Ask Matt Mentor
Dear Matt, Many of my students have trouble working with fractions and it is hindering their work in
algebra. How can I get them caught up on fraction concepts and operations without getting
behind in my schedule for the algebra course?
My Algebra I students (mostly 9th and 10th grade) struggle with computation issues (add,
subtract, multiply, divide integers). How can I best help them learn algebra when they seem
to be hung up on computation?
Response: Dear Algebra Teachers: These two related questions are asked frequently by many teachers. The challenge posed in these questions is only heightened by new Minnesota Standards which lay out high expectations for all students in algebra. Algebra teachers do not have the leisure to pause instruction to re-teach topics from intermediate and middle school, despite the fact that some students may not have functional mastery of these topics. So what teachers need are strategies to move forward with algebra while enabling students to avoid being held back by their skill gaps. The first strategy is to put aside the widely held belief that computational fluency is a prerequisite to algebraic thinking. When algebra is approached as a study of patterns, generalizing quantities and representing relationships, good thinking can happen despite lack of computational proficiency. Certainly in practicing procedures, lack of skill can be a handicap. But we have many powerful tools for coping with this. More on that later. A second strategy is to give a diagnostic assessment of prerequisite skills early in the course. This enables you to identify exactly which students have gaps in exactly which skills. This will make it easier to know when and how often to insert opportunities to address these topics in warm ups, lessons, differentiated assignments, and questioning, among other aspects of instruction. Another strategy, depending on the results of the diagnostic assessment, is to offer some students intervention support through an additional class or after school session. There are several new programs designed just for this, addressing common misconceptions and offering fresh opportunities for extra practice. Another strategy which can be useful for all students, whatever their skill level, is to take opportunities to build on the fundamental concepts underlying all number work. Two of these are number sense and operation sense. Number sense includes an understanding of the behavior of numbers in families, their size or magnitude and relationships to other numbers. So, for example, students need to understand how fractional numbers are related to whole numbers. For integers, they need to understand that the sign before the number indicates both direction and magnitude. In both cases additional experience placing fractions and integers on a number line can be beneficial for all students. Comparing and ordering numbers in multiple contexts can be integrated into warm ups, lessons, or spiral reviews. Estimating the results of fraction operations provides another venue for illustrating and practicing number sense. Another activity involves giving students a set of numbers and having them place them in order. You might give them the set of atypical fractions such as 7/13, 13/15, 2/17, 2/19 and 6/13. Do students realize both 6/13 and 7/13 are near 1/2 and that 6/13 < 7/13 as the denominators are the same? Do they realize 13/15 is the largest as it is almost 1 and that 2/17 and 2/19 are the smallest fractions as there are few parts of the whole and that 2/19<2/17 since nineteenths are smaller than seventeenths? These fractions do not lend themselves eas- ily to decimal conversions so students need to draw on benchmarks that are grounded in their understanding of fractions to make comparisons. Another related fundamental concept is operation sense, understanding deeply the meaning of operations. Occasionally students bring misconceptions about operations that get in their way. A common example is the belief that the product of multiplication is always larger than either factor, which is not true for all numbers. Again, activities which reinforce the meanings of operations and models for operations are useful for all students. This idea can then be extended to algebraic operations. So, for example, considering an array model for multiplication can lead into an activity with algebra blocks or tiles, enabling students to make connections while demystifying algebra. This can be done without interrupting the flow of algebra, since it is good pedagogy to build on what students know, their prior knowledge, using that as a bridge to new learning. An excellent example of an activity that emphasizes the meaning of operations can be found on page 253 of Principles and Standards for School Mathematics (NCTM, 2000). In this activity students are asked to examine points on a number line with no numbers associated with them, and analyze the results of operations on them. It is a rich and complex task. Again, this is appropriate for all students, not only those with less computational proficiency. For the next strategy, we return to the idea of using tools for helping students cope with gaps in prior learning. If necessary, students should be able to use a calculator with fraction capabilities to compensate for lack of proficiency with fractions. Provided students have some number and operation sense, they could also use a calculator for integer computation. It is important for teachers to help students see that they are successful students despite gaps. Meanwhile, students will often realize themselves that even if they are reasonably successful in algebra, their computational deficiencies are slowing them down. Teachers can use opportunities for differentiated assignments or projects to help some students fill in these deficits, and become fluent to the point of no longer needing the help of a fraction or integer calculator. A wonderful story related to the idea of compensating with tools can be found in Cathy Seeley’s story of “Crystal”, found on page 159 in Faster Isn’t Smarter (Math Solutions Press, 2009). This entire book is a wonderful resource for teacher conversations around math teaching. Finally, an overall strategy involves good questioning. Once you are aware of student strengths and weaknesses, you can use that to inform your practice of posing effective questions in your classroom. All students should have their thinking probed, no matter their strengths or weaknesses. Once students become accustomed to having to explain their thinking, and being responsible for carefully listening to other students’ explanations, they have a new means of clarifying and extending their own understanding. To summarize, Matt |
Matt Mentor, a wise and experienced teacher, offers advice about teaching mathematics topics to beginning teachers. Of course, experienced teachers can join in as well. Here’s how it works: Send your answers to MattMentorMCTM@aol.com and Matt will post as many different solutions that adequately address the question as are received. Have a Question for Matt? |
