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Who's been here?

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Kid Snippets: "Math Class" (Imagined by Kids)

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Like learning about circles with actual examples of the shape, we need to approach all math concepts similarly.
Math Vids - Why concrete examples?
The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

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Math No Problem

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Concrete Examples

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By NCTM President Linda M. Gojak
NCTM Summing Up, October 3, 2013

One of the most memorable moments I had in teaching mathematics occurred in a fifth-grade class. We began the year using rectangular arrays as a model to develop the concept of prime and composite numbers. We hung student-made posters of the numbers from 1 to 100 with representations of arrays and lists of factors for each number around the room. By the end of that unit all my students had mastered multiplication facts and could factor with facility as we began our work with fractions. The connections among concepts and the use of concrete representations certainly led to deeper understanding. Later that year, students worked with a variety of models to find area and perimeter of rectangles and extended that experience to find the areas of triangles, parallelograms, and trapezoids. Most students were able to generalize a formula, albeit not always the most efficient, for each polygon. One day, a student commented that this was just like what they had studied at the beginning of the year. When I gave a puzzled look, the class pointed to the posters still on the wall from our first unit of study and said, "You know, that factor and multiple stuff." I had a new appreciation for the power of providing experiences that enable students to make connections among mathematical ideas. My students remembered and understood the mathematics that we had studied months earlier!

Since that experience I have given much thought to the Process Standards in Principles and Standards for School Mathematics, and their impact on teaching. With the current focus on progressions and trajectories of content standards, the potential of the Connection Standard (NCTM, 2000) continues to pique my interest. It's a powerful standard:

Instructional programs from prekindergarten through grade 12 should enable all students to-

recognize and use connections among mathematical ideas;

understand how mathematical ideas interconnect and build on one another to produce a coherent whole;

recognize and apply mathematics in contexts outside of mathematics.

Too often, rather than making sense of mathematical ideas, students focus on remembering procedures or tricks. For example, how many students learn "flip and multiply" to divide fractions but have no idea why it works? Often those who understand why the procedure works struggle to apply it in problem situations. The procedure alone often leads to misconceptions. Students who work from rote memory often invert the wrong fraction, forget to change operations, or even apply the rule when multiplying two fractions. The meaning of operations doesn't change from whole numbers to fractions. For example, in the early grades, the understanding that students develop of division of whole numbers often rests on the idea that "9 ÷ 3," for example, asks how many groups of 3 are in 9. As students move to fractions, it is important to provide them with experiences that connect this whole-number understanding to similar examples with fractions: "9/16 ÷ 3/16," for example, asks how many groups of 3/16 are in 9/16. In this way, students gain a deeper understanding rather than depending on a memorized procedure and can apply division of fractions to a variety of problem-solving situations and real-world applications.

Many teachers use manipulative materials to introduce a new concept. Manipulatives themselves, however, do not ensure understanding. We must provide experiences that support students' efforts to make connections between what they are doing with the materials and the mathematical ideas that they represent. This takes time and teacher expertise. Algebra tiles are not an end to teaching basic algebra concepts-when used appropriately, they provide students with opportunities to connect their work to the concepts. And it is these connections that enable students to make sense of the abstract representations.

Although it is important to think about the connections among concepts within the grade level or courses that we teach, it is also important to reflect on the connections across grade levels. This work involves thoughtful discussions with other colleagues about the way that concepts are taught and the potential linkages among those ideas. Many of us learned mathematics as isolated pieces of information. Taking a mathematical concept and considering how it originates, extends, and connects with other concepts across the grades will help teachers to develop a deeper understanding. It is then that we can plan instruction that ensures that our students regularly make connections to help them make sense of the mathematics they are learning.

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Making Mathematical Connections - National Council of Teachers of Mathematics

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Linda M. Gojak One of the most memorable moments I had in teaching mathematics occurred in a fifth-grade class. We began the year using rectangular arrays as a model to develop the concept of prime and composite numbers.

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Article from youcubed

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Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered " How Close to 100 " as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

Research shows the importance of visual thinking for mathematicians.

When researchers have considered the brains of mathematicians, and compared them with similarly high achieving academics who worked in non-mathematical fields, they find something fascinating (Amalric & Dehaene, 2016). We might assume that when people are thinking numerically, they are thinking "with language" and that high-level mathematical reasoning would draw from language processing areas of the brain. What the researchers found was that the brain activity that separated mathematicians from other academics came from visual areas of the brain - and this was true whatever the mathematical content. It was not only subjects like geometry or topology that caused activity in the visual brain areas, but algebra and calculations, with minimal use of any language areas. This led the researchers to offer the possible explanation that the mathematicians' achievement in math came from early childhood experiences with "number and shape."

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics.

In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). (for math questions that encourage this use of visual and symbolic representations see our Tasks). Additionally, they found that training students through visual representations improved students' math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.

What is Visual Mathematics?

At youcubed we provide many different mathematics tasks that engage students in visual mathematics. Through decades of work with students, teachers, high-tech companies, politicians and others we have learned that people are excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive.

Teachers can create such mathematical excitement in classrooms with any mathematics question by asking students for the different ways they see and can solve the problems and by encouraging discussion of different ways of seeing problems.

For an example of visualizing algebra see here.

When we don't ask students to think visually, we miss an incredible opportunity to increase students' understanding and to enable important brain crossing.

This extract contains excerpts from Jo Boaler's book Mathematical Mindsets, and her new, forthcoming book: Mathematical Diversity.

References:

Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 4909-4917

Park, J., & Brannon, E. (2013). Training the approximate number system improves math proficiency. Association for Psychological Science, 1-7.

Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered " How Close to 100 " as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

Research shows the importance of visual thinking for mathematicians.

When researchers have considered the brains of mathematicians, and compared them with similarly high achieving academics who worked in non-mathematical fields, they find something fascinating (Amalric & Dehaene, 2016). We might assume that when people are thinking numerically, they are thinking "with language" and that high-level mathematical reasoning would draw from language processing areas of the brain. What the researchers found was that the brain activity that separated mathematicians from other academics came from visual areas of the brain - and this was true whatever the mathematical content. It was not only subjects like geometry or topology that caused activity in the visual brain areas, but algebra and calculations, with minimal use of any language areas. This led the researchers to offer the possible explanation that the mathematicians' achievement in math came from early childhood experiences with "number and shape."

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics.

In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). (for math questions that encourage this use of visual and symbolic representations see our Tasks). Additionally, they found that training students through visual representations improved students' math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.

What is Visual Mathematics?

At youcubed we provide many different mathematics tasks that engage students in visual mathematics. Through decades of work with students, teachers, high-tech companies, politicians and others we have learned that people are excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive.

Teachers can create such mathematical excitement in classrooms with any mathematics question by asking students for the different ways they see and can solve the problems and by encouraging discussion of different ways of seeing problems.

For an example of visualizing algebra see here.

When we don't ask students to think visually, we miss an incredible opportunity to increase students' understanding and to enable important brain crossing.

This extract contains excerpts from Jo Boaler's book Mathematical Mindsets, and her new, forthcoming book: Mathematical Diversity.

References:

Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 4909-4917

Park, J., & Brannon, E. (2013). Training the approximate number system improves math proficiency. Association for Psychological Science, 1-7.

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Visual Math Improves Math Performance - YouCubed

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Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered "How Close to 100" as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

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Tens Frames - creating numbers and adding to 20
Counting - One to One correspondence
Comparing Numbers - using unit squares and base ten blocks

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One inch squares - multiplication, division, area and perimeter
Base Ten Blocks - addition, subtraction, comparing numbers, partial multiplication, division, decimals

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Progression Videos

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The progression videos are a series of videos that show how math concepts build and grow through the grade levels. Here is a post that gives a "behind the scenes" of what it takes to make the videos.

Click on the image below to watch the progression video and download the resources.

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Ted Talk - How to be Good at Math and other surprising facts about learning

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Concrete Pictorial Abstract

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Teaching Student-Centered Mathematics
Number Sense Routines
Number Talks
Mathematical Mindsets
Math Workshops in Action

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Refer to the checklist to make a final reflection of any new learning or inspiration.

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What is Number Sense?

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Visual math article in Journal of Applied & Computational Mathematics

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How visual is your math teaching?

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Online Personalized Learning for Grades K-8 Math | Wowzers

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Engaging online math lessons, games, and assessments for grades K-8. Teachers can easily create custom learning paths for their students.