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 her students couldn’t memorize their multiplication facts – and not for lack of trying, on their part or hers. But she was determined that they would succeed. So every day she gave her students timed tests on multiplication, because she knew they needed lots of practice. And every day certain students failed to finish on time – so they didn’t get to practice some of what they needed the most. And every day certain students missed the same problems that they had missed the day before. It seemed that they were practicing the wrong answers over and over. And every day certain students left the same problems blank that they had left blank the day before. They still didn’t 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 weren’t all of the students making the progress she expected?
One evening she went out to her corn field and noticed that the corn was not growing very well – but she didn’t have time to do anything about it right then 4th grade go math. 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 same a couple of mornings later. That evening while correcting math papers, she stopped to think about her corn field, and determined that she would have to get out there and do something about it the next weekend – perhaps start irrigation, or apply a different fertilizer, or…. It suddenly struck her that her corn was a lot like some of her 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 her 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 her students’ natural ability to think, remember, and perform mathematically. She felt that giving up on daily measurement would not improve the situation for those children, but continuing it would not improve it either. 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 develop fluent memory for many of her 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 them, the children certainly would not know which French words corresponded with the English words they knew. They had fluent recall of the words, but could not use them to develop their understanding of the French language. Come to think of it, her little daughter had fluent recall of most of the words to the Pledge to the Flag-which is in English-but she didn’t know what most of those words meant, 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 when singing 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 her math students, who were quite successful on the timed memory tests, but didn’t seem to connect their memorized facts to the concepts that they understood, or to the story problems they perpetually struggled with.