What are the main areas of difficulty and misconceptions that your students have with fractions? What do you think are the promising approaches to address these areas in terms of prior math fundamentals and the core concepts of fractions?
We spent a lot of time last year talking about fractions as numbers when students were first introduced to fractions. We did similar things with fractions with our 3rd and 4th grade students that we do with kindergarten students when they are learning about whole numbers. We counted by fractions- 1/5, 2/5, 3/5, 4/5, 5/5, 6/5... We built fractions using Cuisenaire Rods. Asking- If this rod represents one whole, which rod would be 1/4? We composed and decomposed fractions using part-part-whole maps. This was a big change from our old way of teaching fractions. We found that our student had a much better understanding of fraction after this type of instruction.
Have you ever tried having the students create fraction strips? I did this when I taught 6th graders using a program called "Algebraic Thinking". But it can be done with younger students; probably just using 1 whole, halves, fourths, and fifths. With the 6th graders we did out to twelfths, which was difficult for many of them to build "evenly". We had the students put their initials on the back of theirs and then partner with friends when we would do improper fractions or mixed numbers.
Children need many opportunities to represent fractions - initially through partitioning or sharing and then within varied shaped regions and as collections of objects AND on a number line. Far too frequently children move to work with operations with fractions without experiences representing and comparing and ordering fractions to the point where they are quite comfortable with equivalence and comparing and ordering. We do analogous pre-work with place value as a precursor to work with addition and subtraction of whole numbers, similar grounding is needed with fractions.
I totally agree. I can still remember my first experience with using manipulatives for multiplication and division of fractions and that "Aha!" moment. I have seen similar responses from my peers in staff developments. I definitely think students get more concrete examples and are at least exposed to these types of models (more now, than when I was in school), but they still need more time to master the early models and become comfortable in their understanding of what a fraction means.
Effective Fractions Instruction for K-8 (Doing What Works)
Effective Fractions Instruction for K-8 (What Works Clearinghouse)
The 2nd resource, "Effective Fractions Instruction for K-8" is already in our catalog. The first resource I don't believe is cataloged yet, but I will look into it. As always welcomed to suggestions!
I think it is essential to give students opportunities to build understanding and practice understanding with concrete models before moving to algorithms. Too often students are just told that this is the process and just do it. Students need to know why the algorthm works, possibly even discovering it on their own, in order to remember it and apply it. The problem with this approach is that it requires time and practice, both of which are in short supply in our classrooms.
The goal should be to effect math understandings and abilities in more students. We have them write, draw, and recite to address their aural and visual modes of learning. Providing manipulatives and tools takes advantage of the kinesthetic and possibly other modes. Things to consider:
Do capable students who just learn the rules burn out of advanced STEM, etc. courses and programs due to a lack of deep appreciation and understanding?
Will students, who don’t get the one method presented, invent their own, possibly illogical algorithm? Doesn’t this justify providing them mathematically sound alternative ways.
Are reform-type programs taking enough care to wean capable students off the concrete and onto the abstract understandings necessary for advanced work?
I totally agree with you. Concrete models allow students to deepen conceptual understanding and I believe that if teachers devote the extra time allowing students to explore, less time will be spent on re-teaching. Just a thought!
In reading Steve’s prompt and the Ann Arbor Workshop summary (cataloged at this site), several questions come to mind. Here are two.
Why do we cover the fraction operations in the same order as we do whole numbers? This gets a brief mention by one of the participants, but not much response. Why not teach multiplication ahead of addition? Consider:
Students know from earlier work that multiplication is repeated addition.
Students need to use multiplication to get equivalent fractions in order to add.
Students quickly forget the adding algorithm and adopt a ‘multiplication’ imitation after they see the easier algorithm. Could changing the order of presentation help?
Second question. Should we be emphasizing more than we do the fraction connection with division? Why are fractions treated as a brand new thing?
We form remainder fractions before we get to fractions – 37 brownies / 8 is 4 brownies for each and 5 more divided 8 ways, if someone has a cleaver.
How are two individual pizzas shared by three people? Is taking one-third of each the same as two people taking 2/3 of one and the third person getting 1/3 of each one? Liping Ma and others report U.S. teachers really wrestling with questions like this. It seems the answer is 2/3 of a pizza no matter how you slice it, but if you draw 2 circles and shade in one-third of each one, will people accept that as a model for 2/3? Even after emphasizing what the ‘unit’ is, I’ve met with resistance.
According to several recent articles by H. H. Wu, a mathematician at Cal-Berkeley who has long been weighing in on issues of K-12 mathematics education and who served on the recent presidential mathematics panel, there is only one correct way to represent and teach fractions and their arithmetic, and that is as points on the number line. He takes issue in these articles with a number of claims Steve makes in his piece here. I was wondering if others were familiar with Wu's point of view on fractions and could offer comments, particularly in light of what Steve has offered and the work of such well-known elementary math ed authors as the late John van de Walle.
Mathlanding has cataloged an article "The Ann Arbor Workshop on Fractions - Summary by Alan Tucker" that contains many points of view about foundational issues of how to define and represent fractions.
One of the main areas of difficulty that I have seen with my students is simplifying fractions and finding equivalent fractions. I think one of the best ways to teach this to students is to use manipulatives like fraction tiles (store bought or hand made). If students can build a strong conceptual knowledge of equivalent fractions before moving to short cuts and paper/pencil strategies, then they will both remember the skill and understand the reasonableness of their answers.
I have two other suggestions for helping students bridge from the depicting of the equivalence of fractions to paper/pencil. These are manipulatives that the students make and use until they can proceed with the formal algorithms. Give each student a 4-foot strip of adding machine tape. Have them measure and mark off 3” sections, Assign each a different lowest terms fraction and ask them to generate a set of 12 equivalents, multiplying the assigned fraction by 1/1, 2/2, 3/3, etc. They soon use the shortcut of skip counting all the numerators, then the denominators, but that reinforces the need for multiples for equivalents and for factors when reducing.
Example: on one strip 2/3. 4/6, 6/9, 8/12, 10/15, 12/18, ....
Hold a Fraction party. Give students a sheet of practice exercises, getting equivalents, or adding or subtracting with unlike denominators and have them ‘mix and mingle’ to find other students who have the equivalents needed to complete the exercises.
After the party, make a bulletin board of the strips in top/down order, 1/2 then 1/3, 2/3, 1/4, 3/4, etc. that they can use to complete other exercises in the next few days. Offer a series of short quizzes that wean them away from using the strips. Tell them in advance which strips will be hidden, until they are all hidden.
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