Enhancing Mathematics Instruction for Students With Learning Difficulties
The Enhancing Mathematics Instruction for Students with Learning Difficulties course was developed at the Education Development Center (EDC) in Massachusetts with funding from the National Science Foundation. The course goal is to make mathematics more accessible to a range of learners; particularly students with learning disabilities who study the general education mathematics curriculum. The course is designed for teams of math and special education colleagues to attend together. This promotes the collaboration needed to address challenges related to the teaching and learning of mathematics for students with disabilities.
The EDC designed this course to address three challenges:
- Many students are not proficient in the Number, Operation, and Algebra standards. This plays a critical role during middle school, bridging from arithmetic to algebra.
- Middle school math teachers may feel underprepared to adapt lessons to meet student needs. Such teachers are content experts who may have limited training in meeting the needs of students with disabilities.
- Special education teachers are required to take few, if any, mathematics classes as part of their teacher preparation.
This course draws from research related to:
Student learning disabilities.
How children learn math.
The barriers that can arise when students who struggle in math, as well as reading, are not properly considered.
Accessibility Strategies for Mathematics - The course focuses on barriers students may encounter when approaching mathematics learning and problem solving. The course aligns strategies to address those barriers when planning for student learning and assessment.
Math Learning Disabilities - Math learning difficulties are common and range from mild to severe. This course includes learning opportunities accessible to people with a variety of learning disabilities. It gives an approach to providing accommodations for students who struggle.
Understanding Co-teaching Components - Much of the course revolves around creating opportunities for teams of math and special educators to plan and discuss how to improve instruction for students with learning difficulties
How to Receive This Training
You can choose to partner with Alt+Shift or attend a scheduled Statewide Event. Read more about each option below. If you’d like to be notified when the next statewide event becomes available, you can request to be notified.
Partner with Alt+Shift
Training is provided to ISD staff as part of an ISD partnership. Training is typically provided to the entire district, building, or program staff. This depends on the specific ISD's implementation plan. Training is one piece of the partnership. Strategic planning, implementation support, and capacity building are also addressed through the partnership.
Attend a Statewide Event
Statewide events are opportunities to receive training, but with limited opportunities for follow up support. Participants can expect to gain ideas and strategies that would be usable immediately in their practice, and to gain a better understanding of the nature of the training as part of an exploration process for sites considering a partnership with Alt+Shift.
There are no upcoming events for this training.
Alt+Shift asked partners around the state to share their implementation experiences and the impact on adults and students where they work.
Gregory White and Michael Corridor
Mr. Corridor and Mr. White discuss below how their implementation of the course is going. The teachers drew on ideas in the course based in Universal Design for Learning to teach their unit on Expressions and Equations.
Corridor: I have really enjoyed watching students who typically struggle with math computation work on figuring out visual patterns.
White: We have been focusing on students who struggle with the procedures and algorithms and have been focusing a little more on the thinking and how you figure out the patterns, often asking them to identify what’s changing from one step to the next.
Corridor: I know we both have been discussing and working on making the shift to student-centered instruction. I was wondering, how is it coming along?
White: It’s going well. However, it is a difficult shift for a lot of students. They are so comfortable looking up at the teacher, who is sharing knowledge with them, and taking notes. Since I have transitioned to student-centered instruction, I am trying to get students to interact more. The goal is to get students to engage with each other and the concepts that we are talking about. I want them to start asking questions and sharing what they know. I want them to share what ideas they have about those different concepts instead of just sitting quietly and taking notes. It is a shift for kids, and it takes them out of their comfort zone. But once I get some kids together, they get more comfortable and it starts to become a classroom norm.
Corridor: One of the things I really liked about Enhancing Mathematics was the instructional strategy of using visual patterns and visible thinking routines to better understand linear equations. I have been using figure zero when analyzing visual patterns. It has enhanced instruction, and my students really enjoyed figuring out and drawing how the patterns began. One of the major concepts that my students struggled with was they always assumed all patterns begin at zero and should all be graphed by using the ordered pair (0,0). Therefore, they often were confused when they had to graph linear equations on the coordinate plane. Teaching the students how to go back to figure zero really had a huge impact on how quickly my students were able to master the concept of writing equations in slope-intercept form. It was also helpful to have the students describe the patterns using written language as well as drawings.
White: I have noticed the same thing with many of my students. Students see figure zero as “step zero,” or the beginning point, so they assume it must be always be zero. However, once they do it multiple times and they can work the pattern backwards using their rate to go in reverse order, it becomes kind of like second nature to them.
Corridor: The whole idea of visual patterns really motivated us to start thinking about other creative ways to teach students. I believe that other teachers may find these strategies as helpful as we have. Using these activities to teach students who are easily distractible has helped me increase engagement. It has really helped me to break up instruction. My students have more opportunities to think and do not have to listen to long lectures. It really helps to keep the class moving. However, the most powerful thing about these strategies is when kids start to make the connections between the math and the activities.
The strategies are below:
- Math Twitter Blogosphere
- Estimation 180
- Fawn Nguyen’s blog
- Weigh the Wangdoodles
- Math Playground
- Global Math Department
Corridor: Giving students different ways to explore and view equations and systems of equations is not just fun, but also meaningful. It makes the math concrete and easier for students to relate to. This reminded me of a strategy I used in which I used picture books as an entry point to teaching students linear equations (The Twelve Days of Kindergarten by Deborah Lee Rose and Counting Sheep by Julie Glass).
White: Making math accessible to all students has helped shape students who are less dependent on me for answers. It goes back to the idea of student-centered instruction. The teacher is talking less so there is more time for inquiry and students engaging with their peers.
In our first year of offering this course in the Bay-Arenac Intermediate School District (ISD) area, it was unknown to nearly every teacher I came into contact with. Therefore, getting the word out became our number one priority. With the help of our administrative staff, we created a brochure detailing the content of the course and the benefits to all students, not just those identified with special needs. We sent the brochure to the 6th through 10th grade teachers.
Since this was only somewhat successful, I called the principals of most of our districts to see whether the information had reached their desks. Sadly, few had heard of the series. From there, we sent the information directly to curriculum directors, principals, and even most superintendents.
With the information “out there” we saw a positive uptick in the number of districts responding. Our first four series during the 2014-2015 and 2015-2016 school years (one in fall and one in winter for each year) saw nice size classes (12-24) and representation of 20 area schools. This past year, class sizes were in the teens and seemed to be more individually attended rather than the general and special education pairs we were seeing initially.
Over the three years of putting on this training course, the focus has morphed. It’s moved from supporting only students with identified disabilities to those who experience any type of difficulty in acquiring the needed math skills to ensure a positive Algebra experience in high school.
This is exemplified in a co-taught 9th grade classroom, with a typical class of 28 students. Such a class includes both identified special education students and students who are behind their peers mathematically, and where 0 percent of the students met proficiency benchmarks in math last year. After experiencing the “Step Zero” strategy for writing equations in Enhancing Math and incorporating practices from “Building Thinking Classrooms” (a separate math training that we shared during Enhancing Mathematics), the co-teachers took the leap and changed the culture of their classroom completely.
Once saddled to individual desks, the students were now part of a program designed around discovery and collaboration. The instructors’ initial fear of “losing control” was replaced with students eager to get to class and see what was next. The teachers found the students were agreeable to work in any group, social barriers came down, co-construction of answers increased, knowledge became mobile, and reliance on the teacher decreased. As they continued to refine their new teaching style, these teachers began to give presentations about what they do and spread the word about their success.
I have kept in touch with many of the teachers who have gone through the training. We share topics, research, and resources that reinforce the content of the program. Some teachers have emailed me copies of their assessments they have modified or made more accessible as part of the work on the third day of the training. I have passed these along and received good feedback from their peers. Another plus I have noticed in classrooms is the revamping of teaching slope with an emphasis on scaffolding from sixth grade’s emphasis on proportion and rate of change.
Although I have kept in contact with quite a few teachers, I do not get the opportunity to visit most of their classes. Beginning this year, facilitators of the course are required to have three follow-up contacts with participants through email and/or visits. This is a huge step toward to make sure the good strategies we cover are purposely being used to help students. I look forward to this addition.
As we move into our fourth year of offering this program, I am more positive than ever that it will have a lasting effect on instruction. Hopefully, with the accompanying research and strategic instruction, one day we can see the belief that ‘I am not good in math’ is no longer an acceptable crutch for students.
I have 27 students in my co-taught 6th grade math class, 7 of whom have individualized education programs (IEPs). Implementing the IEPs and providing direct and individualized instruction is a daily challenge. Enhancing Math presents six “areas” or aspects of math learning involved in learning math, and, when affected by a disability, can be a challenge for students. Two common challenges for my students are visually manipulating objects and getting started on the problem.
These challenges are evident when my students struggle to draw the flat surfaces of three-dimensional figures during our surface area unit. If students cannot accurately visualize or draw flat surfaces of three-dimensional shapes, they will have difficulty conceptualizing and calculating measurements. When teaching the lesson, visually showing students a real-life example of a surface area cut out so they can see the flat surface area is vitally important. Then when drawing the flat surface, students need to see how to draw it. I use a color visual to help students see each surface clearly. This helps students organize their work and develop accurate results.
What does this look like in a lesson? When working with rectangular prisms, I first tell the class to color the base green. As they color it green they can clearly see the base and the height of the base. Then I ask, "How many parts are the same size?" The students see the base and the top are the same size and know they can take the area of the base and multiply it by two.
Next, I tell them to color the side yellow, which helps them see the base and height of the side. I then ask, "This is one side, is there another the same size?" Then they see to multiply the area of that side by two.
After that, I tell them to color the front blue, helping them to see the base and height of the front, and I ask, "Are there any other sides that are the same?" The students see that the back of the prism is the same and multiply the area of the front by two.
Finally, the students see that they need to add all three totals together to get the entire surface area of the rectangular prism. When given the visual pictures and guiding questions, students are able to accurately visualize and draw the surfaces of a three dimensional figure.
After completing one problem, the students were very excited and wanted to continue on their own saying, "We got this!" As I walked around the room, students were able to complete the rest of the problems on their own with great success and confidence. Most students continued to use the coloring method to finish their work on their own. The student success rate for the class was 90 percent for surface area of a rectangular prism.
In Enhancing Math training, we look at potential barriers to student learning and use students’ strengths to tackle math problems. If I had not had the student visually color code the parts and give prompts to the problem, the students would have struggled finding the correct base and height for each given part.
Many times, as teachers we misdiagnose the problem when we do not consider all the barriers, which means getting inside students’ minds to see what they are thinking. We need to stop, make the time, and see what they do know!
After attending Enhancing Mathematics Instruction for Students with Learning Difficulties, my co-teaching partner and I immediately began working on a plan to make our math classroom accessible for all. It's about equity, not equality. Our students come to us with a variety of different backgrounds and learning needs. All of our students have the right to an education where they can succeed.
We first looked at the six different models of co-teaching and realized we needed to make some changes. We felt just one model alone did not work all the time. We began to incorporate the different models when we saw them best fit for students. For example, we felt our students would best benefit from alternate teaching during our solving systems of equations unit. During “out learning,” there were a group of students who needed extensive intervention. This model allowed one of us to facilitate our performance task while the other helped students reach a basic understanding. Our collaboration has been strengthened in a way where each of our students have more of an opportunity to have their needs met.
What stood out to us most in our learning about accessibility was understanding the equal sign. When we teach solving equations, we assume students understand the language we use such as "the left and right side of the equation." This assumption is not always correct. Students with language barriers may struggle. Ways we might help students are to use algebra tiles as manipulatives where students have a mat and can visually and tactically move the tiles from side to side to balance them.
Not only did we learn a variety of accessibility strategies that help students with specific needs, but we learned how to put them into action. We chose a student with special needs as our caseload study student. Once we determined what strategies would best help this student, we put the plan into action. We chose a student who attended multiple schools in her younger years. She was missing multiple basic grade-level skills. She struggled with short- and long-term recollection of basic facts. In planning how to help assess this student’s skills in a more accessible way, we provided her with a resource sheet and allowed her to use a graphic organizer. The graphic organizer displayed the rules of exponents. This helped her succeed in proving her mastery of these skills.
Making math more accessible to our students means we need to provide multiple ways in which our students can interact with math. We plan to incorporate more manipulatives (such as Algebra tiles and virtual manipulatives), incorporate other models of collaborative teaching, help better build students understanding of the language of math, make our assessments more accessible, and help students to better think aloud.
Kendall Root & Elaine Mahabir
One of the largest barriers we face with our students is apathy. A lot of the strategies we utilize from Enhancing Math are designed to increase engagement and participation in an effort to overcome this barrier. We work hard to find opportunities for student discourse. We also try to limit the amount of time our students spend in their seats. While these strategies have other benefits, the biggest benefit we see is students who are engaging with course content when they would otherwise be trying to hide at their desks thinking about anything other than math.
We’ve made a couple of changes to our teaching as a direct result of participation in Enhancing Math. The most regular tool we’ve started to use in our class is vertical, non-permanent surfaces*. This was a strategy we first learned of in the course. We’ve found it to be very effective. Another strategy we’ve used is moving from the concrete to the abstract. For example, we completely rewrote our linear functions unit (in Algebra 1) to start with an exploration of arithmetic sequences using manipulatives and then progressed to the concepts of rate of change and initial value. Once students had these down, we were easily able to launch into discussions about lines, slope, and intercepts. The students found the concepts much more intuitive. While the book introduced lines first, and considered arithmetic sequences as an example of linear functions, we found that reversing the order made the material easier for students to understand.
When we went through the Enhancing Math course, we were teaching freshmen in their first-year Algebra course. This year, we have moved to teaching juniors in Algebra II. This change has caused us to consider the concepts and strategies we learned in the course, applying them to new content and different students. To this end, we are revisiting the engagement and content delivery strategies we learned from Enhancing Math and trying to apply them in this new setting.
*The use of vertical, non-permanent surfaces was added to the Enhancing Math course by their instructors who had been to a Building Thinking Classrooms training. For more information, read Peter Lilijedahl’s Building Thinking Classrooms:Conditions for Problems Solving article.
In what ways have you implemented Enhancing Mathematics in your job?
On a big picture scale, I have thought more about the demands of the math tasks and about the specific challenges my target student brings to class. Specific strategies I have used include multiplication charts in the past (but I narrowed down the chart to include only the facts that the student doesn't know), a template (flowchart) for a complicated process, quick checks of concepts to focus instruction, and have students explain their reasoning or their problem solving process to other students, and a new way of checking for understanding.
Describe one implementation challenge and how you overcame it or are working to overcome it.
The process of factoring quadratic expressions includes a lot of decision making, so I decided to create a template with the students. The process of creating the flowchart did not go well, and students left feeling more confused about both flowcharts and factoring! This was due to my inexperience in introducing flowcharts and also because I had not created one myself beforehand! I worked after class, using what we had created as a start, and had the students use it with me during subsequent classes. I told them they would be able to use it on tests, and interest in using it increased.
Describe one implementation success or highlight.
When the class went over the factoring test, I had a different student talk us through the flowchart, reading and answering each question and doing the specified process/step for each box. As we did this, I realized that in our earlier work using the chart, I had been leading the students. They had followed along, but had not really been answering the questions themselves in that time. Going over the test in this way had a number of benefits.The students really started to understand how the flowchart could be helpful. The lead student had to figure out how to answer each question and justify his or her responses. The other students got to hear the student's answers and reasoning. It highlighted that some students are still struggling with parts of the process. My target student did not want to lead the class through one of the problems, as he hadn’t done much on the test. After seeing other students do it, though, he was able to work through his problem just fine. This student has begun contributing more in class as the year has progressed and has had successes.
One recent positive experience related to the factoring flowchart: Another student, who is very resistant to using new strategies, recently asked if the students would be allowed to use the factoring flowchart on the pre-Algebra final exam. She found value in this new strategy!
What is your next step for implementing Enhancing Math?
This summer I plan to spend more time understanding the task demands of the content I teach and getting to know strategies I can implement to enhance instruction in the future. In particular, I may create templates, flowcharts, vocabulary visuals, graphic organizers, and other tools that I may need going forward.
What Others are Saying
We asked partnership sites to share their experience related to the training.