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.

Watching Our Language

I often give talks about math to groups of parents.  So often parents come up to me afterwards with a question or a comment.  They preface it by saying, “I’m not a math person.”  This comment always makes me wonder, “Have you figured out how to double a recipe?  Do you manage your bank account?  Can you tell how much the pants will cost if they are 25% off?”  Yes, you are a math person.  You may not love math, but you are a math person.

We cannot deny how much math affects our modern lives.  Each morning when my son wakes up he checks the temperature for the day and compares it to the day before.  “Oh, mom, it is going to be 13* cooler than yesterday.”  My daughter collects snacks for her Girl Scout troop.  “If we have 12 girls and each box of granola bars holds 8 snacks then I need to bring two boxes.”  We get in the car and I calculate if ¼ of a tank of gas will get me the 30 miles to school and back.  Every hour of the day we are involved in some sort of math.

Two hundred years ago women were considered too delicate to think about math.  We need to stop perpetuating this idea and give our daughters (and sons) the confidence that they CAN do math.  It is OK to say, “I don’t know how to do this problem,” but follow it up with, “Let’s figure it out together.”  Your child will get the idea that they can do math if they stick with it and talk about it.

Most parents know that it is important to read to their children at night but so few parents know that it is also important to do math with their children.  No, I don’t mean worksheets.  I mean, real-life math.  First you need to realize when you are using math in your life and then you need to talk about it with your children. 

Has the price of gas gone up?  If it goes up five cents per gallon how much does that change your weekly gas bill?  Talk about it. 

Are you wearing a fitness tracker?  Have you hit your 10,000 steps for the day?  How far do you have to go?  Talk about it. 

Just this morning the DJ on my radio said that she didn’t like doing math.  Talk about it.  Break down those perceptions.

My children are now in middle school so they are studying decimals and percents.  I have been using that language to give them answers to their questions.  How much longer until we get to the birthday party?  We are 50% of the way there.  Do we have any cake left?  75% of it is gone.  Talk about it.

So many of the jobs of the 21^st century require math skills.  We are doing our children a disservice when we set them up to think that they are not a math person and thus cannot do many of our modern jobs.  Change the language in your house.  Change your child’s attitude and tell them we are all math people.

Building a Strong Mathematical Foundation

Before the days of GPS systems if you needed to go somewhere in a new town you would get out a map and take a look at what route you might take.  You might decide to take the freeway.  You would do that route for many days since that was the one route you had learned and each day it would get a bit more comfortable and familiar. 

After many days of following the same route, you would move to an abstract level.  You wouldn’t need to consult your map anymore to double check the name of the exit.  You would be sure to turn right at the T in the road rather than turning left.  If you wanted to stop and get a coffee on your way, you would know how to alter your route slightly to hit the Starbucks.   If you were on the freeway one day and there was a traffic jam, chances are you would have a good sense of how to exit and take some side streets because you had studied the map and had an idea of the surrounding area. 

Building a strong mathematical foundation with students proceeds in much the same way.  We want them to move from a concrete understanding to an abstract understanding but this process takes time.  We start with the students using tools and building models to develop their understanding of a concept just as you used a map to plan your route.  Once students develop some comfort and skill at this concrete level we can start moving them to more and more abstract levels of understanding just as you did when you were able to alter your route to grab a coffee on your way.

We want students to have a good sense of the surrounding area.  If they get stuck in one place, they should have ideas for how to go in another direction.  The new math practices call for students to “Use appropriate tools strategically.”  In order to chose a route and decide upon the best course of action, students need to have options available.  When we were in school most of us only learned one method for solving a problem.  Teachers have learned, however, that we do students a disservice when we only teach them one method of solving a problem.  If they get stuck or want to check their answer, they don’t have any other tools to choose from.

Additionally, teaching multiple strategies helps to meet the diverse needs of a class.  At any point in time students will be spread out along the continuum of understanding of a concept from concrete to abstract.  All students can be working on the same problem but they might not all be working on it in the same way, at the same level.  Students can learn from their classmates’ methods and ideas about solutions.  The whole group grows mathematically stronger as their consider the problem from different perspectives.

Recently I have been studying some of the math programs that are available for elementary schools.  Though Common Core standards are very clear about having students work at a concrete level before moving on to an abstract level, most of the math programs are not aligned with this thinking.

One program I reviewed had pictures of a few models but then when it came time for students to work independently, all of the work was abstract.  It takes time for students to move from a concrete to an abstract understanding.  Considering the example of the route in a new town again, you might have needed to consult your map each day for a few days before you could travel the route unassisted.  You traveled the route over and over and soon you became very comfortable with it.  We cannot just show students a few pictures of objects and assume that they are now ready for abstract thought about the topic.  Students need to build, model, and discuss their ideas.  They need to consider other students’ models of the concept and look at theirs in light of the new information.  The only way that students can build a strong foundation is to move through these steps from a concrete to an abstract understanding of the concept.

I see too many fifth graders who tell me they hate math.  It doesn’t make sense they say.  It isn’t interesting to them.  When I talk with them further I realize that most of these students never had an opportunity to explore concepts at the concrete level.  They were given algorithms and told to memorize them.  They were then able to apply those algorithms if the new problems looked just like the problems they had practiced.  If they hit a problem that looked slightly different or asked them to use their skill in a new way, they did not have enough of a mental image of the concept to be able to choose a new route.

Teaching students multiple ways to solve problems gives them a stronger mathematical foundation.  They can consider new problems from different perspectives and then choose the best method for solving them. They are flexible thinkers who can try new approaches when the one they are using doesn’t work out.  Teaching in this way also helps to meet the diverse needs of a group of students.  This is one of the strengths of the new Common Core Standards.

Differentiation in First Grade Math

Sometimes it is hard for teachers to know exactly how to differentiate in a classroom. This post explains what a series of differentiated lessons in math looked like in a first grade classroom.

I have been working with a group of gifted first graders for a few months now. From some early assessments, their teacher and I knew that they were working at a higher level than their classmates. We decided to put seven students in my group. The teacher took the rest of the students. The teacher’s group has been walking through the textbook. In this particular unit on addition, they are solving problems with regrouping such as 14 + 7= __________.

My group of students needed more challenge but I didn’t want them to stray too far from what the topic their classmates were studying so we started by solving regrouping problems with larger numbers such as 48+37= __________. We used place value mats and base ten blocks to model each problem. We have been doing lots of trading this year (trading pennies for dimes, dimes for dollars, etc.) so once I reminded them that they would need to trade, it was easy for them to solve these problems which all involved trading units for longs (ones for tens).

The next day the teacher’s group practiced different strategies for solving their two digit plus single digit problems (counting on and building with blocks).

My group was already proficient at counting on and building the easier problems with blocks. The next challenge for them was to solve problems that involved two trades – units for longs and longs for flats. These would be problems such as 129 + 83= ______________.

Before we started, however, I wanted to check that the students knew how to build the larger numbers. The students were confident using the base ten blocks to build numbers up to 100. With larger numbers, their understanding was not as solid. We practiced building and writing numbers such as 110. Many students built it correctly but then wrote it as 10010. Once they saw that their model only used three columns, they realized that the numbers they wrote down needed to correlate to their place value mat. They had one flat in the hundreds column, one long in the tens column and zero units in the ones column. They could then record this as 110. Next they were challenged to build 203. Some built 230 and some built 203. We continued to practice until they could build numbers up to 999 with accuracy. They were then able to solve the more complex addition problems confidently.

Throughout this process, one student correctly built and recorded each number without fail. I made a note of this student’s solid understanding of place value. This would help me plan next steps for him in math.

The next day the teacher’s group continued to need more practice with regrouping. It was clear, however, that my group did not need this practice. What they did need was to solidify their understanding of the place value of larger numbers that they had been building over the past few days.

I decided to have them work on counting strips. If it had been the beginning of first grade, I would have had them start their counting strips at number 1. Since we had been working on numbers over 100, I had them start with 100. They placed one flat on their place value mats. Then they added a unit and recorded 101. They then added another unit and recorded 102, etc. Soon the students were able to work abstractly and did not need the blocks to represent every number but used them at tricky times such as going from 149 to 150.

For the past few days, the class had been benefitting from the two math groups. At this point, it became clear that we needed to split yet a little bit more to meet everyone’s needs. The one child who had a solid understanding of place value from the start did not need to do a counting strip in base ten. I did, however, want to keep him on the same topic so I started him on a counting strip in base five. This would test his understanding of place value and strengthen it. He was thrilled to be working on something at his level and actually went home and asked him mom if he could do counting strips in bases two and three as well!

The day we started working on counting strips, the students in my group were beaming. They were so happy to be working on work that was just at the right level for them. They asked me why the one child was working in base five and the others were working in base ten. I told them that everyone’s brain was getting what it needed. They accepted this idea and continued to work quietly and happily for the rest of the period.

While this level of differentiation was achieved by having two teachers in the classroom, it could have easily been done by one teacher. One group could have worked with the teacher while the other group did some independent work (Marcy Cook math tiles, playing math games, reading, etc.). The teacher could then switch groups.

The biggest hurdle to differentiation is realizing that different students have different needs. Once a teacher acknowledges this, meeting those needs does not mean creating twenty different lessons. It does mean having two or three different levels of a topic available to meet the range of needs in the class.