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Innovative Practices That Increase Mathematics Achievement. by Joan A. Cotter, Ph.D. JoanCotter@ALabacus.com. Slides/handouts: ALabacus.com. Cotter Tens Fractal. FCSC Orlando, FL November 17, 2009 12:30 - 1:30 p.m. Cape Canaveral Volusia. How many little black triangles do you see?.
Innovative Practices That Increase Mathematics Achievement by Joan A. Cotter, Ph.D.JoanCotter@ALabacus.com Slides/handouts: ALabacus.com Cotter Tens Fractal FCSC Orlando, FL November 17, 2009 12:30 - 1:30 p.m. Cape Canaveral Volusia How many little black triangles do you see?
Math Crisis • 25% of college freshmen take remedial math; 38%, in California. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30 percent of the world’s college grads; today it’s 14 percent. CSM 2006 • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th. • Close to 60% of those in jail under the age of 30 have no high school diploma and math is often the reason.
What Makes Little Difference • Class size: engagement rises, but achievement gap remains. (40 in Japan, 50 in China, 26 in Singapore) • Amount of homework. • Counting ability. • Poverty makes greater difference in US than in other countries.
Finland • Teachers from top 10% of undergraduate class. Need master’s to teach. Held in high esteem. • Teachers work together on lessons and visit each other’s classrooms. Half day/week for PD. • Work with students as soon as they fall behind.
Singapore • Although highest scorer in recent TIMSS, Singapore scored 16/26 in science in 1983-84. • In 1990 curriculum changed to emphasize math concepts and problem solving, rather than rote learning. • Stress visualization, patterning, number sense. (Not so much in US versions.) • National curriculum.
China • Math specialists starting at grade 1. • Teach 2 classes/day with 50 students/class. • Teachers’ desks are near other math teachers in workroom to encourage collaboration. • Half day every week for PD. • Standard national curriculum.
Japan • Teacher stays with the same class for 3-4 years. • Teachers’ desks in a huge room with references. • Goal for math lesson: the class understands a new concept, not done something (worksheet). • Teachers emphasize visualization; discourage counting for computation. • Groups quantities into 5s as well as 10s. • Uses part/whole model for problem solving.
What Does Matter • Knowing that learning math depends upon hard work and good instruction, not genes or talent. • Having teachers who understand and like mathematics. • Teaching for understanding. • Supporting children who fall behind.
Innovative Math • Teach for understanding, not rote. • Minimize counting; group in fives and tens. • Practice facts with games; avoid flash cards. • Use part/whole circles. • Use math way of number naming initially. • Teach visualizable strategies. • Teach algorithms with four-digit numbers.
Time Needed to Memorize According to a study with college students, it took them: • 93 minutes to learn 200 nonsense syllables. • 24 minutes to learn 200 words of prose. • 10 minutes to learn 200 words of poetry. This shows the importance of meaning before memorizing.
Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp
Flash Cards • Often used to teach rote. • Liked only by are those who don’t need them. • Give the false impression that math isn’t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols.
Rigorous Mathematics • To develop deep understanding. • To justify reasoning. • To connect ideas to prior knowledge. • To explore concepts.
Because we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth Adding by CountingFrom a Child’s Perspective
A B C D E F A B C D E Adding by CountingFrom a Child’s Perspective F + E
A B C D E F A B C D E Adding by CountingFrom a Child’s Perspective F + E What is the sum? (It must be a letter.)
Adding by CountingFrom a Child’s Perspective F + E K A B C D E F G H I J K
H + F E + I G + D D + C C + G Adding by CountingFrom a Child’s Perspective Now memorize the facts!!
Place ValueFrom a Child’s Perspective L is written AB because it is A J and B A’s huh?
Place ValueFrom a Child’s Perspective (twelve) L is written AB because it is A J and B A’s (12) (one 10) (two 1s). huh?
Subtracting by Counting BackFrom a Child’s Perspective H – E Try subtracting by ‘taking away’
Skip CountingFrom a Child’s Perspective Try skip counting by B’s to T: B, D, . . . T.
Calendars A calendar is NOT a number line: day 4 does not include days 1 to 4.
Calendars September 1 2 3 4 5 6 7 8 9 10 Always show the whole calendar. A child wants to see the whole before the parts. Children also need to learn to plan ahead.
first, second, third, fourth Calendars
Poor concept of quantity. • Ignores place value. • Very error prone. • Inefficient and time-consuming. • Hard habit to break for the facts. Counting Model Drawbacks
5-Month Old Babies CanAdd and Subtract up to 3 Show the baby two teddy bears. Then hide them with a screen. Show the baby a third teddy bear and put it behind the screen.
5-Month Old Babies CanAdd and Subtract up to 3 Raise screen. Baby seeing 3 won’t look long because it is expected.
5-Month Old Babies CanAdd and Subtract up to 3 A baby seeing 1 teddy bear will look much longer, because it’s unexpected.
Recognizing 5 5 has a middle; 4 does not. Look at your hand; your middle finger is longer as a reminder 5 has a middle.
Ready: How Many? Which is easier?
Visualizing 8 Try to visualize 8 apples without grouping.
Visualizing 8 Next try to visualize 5 as red and 3 as green.
Grouping by 5s I II III IIII V VIII 1 2 3 4 5 8 Early Roman numerals Romans grouped in fives. Notice 8 is 5 and 3.
Grouping by 5s : Who could read the music? Music needs 10 lines, two groups of five.
Representative of structure of numbers. • Easily manipulated by children. • Imaginable mentally. Materials for Visualizing Japanese Council of Mathematics Education Japanese criteria.
Materials for Visualizing “In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (Montessori Elementary Teacher)
The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives. Ginsberg and others Manipulatives
Visualizing Needed in: • Mathematics • Botany • Geography • Engineering • Construction • Spelling • Architecture • Astronomy • Archeology • Chemistry • Physics • Surgery
Manipulatives A manipulative must not only be visual, but also visualizable. Can you visualize this rod? Most countries stopped using these by early 1990s.
Colored Rod Drawbacks • Young children think each rod is “one.” • Adding rods doesn’t instantly give the sum; still need to count or compare.
Manipulatives The 4-rod plus the 2-rod does not give the immediate answer. You must count or compare.
Colored Rod Drawbacks • Young children often think each rod is “one.” • Adding rods doesn’t instantly give the sum; still need to count or compare. • 8% of children have a color-deficiency; they cannot see 10 distinct colors. • Many small pieces hard to manage.
Quantities With Fingers Use left hand for 1-5 because we read from left to right.
Quantities With Fingers Always show 7 as 5 and 2, not for example, as 4 and 3.
Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. –Joan A. Cotter Also set to music.