Top left is a scaffolding I use to help students learn to solve math problems using multiplication (3rd grade). The situations are typically rate problems (e.g., 5 pumpkins per plant or $2 per slice of pizza) although the term “rate” is not used yet. The same concept of rate plays out in high school with slope of a line, applied to real life situations (top right).
These types of problems start in 3rd grade (below, top left), play out in 6th grade (below, top right), into 8th grade (below, middle), and into high school algebra and statistics (below, bottom). I referenced this connection previously regarding word problems and dominoes. This highlights how crucial it is that strategically selected gaps in a student’s math education be addressed in context of long range planning.
When I train new math and special education teachers I explain that teaching math should be like feeding a hot dog to a baby in a high chair. Cut up the hot dog into bite-sized pieces. The baby will still consumer the entire hot dog. Same with math. Our students can consume the entire math topic being presented but in smaller chunks.
My approach to doing this is through a task analysis. This is very similar to chunking. It is a method to cut up the math into bite-sized pieces just as we would break up a common task for students with special needs.
While waiting for my coffee order at a Burger King I saw on the wall a different version of a task analysis. It was a step by step set of directions using photos on how to pour a soft cream ice-cream cone. I thought it was amazing that Burger King can do such a good job training its employees by breaking the task down yet in education we often fall short in terms of breaking a math topic down.
One of my beliefs about the education is that teaching is built on a delivery based model. If teachers take certain steps the learning will happen – an educator’s version of Field of Dreams. Often the result is a focus on having students assimilate into the teacher’s class environment.
I subscribe to the exact opposite approach. Teacher’s should accommodate student needs as the focus of the classroom environment.
Below is a quote from a parent whose child benefited from my effort to be hyper responsive to her daughter’s instructional needs. The child had veto power over any activity or strategy I attempted. If what I used didn’t work for her I would try something else.
“Working with Randy has been life changing for my daughter.
Math was her biggest source of frustration and no matter how hard she worked it never made sense. Teachers would tell me she was ‘doing awesome’ but she was really just following steps without understanding any of it. I thought she was going to go through life unable to even buy a candy bar without being taken advantage of.
Randy changed all that. He is able to break math down in a way that makes sense. He is able to identify what is confusing her and find different ways to explain it. He makes it meaningful for her.”
Ask employers what skills are desired in graduates and you will not see academic competence at the top of the list. In schools we talk about creating life long learners and similar qualities but the major focus in the 7+ K-12 schools in which I have served is academics, or more appropriately grades as a proxy for academic mastery. Add to this the focus on exit exams for graduation and you see major disconnect between the desired outcomes and the focus.
I have taught math at 5 colleges or universities and have seen first hand students struggle with content but also with independent study skills. Manchester Community College in Connecticut conducted a survey of students and asked students to cite reasons why students struggle in their classes. The second most commonly cited responses by students themselves is that students don’t know how to study (see below). In high school we talk about study skills. Teachers will share they expect students to be independent but often the focus is on academic mastery and not the study skills.
At Manchester Community College I serve as an instructor at a highly successful (based on objective outcomes) bridge program for first generation students. A major emphasis is a focus on student academic discipline with a mantra that discipline is the bridge between goals and accomplishment (see below). Learning how to BE a good math student, especially academic discipline, is as important as developing the prerequisite skills to be successful. This could be a major focus in the IEP for students who have a goal of college or post-secondary training..
I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.
For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math.
This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.
A widespread problem at the secondary level is addressing basic skills deficiencies – gaps from elementary school. For example, I often encounter students in algebra 1 or even higher level math who cannot compute problems like 5÷2. Often the challenges arise from learned helplessness developed over time.
How do we address this in the time allotted to teach a full secondary level math course? We cannot devote class instruction time to teach division and decimals. If we simply allow calculator use we continue to reinforce the learned helplessness.
I offer a 2 part suggestion.
Periodically use chunks of class time allocated for differentiation. I provide a manilla folder to each student (below left) with an individualized agenda (below right, which shows 3 s agendas with names redacted at the top). Students identified through assessment as having deficits in basic skills can be provided related instruction, as scheduled in their agenda. Other students can work on identified gaps in the current course or work on SAT problems or other enrichment type of activities.
Provide instruction on basic skills that is meaningful and is also provided in a timely fashion. For example, I had an algebra 2 student who had to compute 5÷2 in a problem and immediately reached for his calculator. I stopped him and presented the following on the board (below). In a 30 second conversation he quickly computed 4 ÷ 2 and then 1 ÷ 2. He appeared to understand the answer and this was largely because it was in a context he intuitively understood. This also provided him immediate feedback on how to address his deficit (likely partially a learned behavior). The initial instruction in a differentiation setting would be similar.
In math, many students with special needs fall behind. What results is a Catch-22 in programming and services. If the student is provided extra time to work on the gaps, he or she likely falls behind with current content. If the student is provided extra time to receive support for current topics, the gaps are not addressed
In both cases the extra support time can actually be counterproductive.
The focus on gaps likely results in the student working on different math topics which in effect means the student has TWO math classes – just what a student with math anxiety doesn’t need.
The focus on current topics means the student is trying to learn math topics for which he or she doesn’t have the prerequisite skills needed.
I recommend identifying the prerequisite skills for a current math topic and address ing these skills concurrently in math support or during the summer. For example, I used a Common Core coherence map (top photo below) to identify Common Core prerequisite standards for the standards a student faces in her upcoming school year. Then I listed these with each grade level standard (bottom photo below). The prerequisite skills can be identified using a task analysis approach as well.
This approach allows for a systematic approach to fill in gaps and to prioritize when they are to be addressed. When implemented effectively, the student can see the immediate benefit of the support time – it helps them in math class. Even better, the support teacher can match instruction and work with what is covered in math class.