David Chard, dean of Southern Methodist University’s Annette Caldwell Simmons School of Education and Human Development, is the author of the “Middle School Mathematics” textbook. He also has written numerous math instructional programs.
In this interview, Chard addresses the challenges of getting middle school students interested and proficient in the critical subject. He will speak on this topic this week at the Middle School Matters conference. The George W. Bush Institute and The Meadows Center for Preventing Educational Risk are hosting the meeting on the campus of the University of Texas at Austin.
As part of the Institute’s team of researchers around the country, Chard has advised schools that are part of the Middle Schools Matter network of campuses.
Math obviously requires a considerable amount of reasoning. How do schools dealing with students in sixth, seventh and eighth grades create a culture of reasoning, especially when their students are dealing with the physical and even emotional changes associated with those years?
You are correct. The middle school years represent a tumultuous time for many, if not all, students. They are growing increasingly independent, gradually distancing themselves from adults, but learning to think more abstractly and questioning everything!
This provides a fertile environment for teachers to help students in these grades to learn to reason. I’m not suggesting that it goes smoothly…many students resist thinking too much about anything. But, if they have the right skills and knowledge in place, are provided interesting problems to consider, and given room to think these problems through, their reasoning will grow.
Where we have challenges in mathematics in the middle grades is when teachers and/or parents have given students the message that they are not successful at mathematics and will or cannot be successful moving forward. This message comes through when students struggle with particular concepts or processes and are not encouraged to work through those struggles, to remain persistent, and to think like a problem-solver.
To give a specific example, as students move into the middle grades, they are expected to work more and more with rational numbers. These numbers are conceptually challenging and very different from the tidy world of whole numbers.
With rational numbers you can divide things into infinitely many pieces. And unlike whole numbers even the typical operations change dramatically when applying them to rational numbers. For example, when you multiply two numbers that are greater than 0 and less than 1 (two simple fractions), you get a smaller product.
These insights have to be developed in children so that they have the knowledge to reason about and solve complex problems. I’ll say it again, learning increasingly abstract mathematics is not easy. It can frustrate students who are already experiencing a lot of changes in their lives. However, if we insist on helping them to be successful and give them the tools to do so, they will learn to reason.
How much of learning math at this age, or any age really, is related to vocabulary? In other words, if you have a weak vocabulary, you might be behind in math, too?
As human beings, everything we learn depends on language. Number names, positional concepts (first, next, last), comparative concepts (e.g., taller, fewer) all play a major role in our development of mathematical thinking. Our vocabulary is particularly important in early childhood as we learn a lot of important ideas about magnitude, counting, patterns, and relationships.
However, it is often the case that when students enter school mathematics, we divorce language and vocabulary from mathematical development. Rather than build mathematical vocabulary, we give students the impression that mathematics is about computation only and that words about mathematics are somehow not important.
In other cases, we substitute words that we think make the mathematics easier (e.g., plussing rather than addition). This makes no sense when young people are capable of so much more.
For example, when motivated, they can learn multi-syllabic dinosaur names with ease. So, we need to be more deliberate about using precise mathematical language, encouraging students to talk about their mathematical thinking, and holding ourselves accountable for talking mathematically.
As students grow older, this will help them as they translate words of problems into mathematical sentences that can be manipulated and then back to words. This process can often be an obstacle to students doing mathematics successfully.
You have said that it is not good to teach math at the end of the school day. Give us a couple of other sensible strategies that can help educators reach students at this pivotal point.
Mathematics is challenging, students need energy to focus on challenges. This is why I believe it is better not to do it at the end of the day.
Another important thing to keep in mind is that students need practice with anything they learn that is new. Scheduling time for students to practice new ideas or processes is a very important part of effective teaching.
Sometimes we provide challenging questions that are novel and we don’t provide adequate time for students to develop solution strategies and try them out. We also often don’t give them enough practice with similar problems so that they can develop a schema (map) that indicates the kinds of strategies that might work.
Another key idea is letting students know that they should verbalize their thinking. Problem- solvers talk their way through problems. This talking is a precursor to self-talk that leads to self-regulation. We need to verbalize our thinking when we are faced with challenging situations.
How do you recommend math teachers deal with the pace of their struggling learners? The subject is not always easy to pick up.
This is one of the most challenging aspects of teaching in a classroom with a diverse group of student needs. Mathematics is very hierarchical and mastery of earlier content is necessary for students to learn later content.
By the time students reach the middle grades, there are often five years or more range of knowledge and skills represented in the classroom. Many struggling students do not have the content mastery to learn new content effectively.
Part of the problem historically is that we’ve also focused on breadth of knowledge rather than depth and it was difficult to help students to catch up when the whole class kept moving on without them. However, recent changes in standards and expectations have focused on teaching less and teaching it more thoroughly.
This is only one aspect of supporting struggling learners. Teachers also need to be sure they know the status of their students’ knowledge and skills in the areas that are to be taught.
In some instances, knowing which students have the greatest needs allows teachers to group them for pre-teaching of concepts and processes that represent gaps in their knowledge. If they can be pre-taught these important components of mathematics, they are prepared to learn the grade-level content.
In some instances teachers may need to use supplemental materials or technologies to give lower-performing students practice in areas in which they need support so that they can keep pace with the class. In other instances, students may need outside intervention from a specialist to help them cover necessary content that they didn’t learn in previous experiences.