Curriculum Development and the Teaching-Learning Process: The Development of Mathematical Thinking for all children Genevieve L. Hartman, Ph.D.
Topics for today Part 1: Background and rationale Current teacher preparation & why we should teach math to young children; Children s mathematical capabilities; Our educational goals. Part 2: The development of mathematical thinking for all children
Current Teacher Preparation Training provided to typical ECE teacher far below what is optimal for children s learning and development. Most programs only require one course on mathematics education. ECE teachers focus on literacy and teach very little math, and when they do teach it, they teach it badly.
Government Response We need to be more effective in helping students, particularly disadvantaged students. Doing more early math will promote later school success, especially for low income children.
But, are young children capable? Isn t math too abstract for young children? Maybe they can count a bit, but what about mathematical ideas? Didn t Piaget say the little ones are concrete? Didn t someone say that learning math is not developmentally appropriate for young children?
Let s look at an example: Armando & Keithly What mathematical thinking do you see in the boys play?
Conclusions on free play video Spontaneous engagement in: Operations (like subtraction) Estimation Describing spatial positioning Balancing objects Symmetry Patterns Children enjoy these activities Low-income kids just as likely to explore mathematical ideas as more advantaged peers.
So, Readiness is not an issue, even for lowincome children. What are the obstacles? Lack of sufficient training in math teaching Lack of resources for teachers
Also Some ECE teachers have negative associations with math. My previous history as a poor math student makes me fear teaching math to young children in the future, that being partially my reason for choosing early childhood education.
NAEYC and NCTM (2002) say: Have high expectations and support for all children. Take children s individual interests, backgrounds, and previous experiences in to consideration. Enhance learning By providing opportunities for play By integrating math and other content areas Through intentional teaching Through thoughtful assessment
Include math activities that: Provide a systematic, coherent sequence, build ideas, have specific goals; Build on children s everyday experiences; Engage the children in enjoyable challenges; Foster both informal and formal math; Promote both skills and abstract concepts.
Strong tie to language & literacy Why is communication important in mathematics? Helps children articulate and better understand their own ideas Helps peers understand the ideas Helps the teacher understand what the kids know Particularly important for low-income kids
Our goals for the children Provide enjoyable Pre-K experience, Get the children ready for school, Make sure they look as smart as they are.
In sum, Children are already mathematicians and engineers Readiness for math learning is not an issue; They are already doing real math; Their thinking is more complex and competent than we think; Language and self awareness is crucial. EME is both possible and desirable It can draw on children s abilities and interests It need not be an imposition It IS developmentally appropriate They can enjoy it!
Part 2: The development of mathematical thinking for all children
What should the content be? More than numeracy Exploration of: Number & informal operations Geometry/Shape/Space Patterns Measurement
Number & operations Magnitude (awareness of relative quantities) Verbal counting Enumeration (counting things) Informal addition & subtraction
What do they know already? Some knowledge of number words Concepts of more/less/same Concepts of changes in number, such as: Adding makes more Taking away leaves less
Magnitude We see that young children use spatial arrangement rather than counting to compare amounts. In many cases this strategy works, but not for all tasks. So, even though they have some foundational knowledge, there is still a lot more to learn!
Verbal Counting learning the number words in their proper order Let s learn to count! Will someone count in a different language?
What did you notice? Counting involves Memorization Patterns English is harder than many languages
A video illustration Lateek What does Lateek think counting involves? How does he interpret counting higher
What did we learn from this video? Counting is almost like a song to memorize. Once children figure out the patterns within numbers, and if they have some help, they can count very high.
Enumeration (counting things) Learning to use the number words to figure out how many there are in a set of objects But this is strange and operates under special rules Let s look
What does enumeration involve? Say number words in proper order Any discrete unit-- concrete, abstract, large, small, etc.-- can be counted all at once One-to-one correspondence Count each thing once and only once Match up each number word with a thing Order of counting does not matter
So, our goals are to help children learn: One-to-one correspondence (one number word per object); Stable order principle (number words in consistent order); Abstraction principle (anything can be counted); Cardinality (last number tells how many); Conservation (rearranging doesn t change number).
strategies? Enumeration involves lots of strategies: Subitize, Line up, Push aside, Group, Others?
Operations Baby Hope What kinds of mathematical language does Hope use and what does it mean? Does she have any mathematical ideas?
We want to build on these foundational ideas, helping children to: Start to form ideas about relationships between numbers Begin to engage in informal addition and subtraction strategies
Our goals are to have children: Become familiar with the sequence of numbers Learn the principles underlying enumeration such as Cardinality Conservation Understand transformations Explore strategies for adding and subtracting, even if answers aren t accurate.
Geometry Shapes Recognizing, sorting and identifying shapes Learning about characteristics Mostly 2D but also 3D Spatial relations Analyzing where things are with respect to other things Building vocabulary for talking about these ideas
Shapes: What do children know already? Can discriminate between shapes Can identify shapes But, still working on names
Example: Sarah Kate What skills or knowledge does she need in order to succeed at the task?
What did you see? They have no problem sorting shapes this means they can tell the difference between them. Learning the names is a little harder. English names are Greek 83% In other languages, for example Chinese, the shape names are transparent. So, let s teach the names but don t stop there!
We also need to focus on ideas about shapes Need to help children analyze, reason, and talk about shapes Some key issues: Recognizing non-prototypical shapes Being able to talk about the characteristics of shapes
Our goals are to help children Learn the names of shapes Recognize both prototypical and nonprototypical shapes Be able to point to and count the number of sides and corners in shapes
Spatial Relations Spatial relations and the accompanying spatial language are challenging even for adults! It s quite difficult to take the perspective of another person. But, children with good spatial sense do better at mathematics. So, it s important to address this early.
What do children know already? Children have awareness of distance, direction, and even routes. Older children know spatial relationships. But they may not know the names/labels for these.
Particular challenges for children Left and right Why is this difficult? Do you remember when/how you learned your left from your right?
Particular challenges for children Taking into account two relations at the same time
Our goals are to have children: Analyze spatial relationships Learn the vocabulary for these relationships Practice differentiating between left and right
Patterns Why are they important in mathematics? How we can help children learn about patterns?
First: What is a pattern? A pattern is a predictable sequence that results from correctly applying a rule. For example:
Different Types of Patterns Repeating Patterns Green, Blue, Green, Blue, Green, Blue Cow, Pig, Pig, Cow, Pig, Pig, Cow, Pig, Pig 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3 Increasing/Decreasing Patterns AB, ABB, ABBB, ABBBB 2, 4, 6, 8, 10, 12 200, 190, 180, 170, 160
ones 4, 7, 10, 13, 16 1, 4, 9, 16, 25
Why are patterns important in mathematics? Mathematical thinking involves: Noticing patterns and regularity Explaining them, eventually in formal symbols
1 + 2 = 3 and 2 + 1 = 3 16 + 45 = 61 and 45 + 16 = 61 165 + 671 = 836 and 671 + 165 = a + b = b + a
Another example: # of Kids # of Eyes 1 2 2 4 3 6 15 K x 2 = E
What about preschoolers? Children notice regularity and patterns from an early age
Some issues Ben age 3 Ben age 4 Does he understand patterns in each clip?
Our goals are to Introduce children to patterns Visual (using shapes or colors) Physical Auditory Make sure they can go beyond simple AB patterns Help them to describe them
Measurement What is measurement? How is it different from counting, say, a collection of blocks? Assigning numerical values to continuous quantities We must separate the quantity into smaller units and then count them
classroom Many things to measure Various attributes to measure Different methods, or tools, for measuring things Conventional and non-conventional tools
Link to classroom projects Measuring is necessary for scientific exploration and problem solving. For example: Which type of fruit a lemon, an orange, or a grapefruit will make the most juice? Which type of seed will produce the tallest plant? Can this table fit through that door?
But, not so easy! How would you solve these problems? Much more complex than it seems Involves many sophisticated concepts
Direct comparison Simplest form of measurement For length/height, need to establish a common baseline Example: Which is longer: the blue or red line?
Transitive reasoning... Carmen is taller than Harriet
Transitive reasoning Michael is taller than Carmen
Transitive reasoning Who is taller: How do you know? Michael
Transitive reasoning Comparing two objects by using a third object like a stick, a piece of string, or a ruler!
Using consistent units If you need to use smaller units, make sure they are the same. For example, comparing tables using blocks Use same sized blocks!
Partitioning and unit iteration How many yellow squares do you think could make up the blue rectangle?
Partitioning and unit iteration Imagining the big object broken up into smaller parts. Taking the smaller unit and lining it up end to end with no gaps against the bigger unit.
between unit and size
Conventional tools For example, a ruler:
Not so easy! These concepts are very hard for young children. Might not fully develop until age 7 or 8. But, with a lot of guidance children may show competence.
There is a lot of it e.g., height, weight, length, long, short, longer, shorter, near, far, etc. Sometimes confusing-- Length versus height?
Skills we focus on Direct comparison Vocabulary Careful consideration of the attribute in question: remember looks can be deceiving! For example, which is heavier, the balloon or the potato?
Mathematical Content Length/Height Weight Capacity (how much something holds) Temperature Time
How can you teach early math without pressuring children? Video of lesson on measurement
What does it show? Group engagement Children s fascination and motivation Teacher-led lesson Teacher use of language Lots of open-ended questions Kids challenged to think: she assumes they can Real objects Nothing written Can be pursued further in small groups or individually Integrates well with science
Curriculum A curriculum can be a planned sequence of exciting activities, operating every day to stimulate children s thinking and motivation in the spirit of Montessori and the great progressive educators But it s not easy to do this! Therefore
We need to support the teachers! Through...
Professional Development! Professional Development! Professional Development! Professional Development!
In sum, Preschool-aged children are ready, willing, and able to engage in mathematical activities. Activities must be fun! Teachers need professional support in and out of the classroom. Professional development needs to encourage teachers to talk openly with their students. Learn to look for the teachable moments during free play they are rich opportunities to interact with students. Remember to ask, How did you know that?
Recommendations Choose curricula that focus on: Number Operations Geometry Spatial Relations Patterns Measurement Provide PD for all teachers on: Developmental psychology Development of mathematics
Thank you! For further assistance, contact me at GLH2108@tc.columbia.edu