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What Is Conceptual Understanding In Math

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The Edtech Ingredients For Conceptual Learning

Building Conceptual Understanding in Mathematics

What are the most common ingredients in math edtech programs? What combination of ingredients should you be looking for? Well, that depends on what you’re looking to accomplish. Whether you are looking for extra practice, direct instruction or conceptual learning, this downloadable rubric can help you evaluate the content of edtech programs to ensure you have a recipe for success.

Conceptual Learning Introduction Advantages Disadvantages

Conceptual learning is the latest educational approach that focuses more on understanding the concepts and learning how to organize and distribute data. Unlike conventional learning models which focus on the ability to analyze specific facts, , conceptual learning centres around understanding more extensive standards or thoughts that can later be applied to an assortment of explicit models.

Up to some extent, conceptual learning can be seen as a top to bottom approach against the bottom-up approach executed in the rote learning model. For the people who believe that conventional learning as rote remembrance for mathematical data, conceptual learning is a method for getting students to quickly understand the new subjects and conditions they experience.

Math Zombies And Conceptual Understanding In Math

Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner , and those who advocate for progressive techniques. The progressives/reformers argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself failure to do this results in what some call math zombies.

I will state that I, like many teachers, do in fact teach the underlying concepts for algorithms, procedures and problem solving strategies. What I dont do is obsess over whether students have true understanding nor do I hold up a students development when they are ready to move forward.

For many concepts in elementary math, understanding builds from procedures. The student practices the procedure until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain. . Daniel Ansari , maintains that procedures and understanding provide mutual support. Sometimes understanding comes first, sometimes later. And Im fine with that.

How Do You Test for Understanding?One proxy that teachers use for understanding and transfer of knowledge, is how well students can solve problems and their variations. A student in my seventh grade math class recently provided an example of this. As an intro to a lesson on complex fractions, I announced that at the end of the lesson they would be able to do the following problem:

Ending the Fetish over Understanding

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Developing Conceptual Understanding And Procedural Fluency

Which is more important for students to have: conceptual understanding or procedural fluency? Does one have to be taught before the other can emerge?

Some argue that procedure has to come first, because without procedural fluency, students wont be able to do the work they need to notice patterns. Ive seen students who were more procedurally fluent develop deeper conceptual understanding because they didnt get bogged down in calculations. But I had always believed that conceptual understanding was ultimately more important, because procedural fluency is sterile and fragile without the underlying knowledge of how and why the procedures work. I thought procedures shouldnt be taught without at least giving students some idea of why they work first.

As a curriculum writer for IM Algebra 2, this design principle guides me to write activities that involve both conceptual and procedural work whenever possible. One way is to ask students to use conceptual understanding to make a prediction, and then check their answer by doing a procedure. This activity from our Algebra 2 unit on polynomials is a great example:

Using Arrays To Create Meaning Of Multiplication

Conceptual Understanding in Math

Exploring Multiplication With Rectangles

  • Which rectangles have a side with two squares on them? Write the numbers from smallest to largest.
  • Which rectangles have a side with three squares on them? Write the numbers from smallest to largest.
  • Do the same for rectangles with four squares on a side.
  • Do the same for rectangles with five squares on a side.
  • Which numbers have rectangles that are squares? List them from smallest to largest. How many squares with there be in the next largest square you could make?
  • What is the smallest number that has two different rectangles? Three different rectangles? Four?
  • Which numbers have only one rectangle? List them from smallest to largest.
  • A Collection of Math Lessons’ from Grades 3 through 63.OA.1 Interpret products of whole numbers as the total number of objects in a group3.OA.3 Use multiplication & division within 100 to solve word problems3.MD.7 Relate area to the operations of multiplication and division.

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    Procedural Vs Conceptual Understanding At Home

    My Mom was quick to grab a bunch of walnuts to teach me the concept of multiplication. We can do the same for our children at home. The way we pose questions allows our kids to apply procedural or conceptual thinking to everyday problems.

    For example, if you were to ask your six-year-old to help you set the table for dinner, you could ask in a myriad of ways:

    • Request 1: Please set the table with 7 forks, and 7 spoons.
    • Request 2: Please set the table with forks and spoons for 7 people.
    • Request 3: Please set the table with a fork and a spoon for yourself, your sibling, your parents , your grandparent, and your aunt and cousin, who are coming to dinner.
    • Request 4: Please set the table for us and grandma, like you did last night, but your aunt and cousin are also coming for dinner.

    Request 1 is very explicit and is a simple procedural application of a mathematical skill: counting. Your child is instructed on how many of each item to get and can count 7 forks, and then return to the drawer to count 7 spoons.

    Request 2 is also a procedural application, though it can be achieved through multiple skills and thereby extends their application beyond simple execution of an explicit task. Your child could count and carry twice as in request 1. Or they could add 7 + 7, counting to 7 as they remove forks from the drawer, and then continuing to count to 14 while removing spoons from the drawer. Or they could pair each fork with a spoon as they take them from the drawer and count 7 sets.

    Have You Heard About Aim

    Accelerated Individualized Mastery provides a new solution for struggling math students with gaps in their foundational math skills set. The AIM programs use proven Math-U-See strategies and manipulatives in combination with an accelerated approach to help students successfully master math concepts.

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    Follow These Steps To Create Math Experiences That Will Engage And Motivate Your Students

    Use Math to Add INTRIGUE-Ask “How can math add an interesting twist on the activity?” -Put math at the core of the activity–math is not an add-on

    INTERACT With Math in a New Way -Go beyond paper and pencil -Create an experience through which students dynamically interact with math

    Use Math to INFORM Decisions -Don’t start and stop the math at the calculation -Allow math to be a tool for the decisions you are naturally making in the experience Remember: Learning should be active and rich with decision making. Use math to make decisions and solve new problems creatively!

    Schemas Are Key To Deep Conceptual Understanding

    Importance of Conceptual Understanding in Math Accessible Math K-12 Math Project STAIR

    an abstract or generic idea generalized from particular instancesConceptual \ kn-sep-ch-wl \:of, relating to, or consisting of conceptsUnderstanding \ n-dr-stan-di \a : the power of comprehendingespecially : the capacity to apprehend general relations of particularsb : the power to make experience intelligible by applying concepts and categories

    Like many educational buzzwords, conceptual understanding is one that has been defined in enough different ways that it starts to lose meaning. The focus of many conversations around conceptual understanding is also shallow, focusing on the idea that students need it, and less about how it is actually acquired. Which is unfortunate, because the how is the most important part. So were going to talk about the how, because having a deep conceptual understanding of math concepts is critical for problem solving success.

    What is Conceptual Understanding?

    But just so we dont gloss over the what, lets clarify our working definition of conceptual understanding. As we saw in our dictionary definitions above, concepts are ideas generalized from particular instances, and understanding is making sense of experiences by seeing the relationships between concepts and applying them. So we could say:

    How Do We Build Conceptual Understanding?

    Conceptual Learning With ST Math

    You can learn more about ST Math and how it supports deep conceptual learning, as well as experience some of the games yourself at

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    The Importance Of Developing Conceptual Understanding In Mathematics

    Read more

    A series of blog posts designed to help all stakeholders understand the importance of changing how we teach and learn mathematics. The series also provides a comprehensive vision for mathematics instruction around the Number and Operations- Base Ten learning progression.

    Setting the Scene

    Imagine for a second this setting in a first grade classroom:

    Make Ten strategy

    Gavin, Keira, and Lucas each solve 9 + 6 differently. Gavin looks at his fingers and counts on from 9, saying, nine, tenfifteen. Kiera has memorized the answer of 15. Lucas sees two tens frames and imagines moving one dot over, to re-imagine the expression to be 10 + 5.

    The make 10 strategy he is using sets him up to understand and apply these strategies to a variety of settings in the future. Lucas is on a trajectory to be a proficient mathematics student .

    Gavins approach works and students have to go through this stage before they can progress. If Gavin does not learn other, more sophisticated strategies, he may be stuck as a count on student his entire life. There are numerous anecdotal stories of students who in their college algebra classes still use the count on strategy when combining like terms . Continuing to use this inefficient approach limits opportunity in higher mathematics settings as more of his working memory will be engaged in solving this simple .

    The Debate Over Whether It Is Better To Teach Conceptual Or Procedural Math Understanding First Has Been Contested Over The Past Century Significant Research Has Been Done In Attempts To

    …focus needs to be on relationships between conceptual and procedural knowledge in math“Star Star argues that instead of debating over superiority, education needs to consider the relationships that exist between these two approaches to math understanding. In Star’s perspective, conceptual and procedural knowledge exist on a learning continuum and cannot be separated. With distinct differences between each pole of the continuum the aim of research should be to focus on how the relationships, connections, and intersections between these two approaches impact and deepen student learning.

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    Why Teach Mathematics Conceptually

    As educators, it is our responsibility to prepare students for their future. Is there real value in teaching students skills or procedures such as the long division algorithm? If we dont actually use it today, it is even more unlikely that will use it tomorrow.

    We now know that higher-order thinking and creative problem-solving are the new in-demand skills for the 21st century. To build these skills, students need time to develop deep understandings of concepts and the connections between them. In short, we must give students visual models and the opportunity to discuss their thinking, so they can learn mathematics conceptually.

    Take a moment to check out this typical ORIGO Stepping Stoneslesson. Look at the way visual models are used and how students are asked to share their thinking and explain connections between those models.

    Isnt That Why We Have Common Core

    Conceptual Understanding in Math

    In theory, Common Core was supposed to ensure that schools find a healthy balance of procedural and conceptual math. In fact, thats the definition of rigor provided on the Common Core website.

    But over a decade in, and many educators are questioning whether Common Core accomplished its goals. Even supporters of the Common Core philosophy, myself included, acknowledge that changing the culture of math education has proven more challenging than anticipated.

    And there have been a number of unforced errors in how the CCSS were rolled out. We began testing students on the new standards before anyone really knew how to measure things like how well students construct viable and critique the reasoning of others .

    While the test makers were still figuring it out, they rolled out common core aligned high stakes tests, with real consequences for students and teachers. Not only that, they began testing all students across grade levels on the new standards, rather than rolling them out one year at a time. So students who had taken traditional tests throughout their schools, suddenly switched over to common core testing in middle school or high school.

    And lets not forget the textbooks. Im not sure what was required for textbooks to label themselves Common Core Aligned. But the process cant have been too rigorous.

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    Balancing Conceptual Understanding And Procedural Fluency

    In this episode, Joanie and Curtis consider the balance between conceptual understanding and procedural fluency in mathematics teaching and learning. Acknowledging that this idea is like a pendulum that has swung back and forth over several decades, they unpack the meanings of these terms and discuss the impact and implications on student learning.

    They use the 2001 publication Adding It Up to describe how conceptual understanding and procedural fluency come together with three other ideas to form an understanding of mathematics proficiency, and share examples of what happens when students have one type of learning without the other.

    Youll hear our hosts wrestling with differing perspectives to better understand how educators can support students conceptual understanding and their procedural fluency together in ways that support deep learning and readiness for future math experiences.

    Listeners are encouraged to consider these additional resources mentioned in the episode:

    Teaching Mathematics Conceptually Just Makes Sense

    ORIGO Education was founded in 1995 on the premise that mathematics should be taught conceptually. But what does teaching conceptually mean, and why should math be taught that way?

    In elementary school, mathematics has always relied heavily on students remembering procedures and rules, with the occasional trick to help them calculate correct answers. With this teaching method, the focus is on the acquisition of skills where accuracy and speed are rewarded, rather than understanding the concepts behind the math. Students learned how to manipulate symbols, rather than hands-on resources, and there is certainly no need for discussion or collaboration. Quite simply, if you follow the rule, you will get the right answer, and everyone will be happy!

    Fortunately, educational research over the past few decades has provided the necessary data to shift the focus from routinely learning mathematics skills to understanding mathematical conceptsand its for the better!

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    What Is Conceptual Learning

    Students must be able to apply their knowledge in the present condition they are experiencing. This is the reason conceptual learning is the way to fruitful education. At its centre, conceptual learning empowers students to utilize what is important or what they have just experienced to more readily understand the new topics.

    Whenever teachers and students have a strong understanding of the concepts, how they are inter-related with each other, and a couple of models of every idea, they start to build up their very own exemplar that will enable them to reach resolutions about any problem statements all through their career.

    The Benefits Of Conceptual Learning In Math

    Building Conceptual Understanding in Mathematics

    When students only learn procedures, they often find themselves stuck and confused inhigher level math courses. Conceptual math aims to help students understand why the steps work so that they can approach problem solving with a variety of strategies. Further, conceptual math provides a means to communicate mathematical ideas, and the ability to transfer that working knowledge to higher levels of problem solving and to other fields of study, such as science or engineering.

    This is not to say that procedural math is not an important part of the overall picture. In fact, it is integral, working together with conceptual learning to move a student from concrete to abstract understanding.Mathematical competency relies on both conceptual and procedural competency.

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    The Importance Of Math Manipulatives

    Math manipulatives are an important part of building conceptual understanding. Manipulatives arent always purchased. It is something concrete that you provide the student with that can be changed so that students can grasp an abstract concept.

    One of my colleagues used to always say that the brain thinks in pictures. This is very true especially when introducing a new concept or trying to clear up misconceptions. Students MUST have an anchor to refer back to when learning new concepts. Normally in math if the concept is introduced in the previous grade level that instruction will serve as the anchor. If the concept is brand new, then the teacher has to create the anchor with manipulatives.

    Where theres no anchor, students will try to visualize a math concept that doesnt have a frame of reference.

    Using Hands On Math Experiences To Build Conceptual Understanding

    Hands on math experiences offer students unique learning opportunities that paper and pencil tasks simply just cant. For example, have you ever had a student be able to seamlessly recite their multiplication facts, but are unable to 1) explain how they solved it and 2) apply it to a real world problem? This student lacks a conceptual understanding of multiplication. Conceptual understanding has become an essential component of best practice teaching since the Common Core was adopted, as it expects students to do more than rote memorization. Instead, it demands that students be active participants in real-world learning and application to prepare them for college and careers in the twenty-first century.

    You might be wondering, How do I do this? How do I teach in a way that builds my students conceptual understanding in math? One way you can do this is through offering hands on math experiences to your elementary students. Learning math concepts and skills using math manipulatives and tools gives your students the hands on learning experiences they need to make sense of math ideas through exploration and self-discovery.

    This blog post will answer the following questions:

    • What is hands on math?
    • Why is hands on math important?
    • What is conceptual understanding?
    • What is the difference between conceptual understanding and procedural fluency?
    • Why is conceptual understanding important?
    • How do I help my students build conceptual understanding?

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