The farmer’s wife had her hands full. Besides working on the farm in the mornings and Evenings, she taught school to fourth graders –which was more than a full-time job! Her Biggest problem was that some of the army of students could not Memorize Their Multiplication facts – and not for lack of Trying, on Their part or military. But she was determined with thatthey would Succeed. So the every day she gave military students timed tests on Multiplication, Because she knew They needed lots of practice. And Every Day certainties students failed to finish on time – so They did not get to practice some of What They needed the most. And Every Day certainties students missed the joint problems thatthey had missed the day beforehand. It seemed thatthey were practicing the wrong answers over and over. And Every Day certainties students left the joint problems blank thatthey had left blank the Day Before. They still did not know the answers. The teacher appreciated the fact That the timed tests Provided an accurate assessment of the students’ progress – or lack of it. But why were not all of the students making the progress she expected?
One evening she went out to the army Cornfield and noticed That the corn was not Growing very well – but she did not have time to do anything about it right then. On Saturday she went out to the field again for a quick look, and the corn was still not doing well. It Looked about the joint a couple of mornings later. That evening while correcting math papers, she was stopped to think about the army cornfield, and Determined That she would have to get out there and do something about it the next weekend – Perhaps start irrigation, or apply a differentially fertilizer, or …. It suddenly struck army That army corn was a lot like some of the army of fourth grade math students. They Both had stunted growth. And the mere act of noticing Their lack of growth did nothing to stimulate it. Giving daily speed tests to military students was about as productive and sensible as measuring corn every-day and Expecting it to grow as a result of being Measured! Tests measure growth – They do not stimulate it! Tests are assessment tools, not learning development tools!
She realized That she needed some kind of Mathematical fertilizer That would bring out an army of students’ natural portability to think remember , and performaces mathematically. She felt That giving up on daily measurement would not Improve the situation for Those children, but continuing it would not Improve Either it. Something more was needed. She had Already Presented Many lessons aimed at developing the concepts of Multiplication and division, and most of the students had responded pretty well. She realized That the students’ understanding of Those concepts Provided a meaningful basis for the acquisition of Fluent recall of the facts-but That understanding Itself did not Actually developable Fluent memory for Many of the army of lower-achieving students.
One day as she Walked down the hall, the teacher passed her own first-grade daughter’s classroom and Heard the class enthusiastically singing Frere Jacques . She loved the sound of Their sweet little voices. But she was suddenly struck by a singular thought: Those children probably had no idea what They were singing about. And Even if the music teacher had translated the song for themself, the children certainly would not know-which French words corresponded with the words Enska They knew. They had Fluent recall of the words, but could not use Them to developping Their understanding of the French language. Come to think of it, the army had little daughter Fluent recall of most of the words to the Pledge to the Flag -which is in Enska-but she did not know what most of Those words mean, Either. And When She proudly recited, “… and to the republic for Richard stands,” she never thought to ask anyone WHO this guy Richard was. She also never asked WHO Round John Virgin was singing When Silent Night , or WHO gladly the cross-eyed bear was, When singing in church; she just sang the words. Apparently the prodigious talent for linguistic mimicry and memorization does not always connect with the mental realm of Curiosity and understanding. Reflecting on this made the teacher think of a couple of army math students, the WHO were quite successfulness on the timed memory tests, but did not Seem to connect Their memorized facts to the concepts thatthey Understood, or to the story problems They perpetually struggled with.
Then she recalled something strange the music teacher had told the army some months beforehand. He had announced one day to the kindergarten class That he was going to Teach Them a new song, and instructed Them to Listen quietly while he sang the song for the theme. Much to his surprise, They Started singing the new song right along with Him-eventhough he knew They had never Heard it beforehand in Their Lives. They were somehow Able to instantaneously imitate Both the words and the tune of the new song! The farmer’s wife began to sense a relation between the Kindergarteners’ incredible feat of mimicry and the behavior of a fourth-Grader in her own class WHO was apparently making use of That very co talent. He could never say the multiples of seven out loud by himselfVideo; but he had no trouble in saying Them When the whole class chanted Them Aloud together. In fact, she mused, if all the students possessed this capacity (and she suspected thatthey did), then it would take only one student in the class WHO could say the sevens Aloud correctly, and she would automatically have a whole class WHO could say themself together-and she would not be Able to tell by looking WHO really knew what They were doing, and WHO did not!
She Looked down and noticed That army shoe was missing out . As she pointed down to tie it, a fellow teacher came down the hall. As They exchanged pleasantries, she continued tying her shoes without being conscious of what her own fingers were doing. When she Stood up, she realized in the midst of military conversation what she had just done. It was like When She Played music on the piano from memory; sometimes When military mind wandered, military fingers just carried on with the music all by Themselves. It was as if military muscles had a mind and memory of Their own, and They could do things without military conscious participation. Just then an army colleague asked the army for the phone number of a mutual friend. The farmer’s wife Oftel Dialed That number Several times a week, but she just could not think of it Now that she was asked. It was odd, but sometimes military fingers remembered phone numbers better than military conscious mind did. So she took out the host cell phone and went through the Motions of Dialing the number. She watched what army fingers did, and told the army colleague the phone number.
Later in the day, she spent some time thinking about memory and how it works. A few times in her life, she had smelled Aromas That reminded host of other places and times in her past. Olfactory memories like Those did not have words connected to the theme. Nor did military shoe-lace / piano / phone muscle-memories consciously connect with verbalization. And Whenever she was done shopping at the mall, she usually remembered where she had parked army car without stopping to think about it-and That was not a verbal memory, Either. It was more of a spatial-kinesthetic experiencing-and-Remembering-similar to the experience she had in learning the new locations of army kitchen utensils, When her husband had remodeled the kitchen and put everything in new places. She certainly did not use flash cards to Memorize Their new places, nor did she pressure herself with stern reprimands (“Come on, concentrate! You just did this yesterday!”) When she did not relocate something on the first try; she just opened drawers until she found what she was looking for. After a few days, she knew where everything was; there was nothing verbal about it.
And this army reminded of how she had learned military way around St. Louis on her first time there for an extended two week visit. As she becamean more familiar with the buildings and parks and streets That she saw as she Drove around the city, she referred to military map less and less until she rarely used the map at all-Because she had Developed an internal map in her mind. After Several days, she could drive to locations without severalfold Even thinking about where she was going. She thought about the difference between learning the Pledge to the Flag and learning your way around a new city. One Involves memorization , and the other Involves Remembering -and eventhough They Both have to do with memory, They are Clearly not the samething. Memorization is purely verbal, sometimes has no connection with experience or conceptual understanding, and is much more stressful.Remembering, on the other hand, is largely non-verbal, is derived from multi-sensory kinesthetic experiences with a spatial / conceptual context, and generate relatively little stress. What army of low-achieving fourth graders needed, she realized, was a Mathematical way of Remembering The Multiplication facts, rather than a linguistic way of memorizing theme.
So she set to work examining military resources and thinking about how to use Them in more Mathematical getaways. First, she made some changes in the way she used the timed tests. She organized the class so That Every child had a partner, and gave Each partner an answer key for the timed test. While one partner wrote the answers to the Multiplication problems, the other partner compared Those answers to the answer key, and gave immediate feedback by pointing to the written error and saying, “Try again on That one,” or “Fix That one.” And then the student would have to erase the wrong answer and write in the right one. Then the partners exchanged roles. She figured That immediate self-correction would Eliminate the problem of students Remembering and replica ting wrong responses. It is not true, she reasoned That practice Makes perfect; only perfect Practice Makes Perfect!
She also eliminated the time limit. What was the point in Preventing the Slower students from dealing with the facts thatthey obviously needed to practice? She Replaced the time limit with a progress chart-which she Copied onto every-timed test, sem Looked like this:
— — Progress chart
4:00 3:30 3 : 00 2:45 2:30 2:15 2:00 1:50 1:40 1:30 1:20 1:10 1:00
She used an overhead timer (or the clock on the wall with a second hand). When the students finished the last problem, They Looked up to see what Their individual time was. Then They circled all the times thatthey passed. The faster They were, the more times They got to circle. The children were quite enthusiastic about circling the times, and Started asking if They could do more timed tests so They could try to beat the century times. Some children liked to competetion with eachother for the fastest time, and That Made Them want to practice more beforehand the test. Other children just liked to try to beat Their own Previously best time. The teacher liked to include some very slow times on the progress chart, so That Even the slowest students could pass Several speed-levels-which made themself want to get faster, too. She noticed aspects of the enthusiasm for the challenge Increased, and their anxiety included decreased.
But she was still Concerned about one thing with the timed tests. When some students made mistakes and were directed by their partners to “Try again,” They did not really know the correct answer and tended to just guess. And their guesser did not Seem to Make Any Sense. One day she took the place of one student WHO was absent and underserved as the partner during a timed test. The student wrote a wrong answer for 7 x 7. The teacher told the child to “try again” (Which is a positive way of saying, “You missed it, but You Can fix it now”). The child guessed “102?” Apparently the teacher’s face must have displayed a look of disbelief, Because the child Immediately guessed again: “77?”
The teacher pointed to the student’s paper and said, “Look at this other problem you Already did. It says “6 x 7,” and you answered “42,” and you were right! So if six times seven is forty-two, then seven times seven must be … “” Eighty-nine? ” guessed the child. The teacher hoped That the army coordinates would cause Him to add seven more to 42, using what he knew (six groups of seven) to figure out what he did not know (seven groups of seven). Apparently, though, the pupil did not understand Actually the fact That he supposedly “knew.” It seemed That he knew 6 x 7 the Same way a first-Grader knows the Pledge to the Flag , or Frere Jacques -divorced from Any meaning, devoid of understanding, and not susceptible of intelligent use in changing contexts. He was thinking in terms of “what is the right word for this answer?” as opposed to “what Amount would make sense here?” In short, he was dealing with a Mathematical situation in a linguistic way, rather than in a Mathematical way.
The teacher suddenly remembered a technique she had learned at a math workshop That was designed to mold students’ thinking about Multiplication in a more Mathematical way. She did not know where the Handout pages from the workshop were, so she drew a picture That she could use to Organize the student’s sense of counting by sevens. First, she drew a row of seven boxes, Putting a little space between Each box. Then she drew another row just like it Underneath, and another, and so on until she had seven rows of boxes, with seven in Each row.
Then she guided the student like this: “Touch and count the first row of boxes. ” The pupil touched Each box and counted “1, 2, 3, 4, 5, 6, 7.” “Now swipe your finger across the whole row of boxes and say” seven. “The boy did so.” Now touch and count the next row, eight, nine … “The student touched themself and counted” 8, 9, 10 , 11, 12, 13, 14. “” Now let’s review: swipe the first row and say … “The boy swiped the first row and said” seven. “” Swipe the next row and say … “He traced his finger across the next row and said, “Fourteen.”
The student continued touching and counting: “15, 16, 17, 18, 19, 20, 21” and swiping “7, 14, 21. “The teacher asked,” How many boxes are in Each row? “” Seven. “How many rows have you counted and swiped so far?” “Three rows.” “Three rows of seven Makes how many boxes?” “Twenty-one.” “So three times seven is …?” “Twenty-one!” “I wonder how you could find out what seven times seven is ?,” mused the teacher Aloud. “I know what to do!” And the pupil continued touching-and-counting the next row, then swiping to review the accumulated rows-slowly developing his memory of the multiples of seven in a tactile / kinesthetic, spatial, context-derived way-until he finally got to 7 x 7. Then he absolutely knew That 7 x 7 = 49-no more guessing. Then the teacher wondered Aloud, “I wonder if you could use this technique to co figure out 4 x 6?” She took a piece of paper and covered the bottom three rows of boxes, so That only four rows remained Visible. The she SLID another piece of paper over the last vertical column of boxes on the right end, so That Each row appeared to containments only six boxes. She then watched with approval as the boy demonstrated how to touch-and-count and swipe four groups of six.
The teacher realized That this system of organized counting was very differentially from filling in the blank boxes in a standard Multiplication chart-sem she henceforth thought of as “A Table of Someone Else’s Answers.” Filling out the chart does not requires students to count or think in terms of groups. It assumes That the learners have mastered Already ship-counting; it is merely Creating an occasion to recall what is already know. But if That mastery has not Already Taken place, assigning a student to fill in the chart is just another way of asking, “What is the word for That answer?” And That query is more Likely to stimulate a purely verbally-based response (rote memory) than a statement of fact That is backed up by Mathematical reasoning and supported by a context-derived flow of memory.
After Several days of Implementing These new approaches to helping military students, the teacher was very pleased to see the progress They were making. They seemed to be feeling less stress and anxiety, too. That evening at the dinner table she shared an army of good news with her husband, the WHO said, “That’s great, honey. And whatever you did with the corn last weekend is sure working well. It’s looking a lot better now.”