepts are over-arching mental constructs that students need to make sense of the content we teach. Conceptual understanding enables students to apply facts and skills to the world around them. By starting with concepts, we create a curriculum that's worthy of our students' time and effort. The concepts that drive the PYP are timeless (factual examples change, but not the core understanding), universal (so students can apply understandings across cultures, situations and disciplines) and abstract (so students engage in higher order thinking to grapple with central ideas).

By helping students achieve conceptual understanding, we ensure that they can take facts and skills, and do something with them beyond this moment, this lesson and this classroom. I have seen plenty of students who can accurately convert between metres & centimetres but are unable to choose an appropriate length of skipping rope for a 3-person playground game. These students had mastered a mathematical skill in isolation, but not yet developed a conceptual understanding of length.
Why concept-driven teaching is hard
It's just so messy. Concept-driven learning is an iterative process of constantly pulling apart ideas, putting them back together, re-applying them to different situations, finding ways to correct misconceptions and enabling each child to make meaning in their own way. It would be a lot easier to say, "Here's what you need to know, practise that, now you get a sticker." The problem is that you'd have to add, "Good luck applying that to your life". Factual knowledge is easy to teach because it is simple, procedural and locked in situ. But unless your students intend to live forever in their 4th Grade classroom, that knowledge is utterly useless to them.
"How can I make it happen?"
Questions
Use teacher and student questions as a springboard for deep, conceptual dialogue. Sort, unpack and challenge students' questions. Discuss open vs. closed questions, abstract vs. concrete questions, fat vs. thin questions (ie 1 answer or many). Keep asking probing questions (such as How do you know..?/reflection) to prevent shallow understanding.

Problematic Knowledge
Design resources and situations that are problematic. Perhaps the facts do not add up, or there are pieces of information missing. This challenges students to search for connections, inferences and transference to make sense of the inquiry. Anything that is non-linear, with multiple solutions and multiple interpretations helps our students build the flexibility of thinking to grapple with complexity, change and abstraction.

Inquire
Every day! Make sure you devote enough time for students to actuallydo the learning and construct the understanding. They need ample time for wondering, trying, tinkering, playing, researching and coming to conclusions. Don't step away. Be a participant who says, "Why did you...?", "What would happen if...?"

IntegrateWork with single-subject teachers to develop integrated thinking. Students should take their understandings from class to class, exploring concepts through different content and perspectives (eg symphony orchestras and bus timetables both build an understanding of systems). Try to keep concepts at the core of your planning meetings, to avoid creating shallow, thematic units. Avoid throwing together "linked activities". If the students are studying Ancient Greece, don't immediately say, "Let's make pots!". Perhaps the most important concept is "legacies" and students investigate the influence of the golden ratio on both Art and Mathematics. While studying natural disasters, let students make paper mache volcanoes at home. In class, explore the concept of "impact" on geography, individual lives and communities. How do artists help communities remember and heal after a natural disaster?"

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To read the full post, visit Shannon's blog here.

Shannon is a PYP educator, whose passions include curriculum design, literacy and ESOL practices, enrichment and technology integration. Shannon loves working in PYP classrooms, where each day is filled with the questions, problems and discoveries of young minds. She blogs at shannonodwyer.com and tweets @S_ODwyer.

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This post is an excerpt from a post entitled " Don't lock me into 8 key concepts" by Shannon O'Dwyer. The full post explains concepts in a nutshell and why concept-driven teaching is both easy and hard. This excerpt provides an insight into how teachers can make it happen.

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Concept-based learning is about big transferable ideas that transcend time, place, and situation.