When children were studied during free play, six categories of mathematics content emerged.

  1. Classifying
  2. Exploring Magnitude
  3. Enumerating
  4. Investigating dynamics
  5. Studying pattern and shape
  6. Exploring spatial relations

Four year Olds: investigate dynamics (putting things together, taking them apart, explore motions like flipping).
Four and five Year Olds: study patterns and shape; create patterns and shape; explore geometric properties; spatial relations, location and directions.

Preschoolers employ, at least at the intuitive level, more sophisticated geometric concepts than most children experience throughout elementary school through block play.     Douglas Clements and Julia Sarama

Mathematics is about making sense of the world. It is a search for sense and meaning, patterns and relationships, order and predictability” in a world that is increasing not ordered and predictable and often does not on the surface make sense and seem to have meaning.
And critical thinking is learning how to handle the ambiguity, confusion and that will accompany the search for making sense.

MATHEMATICS is a language. (Some would say it is the language of science, but I digress)

Strategies and skills that underlie our ability to learn to read, to write and to comprehend are immediately transferrable to learning math.

Making predictions..

Using Visuals, Models, Pictures, Actions as Clues

Organizing Information, Data, Ideas:

In discussing a story there can multiple ways to describe understandings. We use words to help each other build common understandings but we can still all walk away having enjoyed and interpreted parts of the text in our own personal ways. In math we must encourage children to make the concepts and ideas their own… to discuss, debate, build language to describe, seek common understandings but still develop personally relevant strategies that work.

Performance matters ( both personally and professionally) in both: Act it out, sing it dance, it diagram it, build it, read it, write it, show it to others and get their opinion.
Perform your understandings. It makes the learning emotionally, socially and intellectually more tolerable and less lonely.

Manipulatives are meant to help build meaning, they are not procedures to follow. Diagrams are about making sense, not drawing to keep busy.
If you can explain it without models and diagrams, great. But you will need to interpret both if you wish to prove to me you do understand. If you cannot explain the diagrams and models you do not have the whole learning.


Read Aloud, Think Aloud, Struggle in front of your students.
Exhibit confusion frequently and ask for clarification.
Hand the authority for the learning over to your students.


Students are often better listeners when they are attending to a peer.


LEARN TO SELF REGULATE:  Monitor yourself

How do you Get Started: Look, think, talk, read, build, diagram, list gather information any way you can to focus on what you need to do in order to attack this problem.

Think it through in your head vs jumping in and getting going (which sometimes is what you have to do…

COLLABORATE Learn from Others, save yourself brain power
Think, discuss , explain to others: That is how you Access a Variety of Strategies

Reduce the complexity for you!!
Make it simpler by diagramming, listing… reducing the information to a more manageable form. Try a diagram or acting it out.

Check back often to what the initial task or problem was. Learn from mistakes.
Be wary and alert for mistakes in thinking

Discuss Solutions and Compare Strategies with others. Learn from mistakes.  This is where you learn more, see connections, analyze your learning and improve your brain power.
Find out if you really do know what you are doing by

Make Explicit Connections to Mathematical Notations and Communications
This is the grand finale. Teachers help here by showing examples of conventional notation and thinking. You do not have to change your way of doing but you need to be able to  read, write, interpret and comprehend the solution and the approaches to it when they are presented in mathematical notation.




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