Monthly Archives: January 2013

The Mathematics of the Natural World, Part 1: Attribute Patterns

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I have always been a lover of numbers. Math and science just go together so wonderfully. To me, the idea that most natural phenomena (population growth, diffusion, cell division, plant leaf arrangement, a beautiful vista…) could be explained by a simple mathematical formula or idea, is just mind-boggling and reassuring, at the same time.

This post will provide you, the teacher, with some definitions, establish the relevance of some mathematical ideas to the natural world, and share links to some online resources that will help you plan math connections to your winter study of patterns in nature.

Mathematics in Nature — An Overview

We will review, in brief, a number of mathematical principles in this blog, over the next several weeks. In each post, the concept will be defined, in mathematical terms, then explained as it relates to the natural world. I will share some real-life examples, and then provide helpful links to some classroom tasks to reinforce the idea.

  • patterns
  • order & magnitude
  • symmetry
  • scale & proportion
  • Fibonacci numbers
  • fractals
  • The “Golden Ratio”
  • tessellations

Patterns in the Natural World

When we think of teaching patterns to students, our first thought is usually those patterns we named with letters, back in kindergarten and first grade:




and so on…

In reality, there is much more to the mathematical idea of patternation than this. There are actually three major types of patterns, classified by the basis for the pattern:

  1. Logical patterns
  2. Numerical patterns
  3. Language patterns

All three types can be studied via your science and nature study work, as we will see today.

Logical Patterns

Logical patterns are conceptual patterns based on meaning. There are two main types of logical patterns: attribute patterns and order patterns. Today we will talk about attribute patterns.

Attribute patterns

Children learn, at a very early age, that objects in the real world have qualities, or attributes, some of which can be directly observed (size, shape, color), others which can be determined by the use of simple tools or tests (e.g., floaters and sinkers, magnetism, etc.). When children sort objects into groups based on like attributes, or classify objects into identified groups, they are using attribute patterns as the basis for their work.

Here’s a real-world example of these two types of patterns, based on my son’s homeschool library and room organization. I know that it is easier for children to find things if there is system to organizing them. I have used two different systems over the years, in classroom and homeschool, both successfully. One involves more on my part, one more of the child’s thinking.

Scenario 1: Pre-Determined Classification System (most common)

Before the start of the year, I organize the classroom or homeschool library according to pre-determined categories, based on past experience and curricular needs, label the shelves or explain the system, and guide students to replace materials in the proper category through classification. This is likely the same system most parents use to help kids organize their bedrooms.

I do this based on several attributes, some observable, some based on purpose (not observable). How do you think I organized the two areas in the photos, below?

Logical patterns attributes

Read below for the criteria by which we sorted my son’s homeschool resources.

Here were our categories, based on use:

  • Encyclopedias and Reference Books (1)
  • History Books (2)
  • Today’s Materials (3)
  • Hats (4)
  • Notebooks (6)
  • Science Books (5)
  • Soccer Stuff (7)

Here is another example, using more obvious attributes…

Logical patterns attributes

Sorting books based on more obvious attributes.

Scenario 2: Student-originated Organization System (less common)

In some cases, I let the students organize belongings, then tell me their criteria for arranging them. This is the skill of categorization, the flip-side of classification.This requires the adult to let go of the process, and accept the students’ system of organization.

When we did this with the classroom library, it entailed a huge mess (at first), lots of argument, and some rather clever, kid-friendly categories. This is the system my two youngest boys have employed when making sense out of about a million LEGO pieces, as below. (NOTE: My middle son employed a label maker and made category labels for the compartments of an inexpensive hardware storage box):

Logical patterns attributes

Form and functionality help the LEGO builder sort bricks.





Logical patterns attributes

Kindergartners sort and categorize seeds, providing their own categories.

***There’s Still Time!***

Don’t forget about the Mid-Winter Give-away

Click over for more details on how to enter!




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Working with Analogies: The Analogies Center

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Working with analogies

Classroom Instruction That Works, $22.80 at Barnes & Noble

Researchers have determined that strategies that have students looking for similarities and differences between and among items result in some of the greatest performance gains (Classroom Instruction That Works, Dean, et al, 2013). Among the ways that students can look at similarities and differences is through creating and explaining analogies.  This blog post will explain what analogies are, how they reveal a deep understanding in students, and some ways to help incorporate working with analogies into independent practice opportunities throughout your curriculum.

What is an Analogy?

An analogy is a comparison between two things that are similar in some way, often used to help explain something or make it easier to understand. In an analogy, the student must determine the way in which two things are related, and extend the comparison to two additional items similarly related. Those of us who have taken the SAT are familiar with this device:



This device is customarily read, “Water is to snow as lava is to ____.” This means that water and snow are related to one another, in the same way that lava and _____ are related. In order to accurately complete the analogy, the student must first determine the specific relationship between water and snow, and then apply that to lava.

The Bridge Map: A Thinking Map ®

We know that all bodies of information, and all thought processes, have their own “shape.” That is to say, there is a pictorial way to represent ideas, processes and functions that is not constrained by words, and is easier for learners of all ages to understand. Not only are these graphical representations easier to comprehend, but they also instruct the student regarding the overall structure of the information relayed. David Hyerle developed Thinking Maps ® as a way to simplify the graphical way that learners represent different cognitive processes. In his system, there are only eight “maps” needed to explain the different thought processes that humans use to process information.


Representing an analogy using a bridge map.

When creating and extending analogies, students could use the traditional device with which we adult test-takers are familiar, but the device requires that the learner is a reader, and, therefore, excludes its use by young children, second language learners, or struggling readers. A non-linguistic way to accomplish the same thing is the bridge map.


An extended version of the same bridge map.

The bridge map, extended, can be viewed at right.

Once students see two pairs of items, they can determine the relationship between items in each pair, which is represented by the line between the items:


“Water is the liquid form of snow, as lava is the liquid form of rock…”

Using Bridge Maps in a Learning Center

There are four main ways that students can work with bridge maps in an independent learning center. Using a combination of all the ways ensures that students understand the entire concept of analogies.

  1. What’s My Rule?
  2. Complete the Analogy
  3. Extend the Analogy
  4. Create an Analogy

For illustration purposes, let’s use something easy to understand, geometric figures, to explain these four different learning tasks.


The Basic Learning Center Design

I always like to use a chunk of bulletin board space for learning centers, so that students can manipulate items as they work collaboratively. Alternatively, a table top can be used (especially if you are working with realia as the items in the analogous pairs).

Other Materials:

  • A large copy of a bridge map (for bulletin board or table top) [See Note]
  • Photographs, drawings or real objects to use in analogous pairs
  • Index cards or sentence strips with words to use in analogous pairs
  • Blank cards or sentence strips to complete or extend the analogy
  • Markers
  • Student copies of individual bridge map worksheets
  • A finished work basket

NOTE: When creating a frame to be used over and over again in a center, consider reproducing it on heavier paper or cardstock, then laminating it. Use hook and loop dots to attach items to the frame, and store materials in zipper-style plastic bags.

See the diagram, below, for an example of how the basic center layout might look.

Learning centers

Version 1: “What’s My Rule?”

Learning centers
Students can examine a teacher-prepared bridge map to determine the “rule.”

Use the basic format (shown above). Use photos, realia, or words to complete one bridge map section. Provide sentence strips for students to write the “rule,” or relationship between pairs of items. The relationship between one pair MUST be the same relationship between all pairs in a given map!


To check their work, students place their “rule” strip on the horizontal line between pairs. The sentence must be true. If it is, then they check the remaining pairs. If the sentences are all true, then they can conclude that the relationship is one possible “rule” for this set of items.


Version 2: “Complete the Analogy”

Bridge map

Teachers can pre-fill some parts of the bridge map, and students can complete the analogy.

Use the basic format, as before. Use photos, realia, or words to partially fill in one bridge map section. Provide index cards and markers, or images, or sources of images, for students to complete the analogy. If providing images and words, provide a variety, so that some complete the analogy and some don’t, and so that students may visit the center more than once. Then provide blank bridge maps for students to record their work, and state their rule.



  • Provide resources for students to research the topic
  • Keep sets of related items in labeled baggies for future use

Version 3: “Extend the Analogy”

This example of an analogy center uses real objects instead of word cards. In the example, below, students must correctly determine the “rule” (relationship) between the top and bottom of the bridge map, then use that rule to extend the analogy. NOTE: For this particular example, there is a very specific rule (“If an isosceles triangle is spun around its vertical line of symmetry, you get a cone.”). Watch out for students who only look shallowly at the relationship (e.g., they put a 3D “diamond” under the parallelogram), because they are missing the specific relationship between the top and bottom items (i.e., the 2D form is spun around a line of symmetry, so the resulting 3D form cannot possibly have all those “edges.”).

Provide attribute blocks and 3D geometric figures or real objects of those shapes, to help students to visualize. Then provide materials for them to affix the 3D object to the map. As before, give them the organizer sheet to record their thinking.

Bridge map

Students can use real objects to complete and extend analogies.


Version 4: “Create an Analogy”

In this example, students use real objects from the classroom to create nets, paper models of 3-dimensional objects which, when cut out and folded, form the 3D shapes of the original items. You will also notice that the board space is divided so each group has a portion as a workspace.

Collect a number of classroom items and display them at the center. Provide paper, writing tools and scissors, and allow students to work individually or in pairs to create 2D representations of the objects, or nets. Students create the analogy by mounting the net, over the real object, and stating the rule:  “_____ is the net form of _____.”

Copies of the nets can be provided by teams below each analogous pair, so that their classmates can check the accuracy of their work (i.e., classmates can construct the 3D figures from the nets to determine if they do, in fact, create the shape of the original object).


Bridge map centers

Students can be given a rule and create the analogy.


Other Ideas Using Analogies:

These are just a few ways that analogies can be used in a learning center. Here are a few more… see if you can think of others (the words in bold describe the relationship [“rule”] between the items in italics):

  • Hardware and human joints: “An elbow works like a hinge.”
  • Tiles and tessellations: “This design is a tessellation of this tile.”
  • Form and function: “A bird’s tail steers like a plane’s wing flaps.”
  • Organelles and parts of a factory: “The mitochondria work as the cell’s power plants.”
  • Seed dispersal and package transport: “Dandelion seeds move like air drops from a plane.”


Next Steps… and Sharing!

Please do try out analogies in your classroom. And share your ideas via our Simple Science Strategies Blog Carnival. Happy analogies!


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New England Stone Walls: A Photo Scavenger Hunt

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What is Comprehension?

Recently, I was in a first grade classroom, where the teacher was introducing a non-fiction text about the desert. He began by asking the students to share what they already knew about the desert. The students’ responses were sparse, and not very encouraging to the teacher.

So we took back the readers, passed out big pieces of drawing paper and art supplies, and asked the kids to draw everything they knew about the desert, THEN tell us, instead. The results (in pictures and words) were phenomenal: camels, oases, chameleons, dust devils, heat waves from the sun, and many other details that the students could not articulate before drawing. We were astounded at what these 6-year-olds knew about deserts.

What does this tell us?

This tells us that, in order to understand something, we have to first envision it in our minds, and (sometimes) in front of us. It also tells us that many students can envision something well before they can talk about it.

Continue reading

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I’m always learning new things about blogging. Take a moment to explore my new sidebar and footers. I’m hoping that the cleaner look will make it easier for you to find your favorite parts of the blog, and will make … Continue reading


The second draft of the Next Generation Science Standards are available for public review. If you are interested in examining them, and giving feedback either as an individual or as a representative of your organization, you may read the standards … Continue reading


As promised, the printable copy of the Winter Newsletter is (finally!) here, after holidays, family illnesses and snowstorms. This issue focuses on the mathematics of the natural world – patterns, designs, trends, and the measurement of all things natural. There … Continue reading

Time for a Mid-Winter Give-away!

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It’s time for our …


Mid-Winter Give-Away!

This month, we are offering TWO great give-aways, and TWO copies of each!

Prize #1: Two (2) copies of The Simple Science Strategies e-Book, Fall 2012 edition

Simple Science Strategies, Fall Edition (2012). 120 pages.
Simple Science Strategies, Fall Edition (2012). 120 pages.














This full-color e-Book contains the most popular posts fromSimple Science Strategies, plus other reader favorites from A Child’s Garden and Squidoo.  This 120+ page e-Book contains ten posts, including full lesson plans, materials lists, and electronically linked resources for download. All printables are included as pages in the e-Book, and as links, for maximum flexibility for the user.

Additional material is provided which is EXCLUSIVE to those who purchase or win the e-BOOK, and which was never published! This e-Book is a great resource for those who want quick and simple science projects for homeschool or classroom, as well as teachers who want to learn more of the science background behind the activities.

Two entrants will win a copy of this e-Book.

Prize #2: Two (2) copies of Nests, Nests, NeSTS!

Nests, Nests, Nests! A 25-page e-Book for homeschool or classroom.

Nests, Nests, Nests! A 25-page e-Book for homeschool or classroom.


As described in the promotional post, this e-Book accompanied our winter studies of the nests of different animal classes. This 25-page e-Book contains blackline notebooking and science journaling pages for use with bird studies, ecology work or other nature studies. A great addition if you are using the Burgess Animal Book for Children or the Burgess Bird Book for Children in your classroom or homeschool.

Two entrants will win a copy of this e-Book.

How to Enter

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