Things students learn we didn’t realize they were learning.
What do students learn when they are taught only one way to subtract? In this discussion between two boys who are playing a trading card game and need to subtract 347 from 6000, Linda Levi muses on the things one boy learned and some opportunities involving the development of an algebraic understanding of number that were missed:
I had never thought carefully about exactly what we tell children when teaching the standard algorithm until I heard this discussion. When we teach the standard subtraction algorithm, we tell children to start with the ones. Of course, you don’t have to start with the ones when you subtract. For example, in a problem such as 5,000 – 3,002, it makes much more sense to first subtract 5,000 – 3,000 (the thousands) and then subtract 2 (the ones). We want children to know that 5,000 – 3,000 – 2 is the same as 5,000 – 2 – 3,000. Children with a strong understanding of subtraction know that they don’t need to start with the ones. Understanding how subtraction works will help students solve algebraic equations such as 3x – 48 – x = 49 or 3x – 48 = 49 – x….
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Civil rights.
Birmingham News has just published photos from the civil rights era that were considered too inflamatory to publish when they were taken. I was struck, in particular, by the expressions on the faces of the two young black women as they entered an all-white school for the first time and by the face of the white woman waving a confederate flag.
A segrated school in Birmingham. April 4, 1961.
Integration of a high school in Birmingham. September 1963.
If you are white, would you have been among these people protesting the integration of an elementary school? Would you have been this woman? She looks so sure of herself.
Would you have protested the integration of your high school, as these high school students did? How do you know what you would have done?
What is the civil rights issue of our day?
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Should you show students how to solve problems?
For many people, the answer is “obviously, yes.” But when and how? Research on young children’s mathematical thinking has shown that children can invent strategies to solve problems that are posed within their zone of understanding. Asking children to invent strategies aids the growth of understanding; how a child solves a problems can tell you a lot about what the child understands. Yet certain tools for representing and supporting understanding, such as numerals or number lines, must be shown to children, because they are conventions or because inventing them from scratch is inefficient. But there is a large gray area here where situational variables and teachers’ beliefs and experiences come into play and decisions are not clear cut.
How does a teacher decide when to show and when to not show students how to solve problems? I asked two veteran teachers when, if ever, they show young children how to solve problems. They have been involved in Cognitively Guided Instruction (CGI) for over 15 years each. Both now work as math specialists for the district. Before that, they were classroom teachers. Carrie Valentine taught upper elementary and Mazie Jenkins (sometimes known as Ms. J in CGI writings) taught primary. Both have worked a lot with kids from all kinds of backgrounds, including low income and low achievers in math. It’s certainly a question worth thinking hard about.
Here’s what they had to say:
MJ: The famous question always gets asked. The answer and how you go about getting students to exhibit different strategies is very complex.
Teachers have to learn how to get students to reflect on the strategies that are shared. For example, how is Susan’s strategy like Megan’s strategy? How are they alike and how are they different? Who else solved their problem like Susan’s? Who solved their problem like Megan? Who has a completely different strategy? How is it different than Susan’s? Ask specific questions for the strategy, how are tens used in this strategy? Was this a good way? How do you know? Have we seen this strategy before? When? What kind of problem did we see this strategy used? Did anyone use numbers to solve this problem?
I usually do not show a strategy. I might talk about how I have seen another student solve a similiar problem. (Deborah Ball demonstrates this on a videotape).
I listen and observe to find students who are solving a problem in a different way and build upon that. I ask students to solve problems in at least two different ways (direct modelers show two ways of direct modeling – but they do not know this)
Teachers need to have a good understanding of the development of strategies to know what is developmentally appropriate.
CV: I think it’s possible but requires a great deal of expertise. Skill in problem posing and questioning is essential. I wonder what is meant by strategies? I think of them from a cognitive perspective. Is there something going on internally different in the math understanding? But, I think most people think about the representations that their students use. In that case I definitely see utility in showing ways to represent to hasten the learning. Kids love ‘tools’ such as the empty numberline, arrow language, and ratio tables when they are ready. I would introduce them after some level of understanding emerges. The empty numberline after counting on emerges, arrow language after decomposition and facts of ten are known, and ratio table after kids can double using base ten. There are other ‘tools’ but these seem to capture kids’ attention and are efficient ways to record their thoughts and later to help them solve problems.
You get the gist. It’s actually a complicated question and deserves a nuanced response.
SE: What about upper grades?
CV: Same way. First what is meant by a strategy vs. a tool if there is a difference. Then talk about ways to record thinking.
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Teacher to principal: This prepackaged curriculum doesn’t know my students as well as I do.
I know of many districts who have responded to the increased pressures of high-stakes standardized testing by standardizing the curriculum, in some places, right down to the page number a teacher should be on for any given day. Almost all of the teachers I know realize that this kind of approach goes against what they know about how children learn, but few have responded as dramatically as Ms. S.
Ms. S teaches fifth grade in a small district just outside of a large college town, not unlike Austin, Texas. A couple of years ago, her district adopted Everyday Mathematics, a program developed by the University of Chicago School Mathematics Project and lauded by many. Now, Ms. S has been teaching for many years. Her tried and true approach — whether it’s first grade or fifth grade — is to pose problems that students can solve using a variety of strategies, then help her students to express the concepts behind the strategies they use. She doesn’t show kids how to solve problems; she writes problems in such a way that her students can use what they know to construct their own strategies. In fact, she insists when her students come to her using procedures they don’t understand that they not use those procedures, for the time being. Her highest priority as a teacher is to understand her students’ thinking and to build on it.
As it happened, the year her district adopted Everyday Math, her principal decided to place a large number of students in Ms. S’s class who were considered struggling.
When her district delivered the boxes and boxes of Everyday Mathematics materials to her room, Ms. S took a look at the curriculum. She realized that she could write problems that were better tailored to her students’ needs and understanding and also met district curriculum objectives. So she pushed the boxes aside, closed her door, and began teaching.
Every once in a while, her principal would inquire about what page she was on. Ms. S would tell him, truthfully, “We’re learning a lot.”
All the students in Ms. S’s district take a standardized test at the beginning and end of the year. Students are expected to gain 7-9 points on this test. 10 points is considered quite a respectable gain. At the end of the year, about a week after Ms. S’s students had taken the test, she was surprised to find the principal knocking loudly on her classroom door. When she answered, he told her, with a great deal of excitement, that out of all the fifth-grade classes in the district (about 15 sections), hers was the ONLY class where every single student had met the passing standard for the test. Moreover, he continued, many of these students had gained an astounding 30 or so points between the beginning and the end of the year tests. He asked her, almost rhetorically, “You don’t even use Everyday Math, do you?” “Nope,” she answered. And then she took the opportunity to suggest that perhaps all the money that had gone into the Everyday Math materials may have been better used to support teachers to take the time to learn about their children’s mathematical thinking. Richard Elmore, writing in Harvard Magazine, agrees: “You can’t improve a school’s perfomance, or that of any teacher or student in it, without increasing the investment in teachers’ knowledge, pedagogical skills, and understanding of students” (p. 37).
This year, for the first time, Ms. S has a preponderance of students from the high end of the achievement spectrum. They are challenging in a different way, she says, because they know how to execute procedures such as subtraction with regrouping and long division very well, but often have little understanding of why the procedures work, and almost no flexibility in their choice of strategies. Even with this group, she says, she has to begin the year by posing many types of problems with smaller numbers than one would normally expect at fifth grade in order to help her students develop some flexibility and variety in the strategies they use. It’s worth it; they surpass expectations in the end and their understanding is much, much deeper.
Thank you, Ms. S, for reminding us what teaching is all about.
REFERENCE: Elmore, Richard. (2002). Testing Trap. Harvard Magazine. October, 35-37+.
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