I've been working with a handful of students in my main placement to remediate the way they think and compute fractions. I've come to the conclusion that for these students, it can be very useful to use manipulatives, drawings, and alternative language to foster an understanding about fractions. What I mean by alternative language is that there is often more than one way to say something, and when the language prompt is confusing, then it will be difficult to get at the math concept. Therefore, one must change his or her approach to teaching (revealing) the math/concept!
Unit 5 was all about fractions--as I have already mentioned. Over the past few weeks, students learned to add fractions with common denominators, then they explored improper fractions and mixed numbers, before moving on to studying equivalent fractions. Equivalent fractions proved to be a challenging undertaking for the students with whom I provided additional support. Mostly, I learned that when talking about "like denominators" or the LCD, it was helpful to first ask students about the relationship between the denominators. Often times I would have students "skip count" and compare the number sequences of two fractions having different denominators. This was helpful. Introducing language such as "multiplier" was, at times, confusing for some students, and so I found it helpful to use the language over and over so that they were familiar with what I meant--but I always explained what I meant so that when I used the language, it was not a big deal--it was natural. Students really seemed vague about the idea of the multiplier (3/3, 2/2, 7/7, etc.) They did not understand why you would multiply a fraction by a multiplier such as (3/3) as opposed to just 3. I stressed over, and over, and over again why this was.
I found it a bit alarming or concerning that the reason for multiplying a fraction by a multiplier that was equal to 1 was not as explicitly stated or practiced or investigated. I wonder how students' understanding of how to form equivalent fractions would have been improved if they had investigated, found patterns, and explained their way of thinking to each other. Perhaps taking the time to have a math congress on this topic would have served many students very well, because today, when I graded Unit Tests, I wanted to weep! Some of the students that I had worked so hard with and whom I thought had understood the concepts slid back. One score was as low as 7/20. There were a handful of scores ranging from 9-14.
Now I'm wondering, what do I do to help these students who don't have the basic understandings about fractions? If their foundation is weak, what will this mean as they continue to progress in math--later this year, next year, once they are in junior high and high school and even into college and their professions?
I want them to understand! But I'm also very concerned that they don't and that the PACE doesn't allow for time to slow down and clarify confusion!
What will I do when I'm a teacher and I have an entire classroom of students at varying rates? Meeting with small groups means investing A LOT of time and energy.
I think I'm going to have to find a creative way to introduce the material that meets the learning needs of the students. I think I'll also have to rely on the classsroom community (that might mean students pair up to teach each other and offer their way of thinking to another person).
2 comments:
Jocelyn you're asking really important big questions here. And digging into some real tensions in teaching and learning mathematics. What I hear you expressing is a concern for students having time and opportunities to develop a conceptual understanding of fractions (or portions) and bumping up against the pressures of time. I am curious to hear you talk more about the students experiences when adding fractions with common denominators-- do you know if, previous to this, they had opportunities to explore adding fractions with unlike denominators? The reason I ask about this is... there are some really sensical and typical ways people think about combining portions without having to convert fractions to like denominators. For example, if you eat a half of something and I eat a fourth of something, we can think about that (and kids can think about that) easily without the procedure of changing the numbers.
I especially like your attention to the relationship between denominators. This seems like a powerful terrain to explore for a conceptual understanding of what portion each fraction is representing.
When you say you stressed over and over and over again why you would multiply a fraction by a multiplier, what do you think was particularly confusing about this for students? What did their exploration of this look like? The reason this stands out to me is because of how hard it sounds like you were working and that you were doing the explaining. I always have this saying running through my head, "Who is doing the cognitive work?" As a teacher I could find myself in just this situation you're writing about what I am trying to do and appreciate you're doing is wondering about how students' understanding (in this case of how to form equivalent fractions would have been improved if they had investigated, found patters, and explained their thinking to each other) is just where I think energy should go.
What is your hunch about why students struggled on the assessment?
Allison,
just briefly, and I will try to reflect in greater depth a little later: I hear what you are saying (what you are getting at) when you asked who was doing the majority of the work: the students or the teacher?
Funny how in hindsight or when these things are pointed out to a person, it becomes a bit more obvious! I was the one doing the cognitive work.
We don't do much investigation in our classroom, and students didn't ever have an opportunity to investigate. BUT...that's not to say that when I work in a group next, I'll see how I can structure more investigative practice in small group remedial sessions.
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