| The farmer's wife had her hands full. Besides | | | | without even thinking about where she was going. |
| working on the farm in the mornings and | | | | She thought about the difference between |
| evenings, she taught school to fourth graders - | | | | learning the Pledge to the Flag and learning your |
| which was more than a full-time job! Her biggest | | | | way around a new city. One involves |
| problem was that some of her students couldn't | | | | memorization, and the other involves |
| memorize their multiplication facts - and not for | | | | remembering-and even though they both have to |
| lack of trying, on their part or hers. But she was | | | | do with memory, they are clearly not the same |
| determined that they would succeed. So every | | | | thing. Memorization is purely verbal, sometimes |
| day she gave her students timed tests on | | | | has no connection with experience or conceptual |
| multiplication, because she knew they needed lots | | | | understanding, and is much more |
| of practice. And every day certain students failed | | | | stressful.Remembering, on the other hand, is |
| to finish on time - so they didn't get to practice | | | | largely non-verbal, is derived from multi-sensory |
| some of what they needed the most. And every | | | | kinesthetic experiences with a spatial/conceptual |
| day certain students missed the same problems | | | | context, and generates relatively little stress. |
| that they had missed the day before. It seemed | | | | What her low-achieving fourth graders needed, |
| that they were practicing the wrong answers | | | | she realized, was a mathematical way of |
| over and over. And every day certain students | | | | remembering their multiplication facts, rather than |
| left the same problems blank that they had left | | | | a linguistic way of memorizing them. |
| blank the day before. They still didn't know the | | | | So she set to work examining her resources and |
| answers. The teacher appreciated the fact that | | | | thinking about how to use them in more |
| the timed tests provided an accurate assessment | | | | mathematical ways. First, she made some |
| of the students' progress - or lack of it. But why | | | | changes in the way she used the timed tests. She |
| weren't all of the students making the progress | | | | organized the class so that every child had a |
| she expected? | | | | partner, and gave each partner an answer key |
| One evening she went out to her corn field and | | | | for the timed test. While one partner wrote the |
| noticed that the corn was not growing very well - | | | | answers to the multiplication problems, the other |
| but she didn't have time to do anything about it | | | | partner compared those answers to the answer |
| right then. On Saturday she went out to the field | | | | key, and gave immediate feedback by pointing to |
| again for a quick look, and the corn was still not | | | | the written error and saying, "Try again on that |
| doing well. It looked about the same a couple of | | | | one," or "Fix that one." And then the student |
| mornings later. That evening while correcting math | | | | would have to erase the wrong answer and write |
| papers, she stopped to think about her corn field, | | | | in the right one. Then the partners exchanged |
| and determined that she would have to get out | | | | roles. She figured that immediate self-correction |
| there and do something about it the next | | | | would eliminate the problem of students |
| weekend - perhaps start irrigation, or apply a | | | | remembering and replicating wrong responses. It |
| different fertilizer, or.... It suddenly struck her that | | | | is not true, she reasoned that practice makes |
| her corn was a lot like some of her fourth grade | | | | perfect; only perfect practice makes perfect! |
| math students. They both had stunted growth. | | | | She also eliminated the time limit. What was the |
| And the mere act of noticing their lack of growth | | | | point in preventing the slower students from |
| did nothing to stimulate it. Giving daily speed tests | | | | dealing with the facts that they obviously needed |
| to her students was about as productive and | | | | to practice? She replaced the time limit with a |
| sensible as measuring corn every day and | | | | progress chart which she copied onto every |
| expecting it to grow as a result of being | | | | timed test, which looked like this: |
| measured! Tests measure growth - they do not | | | | --- --- Progress Chart: |
| stimulate it! Tests are assessment tools, not | | | | 4:00 3:30 3:00 2:45 2:30 2:15 2:00 1:50 1:40 1:30 |
| learning development tools! | | | | 1:20 1:10 1:00 |
| She realized that she needed some kind of | | | | She used an overhead timer (or the clock on the |
| mathematical fertilizer that would bring out her | | | | wall with a second hand). When the students |
| students' natural ability to think, remember, and | | | | finished the last problem, they looked up to see |
| perform mathematically. She felt that giving up on | | | | what their individual time was. Then they circled all |
| daily measurement would not improve the | | | | the times that they passed. The faster they |
| situation for those children, but continuing it would | | | | were, the more times they got to circle. The |
| not improve it either. Something more was | | | | children were quite enthusiastic about circling the |
| needed. She had already presented many lessons | | | | times, and started asking if they could do more |
| aimed at developing the concepts of multiplication | | | | timed tests so they could try to beat their old |
| and division, and most of the students had | | | | times. Some children liked to compete with each |
| responded pretty well. She realized that the | | | | other for the fastest time, and that made them |
| students' understanding of those concepts | | | | want to practice more before the test. Other |
| provided a meaningful basis for the acquisition of | | | | children just liked to try to beat their own |
| fluent recall of the facts-but that understanding | | | | previously best time. The teacher liked to include |
| itself did not actually develop fluent memory for | | | | some very slow times on the progress chart, so |
| many of her lower-achieving students. | | | | that even the slowest students could pass |
| One day as she walked down the hall, the teacher | | | | several speed levels-which made them want to |
| passed her own first-grade daughter's classroom | | | | get faster, too. She noticed that their enthusiasm |
| and heard the class enthusiastically singing Frere | | | | for the challenge increased, and their anxiety |
| Jacques. She loved the sound of their sweet little | | | | decreased. |
| voices. But she was suddenly struck by a singular | | | | But she was still concerned about one thing with |
| thought: those children probably had no idea what | | | | the timed tests. When some students made |
| they were singing about. And even if the music | | | | mistakes and were directed by their partners to |
| teacher had translated the song for them, the | | | | "Try again," they didn't really know the correct |
| children certainly would not know which French | | | | answer and tended to just guess. And their |
| words corresponded with the English words they | | | | guesses didn't seem to make any sense. One day |
| knew. They had fluent recall of the words, but | | | | she took the place of one student who was |
| could not use them to develop their understanding | | | | absent and served as the partner during a timed |
| of the French language. Come to think of it, her | | | | test. The student wrote a wrong answer for 7 x |
| little daughter had fluent recall of most of the | | | | 7. The teacher told the child to "try again" (which |
| words to the Pledge to the Flag-which is in | | | | is a positive way of saying, "You missed it, but |
| English-but she didn't know what most of those | | | | you can fix it now"). The child guessed "102?" |
| words meant, either. And when she proudly | | | | Apparently the teacher's face must have |
| recited, "...and to the republic for Richard stands," | | | | displayed a look of disbelief, because the child |
| she never thought to ask anyone who this guy | | | | immediately guessed again: "77?" |
| Richard was. She also never asked who Round | | | | The teacher pointed to the student's paper and |
| John Virgin was when singing Silent Night, or who | | | | said, "Look at this other problem you already did. |
| Gladly the cross-eyed bear was, when singing in | | | | It says "6 x 7," and you answered "42," and you |
| church; she just sang the words. Apparently the | | | | were right! So if six times seven is forty-two, |
| prodigious talent for linguistic mimicry and | | | | then seven times seven must be..." "Eighty-nine?" |
| memorization does not always connect with the | | | | guessed the child. The teacher hoped that her hint |
| mental realm of curiosity and understanding. | | | | would cause him to add seven more to 42, using |
| Reflecting on this made the teacher think of a | | | | what he knew (six groups of seven) to figure out |
| couple of her math students, who were quite | | | | what he didn't know (seven groups of seven). |
| successful on the timed memory tests, but didn't | | | | Apparently, though, the pupil didn't actually |
| seem to connect their memorized facts to the | | | | understand the fact that he supposedly "knew." It |
| concepts that they understood, or to the story | | | | seemed that he knew 6 x 7 the same way a |
| problems they perpetually struggled with. | | | | first-grader knows the Pledge to the Flag, or |
| Then she recalled something strange the music | | | | Frere Jacques-divorced from any meaning, devoid |
| teacher had told her some months before. He had | | | | of understanding, and not susceptible of intelligent |
| announced one day to the kindergarten class that | | | | use in changing contexts. He was thinking in terms |
| he was going to teach them a new song, and | | | | of "what is the right word for this answer?" as |
| instructed them to listen quietly while he sang the | | | | opposed to "what amount would make sense |
| song for them. Much to his surprise, they started | | | | here?" In short, he was dealing with a |
| singing the new song right along with him-even | | | | mathematical situation in a linguistic way, rather |
| though he knew they had never heard it before | | | | than in a mathematical way. |
| in their lives. Somehow they were able to | | | | The teacher suddenly remembered a technique |
| instantaneously imitate both the words and the | | | | she had learned at a math workshop that was |
| tune of the new song! The farmer's wife began | | | | designed to mold students' thinking about |
| to sense a relation between the kindergarteners' | | | | multiplication in a more mathematical way. She |
| incredible feat of mimicry and the behavior of a | | | | didn't know where the handout pages from the |
| fourth-grader in her own class who was | | | | workshop were, so she drew a picture that she |
| apparently making use of that very same talent. | | | | could use to organize the student's sense of |
| He could never say the multiples of seven out | | | | counting by sevens. First, she drew a row of |
| loud by himself; but he had no trouble in saying | | | | seven boxes, putting a little space between each |
| them when the whole class chanted them aloud | | | | box. Then she drew another row just like it |
| together. In fact, she mused, if all the students | | | | underneath, and another, and so on until she had |
| possessed this capacity (and she suspected that | | | | seven rows of boxes, with seven in each row. |
| they did), then it would take only one student in | | | | Then she guided the student like this: "Touch and |
| the class who could say the sevens aloud | | | | count the first row of boxes." The pupil touched |
| correctly, and she would automatically have a | | | | each box and counted "1, 2, 3, 4, 5, 6, 7." "Now |
| whole class who could say them together-and she | | | | swipe your finger across the whole row of boxes |
| would not be able to tell by looking who really | | | | and say "seven." The boy did so. "Now touch and |
| knew what they were doing, and who didn't! | | | | count the next row: eight, nine..." The student |
| She looked down and noticed that her shoe was | | | | touched them and counted "8, 9, 10, 11, 12, 13, |
| untied. As she bent down to tie it, a fellow | | | | 14." "Now let's review: swipe the first row and |
| teacher came down the hall. As they exchanged | | | | say..." The boy swiped the first row and said |
| pleasantries, she continued tying her shoe without | | | | "seven." "Swipe the next row and say..." He traced |
| being conscious of what her own fingers were | | | | his finger across the next row and said "fourteen." |
| doing. When she stood up, she realized in the | | | | The student continued touching and counting: "15, |
| midst of her conversation what she had just | | | | 16, 17, 18, 19, 20, 21" and swiping: "7, 14, 21." The |
| done. It was like when she played music on the | | | | teacher asked, "How many boxes are in each |
| piano from memory; sometimes when her mind | | | | row?" "Seven." How many rows have you |
| wandered, her fingers just carried on with the | | | | counted and swiped so far?" "Three rows." "Three |
| music all by themselves. It was as if her muscles | | | | rows of seven makes how many boxes?" |
| had a mind and memory of their own, and they | | | | "Twenty-one." "So three times seven is...?" |
| could do things without her conscious participation. | | | | "Twenty-one!" "I wonder how you could find out |
| Just then her colleague asked her for the phone | | | | what seven times seven is?," mused the teacher |
| number of a mutual friend. The farmer's wife | | | | aloud. "I know what to do!" And the pupil |
| often dialed that number several times a week, | | | | continued touching-and-counting the next row, |
| but she just could not think of it now that she | | | | then swiping to review the accumulated |
| was asked. It was odd, but sometimes her | | | | rows-slowly developing his memory of the |
| fingers remembered phone numbers better than | | | | multiples of seven in a tactile/kinesthetic, spatial, |
| her conscious mind did. So she took out her cell | | | | context-derived way-until he finally got to 7 x 7. |
| phone and went through the motions of dialing the | | | | Then he absolutely knew that 7 x 7 = 49-no |
| number. She watched what her fingers did, and | | | | more guessing. Then the teacher wondered aloud, |
| told her colleague the phone number. | | | | "I wonder if you could use this same technique to |
| Later in the day, she spent some time thinking | | | | figure out 4 x 6?" She took a piece of paper and |
| about memory and how it works. A few times in | | | | covered the bottom three rows of boxes, so |
| her life, she had smelled aromas that reminded | | | | that only four rows remained visible. The she slid |
| her of other places and times in her past. | | | | another piece of paper over the last vertical |
| Olfactory memories like those did not have words | | | | column of boxes on the right end, so that each |
| connected to them. Nor did her shoe-lace/piano | | | | row appeared to contain only six boxes. She then |
| phone muscle-memories consciously connect with | | | | watched with approval as the boy demonstrated |
| verbalization. And whenever she was done | | | | how to touch-and-count and swipe four groups of |
| shopping at the mall, she usually remembered | | | | six. |
| where she had parked her car without stopping | | | | The teacher realized that this system of |
| to think about it-and that was not a verbal | | | | organized counting was very different from filling |
| memory, either. It was more of a | | | | in the blank boxes in a standard multiplication |
| spatial-kinesthetic | | | | chart-which she henceforth thought of as "A |
| experiencing-and-remembering-similar to the | | | | Table of Someone Else's Answers." Filling out the |
| experience she had in learning the new locations | | | | chart does not require students to count or think |
| of her kitchen utensils, when her husband had | | | | in terms of groups. It assumes that the learners |
| remodeled the kitchen and put everything in new | | | | have already mastered skip-counting; it is merely |
| places. She certainly did not use flash cards to | | | | creating an occasion to recall what is already |
| memorize their new places, nor did she pressure | | | | known. But if that mastery has not already taken |
| herself with stern reprimands ("Come on, | | | | place, assigning a student to fill in the chart is just |
| concentrate! You just did this yesterday!") when | | | | another way of asking, "What is the word for |
| she didn't relocate something on the first try; she | | | | that answer?" And that query is more likely to |
| just opened drawers until she found what she | | | | stimulate a purely verbally-based response (rote |
| was looking for. After a few days, she knew | | | | memory) than a statement of fact that is backed |
| where everything was; there was nothing verbal | | | | up by mathematical reasoning and supported by a |
| about it. | | | | context-derived flow of memory. |
| And this reminded her of how she had learned | | | | After several days of implementing these new |
| her way around St. Louis on her first time there | | | | approaches to helping her students, the teacher |
| for an extended two week visit. As she became | | | | was very pleased to see the progress they were |
| more familiar with the buildings and parks and | | | | making. They seemed to be feeling less stress |
| streets that she saw as she drove around the | | | | and anxiety, too. That evening at the dinner table |
| city, she referred to her map less and less until | | | | she shared her good news with her husband, who |
| she rarely used the map at all-because she had | | | | said, "That's great, honey. And whatever you did |
| developed an internal map in her mind. After | | | | with the corn last weekend is sure working well. |
| several days, she could drive to several locations | | | | It's looking a lot better now. |