Supporting Struggling Students

I was recently reading a teacher’s manual for fourth grade.  It suggested if students were struggling with a particular concept that the teacher should “make the font bigger.”  I don’t think this strategy is going to help for most students but it does bring up the question,  “How can I help students who are struggling?”

Development of math concepts proceeds along a continuum from concrete to abstract.  When students are first learning, they should start with a concrete representation of the concept.  As their skill grows, they can move to a more abstract understanding.  For instance, when working on a subtraction problem such as 32-19, students should first work with unifix cubes or other objects so that they can count out 32 cubes and then take away 19 of them.  As they develop more proficiency, they can move on to using more abstract manipulatives such as base ten blocks.  These blocks include a stick with marks on it to show that it is a stick of ten.  Unlike the cubes, however, one cannot break it apart and take some off.  To solve 32-19, one can take away one stick of ten.  Then to subtract the nine, one needs to take away another stick of ten and trade it in for ten units in order to be able to take away nine units.  Once students have worked with the blocks enough, physically trading in the ten stick for ten units, they will then understand why we “cross out the 3 and make it a two” when using the algorithm. 

Often when students struggle it is because they haven’t spent enough time at the concrete stage or they have been shown the algorithm directly and haven’t spent any time at the concrete stage.  They have memorized a procedure but there is no understanding about why it works.  If they are presented with a problem that doesn’t fit their memorized procedure exactly, they cannot solve it. 

To support these students in building their understanding, you need to go back to a more concrete representation, such as manipulatives.  These will provide a visual model of the concept.  Students can work through problems by creating models.  They will physically move the materials and be able to see what happens to the numbers as they work.  If they do this they will be using their kinesthetic and visual modes of learning.  If they then discuss what they are doing they will be using their auditory sense as well.  By approaching the problem using so many pathways, they will develop a deeper understanding.  They will see that math makes sense. 

Consider an example from middle school.  When I have worked with middle school students and brought up the concept of trapezoids, I am often met with many blank stares.  It is difficult for some students to remember what a trapezoid is and how it is different from other quadrilaterals.  On the other hand, students who have used pattern blocks a lot in their earlier years know exactly what I am talking about.  When they were young, they had the opportunity to hold trapezoids often.  They built designs with them.  They put two trapezoids together to build a hexagon.  They fit the acute angle of a trapezoid between two equilateral triangles to make an interesting design.  (Though they had no idea about this terminology when they did it.)  When I mention trapezoids to middle school students who have had these concrete experiences they have a physical reaction to the word.  They often exclaim, “the red one!” (In a set of pattern blocks the trapezoid is red.)  Many hold out their hand as if they are remembering what a trapezoid feels like.  They can also tell me that it has two sides, two sides are parallel, two angles are acute, two angles are obtuse.  They can work from this specific information to generalize a definition of a trapezoid.  They are building on a solid foundation of experiences with the shape and expanding their understanding with new experiences.  The students’ early activities with the manipulatives have an effect on their math understanding, even many years later. 

Often students begin to struggle with math in fifth grade.  Up until this point these students may have been good at memorizing algorithms and teacher instructions.  Their math experiences did not include early experiences with manipulatives and problem solving that helped them make sense of the math.  Fifth grade is when students need to start applying algorithms, often in sequence.  If they do not have a firm understanding of what they are doing, the process falls apart for them because they do not have a firm enough understanding of what is happening to be able to follow the numbers and make sense of the work.  It is like building a house on sand.  The foundation is not firm enough to support the weight of the heavier math.  More often than not, this results in a frustrated student who says, “I hate math.”  This student often starts to turn away from math at this point and is unlikely to pursue math challenges and courses in the future. 

We need to go back and shore up the foundation for these students by helping them build their understanding from concrete to abstract.  By using manipulatives and helping students make sense of the math then they will be able to move forward with a more positive attitude.  They will be more likely to stick with math and gain skills that are needed for the jobs of the 21^st century.

Communication In Math Class

I once had a parent very upset with me because I gave her son a math assignment to do with a partner.  “Are you trying to teach my child math or are you trying to teach them social skills?”  Well, both.

Times have changed since I was a student and since many of today’s parents were students.  It is true that my middle school math class was silent.  No talking was allowed.  We all sat in rows alphabetically.  I sat behind the same boy for all of 6^th, 7^th, and 8^th grade.  I never talked to him or any other students in class nor did we ever share our ideas in front of the class.

If one talks to adult mathematicians, however, it is clear that the work of a mathematician is social work.  Solutions are rarely arrived at on one’s own.  It is through collaboration and discussion that solutions are found. 

The new Common Core Math Practices reflect this.  One of them reads: “Construct viable arguments and critique the reasoning of others.”  Students need to learn to clearly explain their thinking to others.  They also need to learn how to disagree with and question the work of others in a tactful manner.

My husband and I recently remodeled our 50 year-old kitchen.  He was very invested in the color of the new floor.  He really wanted a deep reddish color.  I wasn’t focused on the floor at all and was ready to go with the first color suggested by the contractor.  I was focused on the backsplash and what kind of tiles to use there.   My husband wanted the backsplash to be plain white.  By bringing our two different perspectives together, we were able to create a finished product that is more complete and more elegant than we could have done on our own. 

I know you have had this experience too.  You worked on a work project, a house project, or had a discussion with someone.  By bringing your ideas together the final result was much stronger.

A math problem flows in the same way.  Yes, we can often solve a problem by using algorithms to get the solution but another person might have a more elegant way of getting to the solution.  Their path might take into account patterns or ways of grouping numbers thus making the problem so much simpler than you originally thought.  They might have attended to a piece of the problem that you hadn’t considered carefully enough, making the final solution more complete.  In addition, the sharing of ideas makes each person reconsider their perspective in light of new information and either reject their first idea or add to it.

By talking about math with our children, we can also help them see new connections in math and think about concepts in ways that may be different or new to our children.  We also help them learn that math is part of the real world.  It is not just something that is done inside of a math classroom.  We can work together on sudoku puzzles on an airplane or play Blockus together in the evening.  It is more enjoyable if it is shared.

It is funny to think back to my middle school math classroom.  In addition to regular math class, I was on the math team.  The same teacher practiced with us after school, solving complex problems together as a group but during our regular class time we were in those rows.  Why did the teacher teach the two groups so differently?  The implication was that kids who were good at math should talk about it and those who weren’t should listen quietly to the teacher’s ideas.  Maybe those of us on the team got better at math BECAUSE we talked about it after school.

Common Core directs all students today to be doing math like mathematicians.  We want them to talk and share ideas and learn from each other.  Not only will this prepare them better mathematically but it will make math feel like math is a part of their whole lives, not just their school lives.