Imprecise Geometry with Doug Clements (Early Math Collaborative at Erikson)

Summary

The script from a lecture on geometry and measurement education reveals a concerning trend in U.S. schools. Research indicates that younger students have a better innate understanding of shapes than those in higher grades, suggesting that traditional teaching methods may be ineffective. The speaker, along with Mike Batista, conducted tests showing that children as young as four could identify basic shapes with high accuracy. The lecture criticizes current educational materials for their lack of precision and poor examples, advocating for a more engaging and precise approach to teaching geometry from an early age. The speaker emphasizes the importance of fostering geometry and spatial thinking to lay a strong foundation for advanced mathematical concepts.

Takeaways

• 📊 Research indicates a decline in geometry and measurement skills as students progress from first to third grade in the U.S. education system.
• 🔢 Young children (ages 4-6) demonstrate a strong ability to identify and differentiate basic shapes, scoring high percentages on related tests.
• 📉 There is a noted decrease in performance when identifying more complex shapes like triangles and rectangles, suggesting a need for improved instruction.
• 👨‍🏫 The script criticizes traditional teaching methods that focus on basic shape identification without fostering deeper understanding or precision in reasoning.
• 🏫 The author argues that children enter school with a solid foundation in basic geometry but that this knowledge is not built upon effectively in the curriculum.
• 🚫 The script points out the inadequacy of certain educational materials, such as a test example that provides poor models of triangles.
• 🎵 It critiques the integration of math and music through songs that incorrectly model geometric concepts, suggesting it does more harm than good.
• 🔍 The importance of precision in reasoning and thinking is emphasized as fundamental to mathematics, which traditional instruction often overlooks.
• 🧩 The Building Blocks project is highlighted as a positive example of engaging children with a variety of shapes to promote spatial thinking and geometry skills.
• 🌟 The script concludes that including geometry and spatial thinking in early education is critical for later mathematical development and has significant benefits.

Q & A

What did the University of Wisconsin researchers discover about first graders' understanding of polygons?

The researchers found that first graders are more likely to count the sides and angles of polygons to differentiate one from the other than third graders, suggesting that traditional teaching methods might be less effective over time.

What did the test given to young children by Mike Batista and the speaker reveal about their understanding of shapes?

The test revealed that children aged four, five, and six did quite well, scoring 92%, 96%, and 99% respectively on identifying squares, even with distractors present. However, their performance was lower for triangles and rectangles, at around 60% and 50% correct, respectively.

Why did the speaker and Mike Batista administer a paper and pencil test with numerals to very young children?

They administered the test to see how children aged four, five, and six would compare to older children. The test was the same instrument given to 1,500 kids from six to twelve years of age.

What was the speaker's experience with presenting his dissertation, and how did it relate to his views on teaching mathematics?

The speaker's dissertation presentation was overshadowed by a more engaging math and music session happening next door, leading to a loss of audience. This experience highlighted the importance of making mathematics engaging and precise, rather than just focusing on basic naming of shapes.

What did the speaker criticize in the math and music integration example provided in the script?

The speaker criticized the math and music integration for providing incorrect mathematical models of triangles, such as referring to a piece of pie or a slice of pizza as a triangle, which does not meet the geometric definition of having three straight sides and three vertices.

What is the speaker's view on the current state of geometry and measurement instruction in the United States?

The speaker views the current instruction as a wasteland, suggesting that it lacks precision and fails to engage students effectively, leading to a flat learning curve and minimal gains in understanding.

What does the speaker suggest as an alternative to the traditional approach to teaching geometry and measurement?

The speaker suggests moving beyond just naming basic shapes and instead fostering precision in thinking about shapes and their properties. This includes making mental images, transforming shapes, playing with shapes, investigating, composing, and decomposing shapes.

What is the 'Building Blocks Project' mentioned in the script, and how does it relate to geometry education?

The 'Building Blocks Project' is an approach that presents children with a wider variety of shapes to talk about, sort, play with, and explore. It encourages building shapes from parts and using various materials, promoting a deeper understanding of geometry and spatial thinking.

Why is it important to include geometry and spatial thinking in early education according to the speaker?

Including geometry and spatial thinking is critical because it provides meaningful and motivating areas for learning number, logic, and eventually calculus. The speaker argues there is much to gain and little to lose by fostering this kind of development early.

What is the 'hypocritic oath' mentioned in the script, and what does it symbolize in the context of the discussion?

The 'hypocritic oath' is a play on words, likely meant to be 'hypocritical', suggesting that the current methods of teaching mathematics are not only ineffective but also potentially harmful to students' understanding. It symbolizes the speaker's strong disapproval of the status quo in math education.

Outlines

00:00

📚 Early Childhood Geometry Education

The paragraph discusses the inadequacy of traditional geometry and measurement education in the United States. University of Wisconsin researchers found that first graders are more likely to differentiate polygons by counting their sides and angles than third graders. The speaker highlights that children enter school with a basic understanding of shapes, but the educational system often fails to build upon this foundation effectively. The narrative includes anecdotes about testing young children and the lack of progress in their understanding of basic shapes, as evidenced by flat learning graphs. The speaker criticizes the simplistic approach to teaching shapes and the lack of precision in reasoning and thinking in mathematics education. The paragraph concludes with a critique of a math and music integration example that inaccurately represents geometric shapes, suggesting that such methods are detrimental to children's learning.

05:02

🔍 Enhancing Geometry and Spatial Thinking

This paragraph emphasizes the need to move beyond basic shape identification to foster precision in thinking about shapes and their properties in children. The speaker advocates for a more comprehensive approach to geometry education that includes a variety of shapes, such as pattern blocks and tangrams, to encourage exploration, sorting, and creative play. The Building Blocks project is mentioned as an example of how to engage children with a wider range of shapes, promoting spatial thinking and laying a foundation for more advanced mathematical concepts. The speaker concludes by stressing the importance of integrating geometry and spatial thinking into early education, arguing that it offers meaningful and motivating contexts for learning and has the potential to significantly benefit students' mathematical development.

Keywords

💡Geometry

Geometry is a branch of mathematics that deals with the properties, measurement, and relationships of points, lines, surfaces, and solids. In the video, the emphasis on geometry is crucial as it forms the foundation for understanding shapes, their attributes, and spatial reasoning. The script discusses how traditional instruction often fails to engage students effectively with geometry, as seen in the lackluster performance on basic shape identification.

💡Measurement

Measurement refers to the process of determining the size, amount, or degree of something, often using standard units. The video script touches on measurement in the context of polygons, where first graders are more likely to count sides and angles than third graders, indicating a decline in these fundamental measurement skills as students progress in traditional education systems.

💡Polygons

A polygon is a closed two-dimensional shape with straight sides. In the script, the differentiation of polygons by counting their sides and angles is used to illustrate the natural geometric understanding that young children possess, which the speaker argues is not being nurtured effectively in traditional educational settings.

💡Precision of Reasoning

Precision of reasoning is the ability to think and argue with accuracy and exactness. The video script stresses that while number and shape are important in mathematics, the core of mathematics lies in the precision of reasoning and thinking. This is contrasted with the 'slop' or lack of precision in the examples given from traditional educational materials, which do not foster critical thinking skills.

💡Distractor

A distractor is an element designed to divert attention or mislead in a test or problem. The script mentions 'real distractors' in the context of a test given to young children, which they managed to navigate with relatively high accuracy, suggesting that even with potential confusion, children have a strong innate ability to understand shapes.

💡Spatial Thinking

Spatial thinking involves the ability to visualize and mentally manipulate objects in three-dimensional space. The video advocates for the inclusion of spatial thinking in education, as it is critical for developing a deeper understanding of geometry and can be applied to various areas of mathematics, including logic and calculus.

💡Pattern Blocks

Pattern blocks are geometric shapes used in educational settings to help students learn about shapes, symmetry, and design. The script references pattern blocks as part of the Building Blocks project, which introduces children to a wider variety of shapes, encouraging them to explore, sort, play with, and create designs, thus enhancing their geometric and spatial understanding.

💡Tangram

A tangram is a dissection puzzle consisting of seven flat pieces that are put together to form different shapes. The script mentions tangram shapes as part of the variety of shapes introduced to children in the Building Blocks project, which helps them to think creatively and explore the properties of shapes.

💡Mental Imagery

Mental imagery is the ability to create and manipulate images in one's mind. The video script suggests that fostering mental imagery is crucial for geometric and spatial understanding. It implies that traditional education often fails to engage this skill, which is essential for precise thinking and problem-solving in geometry.

💡Transforming Shapes

Transforming shapes refers to the process of altering the size, position, or orientation of shapes while preserving their overall structure. The script encourages the practice of transforming shapes as a way to deepen students' understanding of geometry and spatial relationships, which is a key aspect of the Building Blocks project.

💡Decomposition

Decomposition in the context of geometry means breaking down shapes into their constituent parts or simpler forms. The video script highlights the importance of decomposing shapes as a method to teach children about the structure and properties of geometric figures, which is part of the educational approach advocated in the Building Blocks project.

Highlights

U.S. traditional instruction in geometry and measurement is ineffective.

University of Wisconsin researchers found first graders are more likely to differentiate polygons than third graders.

Children's knowledge of shapes is underutilized in traditional education.

Young children (ages 4-6) perform surprisingly well on geometry tests.

Children entering school already know a lot about simple shapes.

Current educational methods do not build effectively on children's existing knowledge.

The importance of precision in reasoning and thinking in mathematics.

Anecdote about a child's incorrect identification of shapes in a school workbook.

Critique of the integration of math and music through songs that misrepresent mathematical concepts.

The need to move beyond basic shape naming to more precise thinking about shapes.

The Building Blocks project introduces a wider variety of shapes to children.

Children engage with shapes through sorting, playing, and creating designs.

Building shapes from parts helps develop spatial thinking.

Geometry and spatial thinking should be included in education from an early age.

Fostering early development in geometry and spatial thinking has significant benefits.

There is much to gain by integrating geometry and spatial thinking into early education.

Transcripts

lesson geometry and measurement from

preschool to high school traditional

instruction in the United States Is A

Wasteland here the uh University of

Wisconsin researchers found that first

graders are more likely to count the

sides and the angles of polygons to

differentiate one from the other than

third graders were we actually teach it

out of them okay and it appears on all

these kind of tests so what do kids know

and why do they suffer like that

well they they know a lot we gave this

test to kids you might think why did you

give them such a test with like paper

and pencil test with numerals all over

to really young kids and the reason is

Mike Batista and I had given the same

instrument to uh 1,500 kids from six to

12 years of age we wanted to see how

kids four five and six would compare

okay and they did pretty well um they

did 9 2 96 and 99% for fours fives and

sixes on on that they did 82 86 and 91%

on squares even with some real

distractors in there right triangles are

a little lower about 60% and rectangles

a little lower again uh but still about

50% correct on on rectangles so what did

we do we found that kids knew quite a

bit when they entered school just about

simple shapes like this this is Mike

Batista in my research out here can you

imagine if you were a principal of a

school and you had to show this graph to

parents and said look at what we taught

your kids this year that's like the

flattest graph you could find and this

is just lowlevel naming of shapes it's a

little better for triangles but not much

better how come because this is what

kids see in school my son brought this

home

okay hey one good thing about it the

triangle doesn't have a horizontal base

up at the top oh okay so that's

different right but you know how the

publisher puts a dted line around it so

that not only do you can you read the

instructions down here but they provide

you a model look at what they chose as

their best example of a triangle it's

got a

hook right

mathematics number and shape are

important constituents of mathematics

but at its fundamental base mathematics

is about Precision of reasoning and

thinking

Precision of reasoning and thinking this

is slop okay uh let's take a look at

some others uh sailboat he got credit

for the sailboat and he might have been

looking at those two triangular s sails

but he circled a

pentagon right and he got full credit

and what's the sandwich doing here all

right no we don't need that kind of

stuff but my favorite is this one okay

my favorite is this one because I

remember well it's it's burned into my

memory I was presenting and if you think

okay he's not a very good presenter now

but I was worse before okay this is my

first time out and I was presenting my

dissertation and I turn my back to the

audience and I have like 500 numbers on

the screen saying L you can see from my

research you know and stuff like that

and next door was a person a guy who was

also talking about Early Childhood math

but he was doing math and music and

there was a partition

you know folding partition between the

two rooms that wouldn't close I know cuz

I grabbed that thing and tried to close

it it would not close cuz his his

speakers were right on the other side of

the opening so the music's wafting in

and everybody's laughing and singing

next door and I'm here and now you can

see from my resarch and every time I

turned around five more people were gone

from the back they went next door with

the other guy so you know when a

presentation's going wrong you start

sweating and and everything you know

you're all nervous and everything and

and then and then at the end Jackie

turns over to to to the person next to

her and say why do we have to sit in the

front we can't even leave now you know

it was miserable miserable finally it's

over I go next door and say be big about

it go next door and and find out what

they're doing so you should do more of

that stuff you know you're just not

motivating nobody wanted to hear what

you had to say so I next door and I got

the song sheets now I don't know the

tune to these because I was trying to

present I was sweating at the time and

um uh but take a look take a look at the

kind math and music they were doing

let's look at the last verse here what's

a triangle with sides of three a piece

of pie for you and

me no it isn't oh maybe they get better

wait a minute wait a minute what's a

triangle with sides of three a piece of

pie for you and me a musical triangle

ding ding

ding let's called the

triangle what's what's wrong with a

musical triangle it's a mathematical

model of a

triangle it doesn't even meet right and

what else they're not it it's not it's

actually doesn't have a corner a Vertex

right the vertex is two straight lines

that come to a point but they're all

curved I know we call it a triangle but

it's not a good example well I'm sure

they get the last one what's a triangle

with sides of three a piece of pie for

you and me a musical triangle ding ding

ding a slice of pizza with everything oh

my God that's a section of a circle they

got three out of three wrong this kind

of stuff breaks the hypocritic oath kids

would be better off if they didn't come

to school the day they did this kind of

song that's no way to integrate math and

music get rid of that too all

right we got to move Beyond naming basic

shapes to getting kids thinking with

Precision about shapes and their

property making mental imag just

transforming shapes playing with shapes

investigating shapes composing and

decomposing shapes so one thing we do in

the building blocks project for instance

is to present you know kids kids with a

wider variety of shapes those of you

that know the pattern blocks will

recognize the pattern block shapes are

here we make them bigger but they're

here but so are tangram shapes so are

long thin rectangles pentagons that look

like this but also pentagons that look

like that so that kids have a wide

variety of shapes to talk about and to

sort and play with and they do they

explore they make pictures they make

designs they play games with

them so we do a lot of that we build

shapes from part from straws from

toothpicks on the computer we build them

from parts and everything else we do a

lot of that kind of stuff so my research

lesson here is include geometry and

spatial thinking in a fundamental way

it's critical it provides meaningful and

motivating areas for number for logic

eventually for later calculus there's a

little to lose and a much to gain by

fostering that kind of development

early