252. Please, could you share more information on creating hands-on math lessons?
YES!!!! One of the key ideas that administrators, curriculum specialists, teachers, instructional assistants, paraprofessionals and aides need to clearly understand is that CONCEPT DEVELOPMENT requires hands-on learning opportunities so students can:
Hear as the teacher describes the observation using
the language of the content area;
See the key content area WORDS that name
the concepts in the content area;
Speak using the key words thus giving evidence of understanding
the concepts in the content area;
Experience through doing, through actions that allow students to apply
the concepts and the key content area language to new situations.
Thus, in Kindergarten and 1st through 3rd grade, students need lots and lots of hands-on learning opportunities to master
Listening with understanding;
Seeing the key concept words and reading and writing them;
Speaking about their observations using key concept words;
Experiencing applying the learned concepts,
BEFORE the same concepts are presented in the upper grades through WORDS and/or SYMBOLS ONLY, that is, abstractly.
For example, in KINDERGARTEN, through hands-on experiences, students can learn addition, subtraction, multiplication, division, and fractions; and, once they learn to count to 10, they can learn decimals. They should also learn, through hands-on experiences, place value, addition, subtraction, multiplication and division of fractions, and addition, subtraction, multiplication and division of decimals. They should learn other key concepts like reciprocal, etc.
NOTICE: ALL these learning experiences are HANDS-ON, concrete, using manipulatives, not abstract experiences using numbers and mathematical symbols ( + - x ./. ).
HOW? Well, here are some examples:
Teaching how to count:
Counting requires that students consider each item to be counted ONE time only.
Teachers need to tie the idea of counting ALWAYS to the idea of each item considered only once.
When teachers are teaching to count, the visual representation of the numbers is NOT what is essential or what is being taught. That comes later; in other words, how do you represent, abstractly, three objects with the abstract representation or symbol of “3” is a different concept than the concepts of counting and naming a bunch, or a number of items.
Thus, teachers always need to COUNT items when they are teaching how to count – whether “beans,” or “chips,” or “pencils;” students need to physically COUNT items and SAY (ONLY) the number of counted items. No written or printed numbers or abstractions at this point; just counting!
Once the children can count items –never counting an item in a group more than once, students can begin to ADD.
Teachers can prepare geometric figures in two different sizes –one size bigger than the other—and in the same color (For example, smaller and bigger RED circles). On one of the smaller geometric figures the teacher places (and the students count) 1, 2, 3 . . . beans or chips or any other items they wish to add. On another of the smaller geometric figures the teachers places (and the students count) 1, 2, 3 . . . . . beans or chips or any other items they wish to add. Now, teacher and students move the beans or chips from the smaller geometric figures to the larger geometric figure, saying, “How many all together?” and COUNTING the total amount.
Once the children can count items –never counting an item in a group more than once, students can begin to SUBTRACT.
Teachers can prepare geometric figures in two different sizes –one size bigger or larger than the other –and in the same color (For example, smaller and bigger BLACK rectangles). On the large geometric figure the teacher places (and the student count) 1, 2, 3, 4, 5, 6, . . . .beans or chips or any other items they wish to subtract. Then they remove to a smaller geometric figure a number of the counted beans or chips saying: “How many are we taking away? 1, 2, 3, . . .” and counting the chips or beans as they are removed. Pointing to the large geometric figure once again, the teacher can ask “How many are left? How many remain?” while the students count the beans or chips left behind.
Once the children can count items –never counting an item in a group more than once, students can begin to MULTIPLY.
Teachers can prepare geometric figures in two different sizes –one size bigger or larger than the other –and in the same color (For example, smaller and bigger GREEN triangles). Teacher and students place 1, 2, 3, . . . smaller geometric figures in a row, counting the small geometric figures placed in a row. Now, the teacher begins by placing EQUAL AMOUNTS of beans or chips in each of the smaller geometric figures, ONE per figure until ALL figures have the same amount.
Thus the teacher and the students place ONE bean in each GREEN triangle and count to ONE for each triangle, pointing out that each smaller triangle has the same or EQUAL amount than the others. Then the teacher adds ONE MORE bean to each smaller GREEN triangle and count to TWO for each triangle, pointing out that each smaller triangle has the same or EQUAL amount than the others –TWO in each. The process is repeated, adding –ONE AT A TIME TO EACH TRIANGLE—until the desired number per smaller GREEN triangle is reached. At this time the teacher asks: “We have (# or beans) in each triangle; we have (# of beans) in each of (# of triangles). How many do we have all together? What is the TOTAL number of beans we have?” Teacher and students place the beans from each smaller GREEN triangle into a LARGE GREEN triangle and count the TOTAL number of beans or chips, or whatever item the chose to count.
Once the children can count items –never counting an item in a group more than once, students can begin to DIVIDE.
Teachers can prepare geometric figures in two different sizes –one size bigger or larger than the other –and in the same color (For example, smaller and bigger PURPLE squares). On the large or bigger geometric figure the teacher places, ONE BY ONE, a number of beans or chips as the teacher and the children count them. (This number of beans or chips must be divisible by the number of smaller geometric figures that the teacher will then place next to the large geometric figure.) Now the teacher places a number of smaller geometric figures next to the large geometric figure. Teacher and students count the smaller geometric figures.
The teacher then begins to take away ONE BEAN at a time from the large geometric figure and places each bean on the smaller figures to obtain EQUAL amounts in each smaller figure. “One bean in each of (# of smaller purple squares).” Then the teacher takes away from the large figure ONE MORE bean and places each bean in each of the smaller figures. “We now have TWO beans in each of (# of smaller purple squares).” This process continues until ALL beans have been taken away from the large figure and there are EQUAL amounts in each of the smaller figures. Teacher says: “I divided (total number of beans) into (# of smaller PURPLE squares) and I see (EQUAL AMOUNT in each square).”
The basic idea in FRACTIONS is EQUAL PARTS of a WHOLE. Thus, paper folding is the best way to teach FRACTIONS. Entire pieces of paper of different sizes can be divided into equal parts: 2, 3, 4, 5, 6, 7, 8, 9, . . . .. Students can divide papers of different sizes and colors and then cut the equal parts and place each equal part onto another whole piece of paper divided (by printed lines) into the same number of equal parts so students acquire the idea of a whole divided into equal parts. Students can learn to say “TWO equal parts,” “THREE equal parts;” etc., AND students also need to learn to count: “one half, two halves,” one third, two thirds, three thirds’” etc.
ONCE students can count to 10, teaching decimals is the same as teaching fractions; the only difference is that the number of equal parts in a whole must be multiples of 10. Folding papers to 10 equal parts, counting the equal parts is ALL that is required. Students can learn to say “ONE of 10 equal parts,” TWO of 10 equal parts,” etc., AND students also need to learn to count: “one tenth,” two tenths,” etc.
Teaching addition, subtraction, multiplication and division of fractions and decimals is exactly the same as shown above. The only difference is that students must ADD, SUBTRACT, MULTIPLY and DIVIDE equal parts of a whole, not beans or chips.
Once students can count “WHOLES” –one whole or entire piece of paper, two whole or entire pieces of paper, three whole papers, for whole papers, etc,-- students can recognize the “reciprocal” of each number: a whole piece of paper divided into TWO equal parts represents the reciprocal of TWO WHOLE entire pieces of paper; a whole piece of paper divided into THREE equal parts represents the reciprocal of THREE WHOLE entire pieces of paper, etc.
TEACHING PLACE VALUE
This is the most important concept in our mathematical system. The place where a number appears has intrinsic value. To teach this concept, small cubes that can be hooked onto other cubes to make a large string or column of ten cubes can be used.
The teacher designs a board with –first—one line above which only single, separate cubes are placed in a vertical row—but separated. These are the units. Students place and count cubes up to 9.
The teacher adds another line to the board, to the left of the first line. Teacher and students place single, individual, separate cubes onto the first line and count:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10. ALL cubes are joined into a single column and now, ALL joined together, the 10 cubes are placed on the SECOND line to the left of the FIRST line. These are the tens.
Teacher and students begin to count how many cubes in a columns = 10, and how many separate cubes they can add on to the first line to make 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. All the 10 UNIT cubes are joined into another single column. This second column is placed on the second line to the left of the first line, etc.
For more in-depth information, classroom demonstrations, and "coaching" of new and/or experienced teachers, Dr. CARMEN SANCHEZ SADEK offers:
1. Cognitive - Academic Language and Vocabulary Development
2. Cross Cultural Diversity - Multicultural Strategies
3. Effective Instruction for English Learners (L.E.P. students) Parts 1, 2, 3, 4
4. Promoting Academic Success in Language Minority Students
5. Cognitive - Academic Language and Vocabulary Development
6. Oral Language / Literacy Skills / Higher Order Thinking Skills
7. 50/50 Dual Language Programs: design, planning and implementation
8. The Structure of English / The Structure of Spanish
9. Transition: Introduction to English Reading
Web Site Programs for Teachers: Numbers 1, 5, 7, 8, and 9.
Web Site Programs for Paraprofessionals: Number 3.
Web Site Programs for New Teachers:
Enhanced Cultural Sensitivity - The Challenge of Students Diversity
Identifying / Responding to Students' Language Needs
Phonemic Awareness: Teaching English phonics to L.E.P. students
Relationship Between Reading, Writing and Spelling
Improving Reading Performance -- Building Oral Language Skills)
Write and e-mail any additional questions you may have, and Dr. CARMEN SANCHEZ SADEK will establish with you, your school or district a Technical Assistance Service Contract. Dr. CARMEN SANCHEZ SADEK will answer all your questions promptly and to your satisfaction.
For information and credentials please click on the link below or contact directly:
CARMEN SANCHEZ SADEK, Ph.D.
Educational Consultant, Program Evaluator
National Board for Professional Teaching Standards, Certification (12/2006)
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Phone and Fax: (310) 474-5605