Square roots can be a confusing concept. They can be especially tricky for younger students (e.g., middle schoolers) who may not be developmentally ready for the concept, for students who have difficulty making connections between concrete and abstract concepts, and for students who have difficulty visualizing information.
As students begin to work with functions, sometimes as early as 6th grade, x/y function tables become a standard way to organize information and derive “y” for given values of “x.” Students are asked to "find" different values of "y" given different values of "x," and to fill-in these values on a table.
Completing a function table requires multiple steps of thinking, which students may or may not be able to complete effectively in their heads. Whereas advanced students and math teachers are able to select an "x" value, create a numerical equation using that "x" value, hold that numerical equation in their heads, and perform one or more calculations to determine the value of the numerical equation, younger students and students with math learning disabilities may become confused, lost, and/or make calculation mistakes trying to maintain the high cognitive demand that this task requires.
Skills & Strategies #2 - Four Non-Traditional Ways a Multiplication Table Can Support Mathematical Thinking
Some upper elementary, middle, and high school students do not know their basic multiplication facts. These students not only struggle with using multiplication to find products, but they also have difficulty with other types of math problems and mathematical thinking that require an understanding of the quantitative relationships between specific factors and products, and between specific dividends and quotients. A multiplication table has become an essential tool for my students who lack mastery in multiplication facts. Once students become familiar and comfortable with using a multiplication table (stay tuned for more on to strategies to achieve this), it becomes an ally and a tool that can support their mathematical thinking in ways that cannot be achieved with a calculator. Here are four examples of non-multiplication problems for which a multiplication table can help a student who struggles with his multiplication facts.
There is a lot that we do not know or fully understand about mathematics learning disabilities. Within academic research and professional practice, you will find multiple definitions and explanations of what makes up a math learning disability, some of which conflict. As an educational psychologist and mathematics learning specialist, I have worked closely with students to assess, diagnose, and remediate mathematics learning disabilities given the current state of the field and what I know to be true about learning disabilities and how kids learn math. Based on research, theory, and my own professional training and work with individual students, here is what I have come to know about students with mathematics learning disabilities:
Many students do math very quickly. For some, a quick pace might be appropriate if they are cognitively engaged and monitoring their thinking at this speed. However, for most students, doing math too quickly means rushing through problems without fully engaging their minds, impulsively writing down answers before they have thought them through, and completing problems on autopilot without taking thought as to what they are really doing. Speedy math often leads to under-learning the material, creating sloppy and often illegible work, and of course, making preventable mistakes.
This summer I had the pleasure of being interviewed by Anne Marie Morey, an Educational Therapist and founder of the Bay Tree Blog, about how to support students who are struggling with math. We had a wonderful conversation about the role of prior knowledge in classroom learning, addressing math anxiety, supporting students with executive functioning difficulties, understanding how visual processing delays can impact learning, and how to cultivate math problem solving skills. To listen to the interview or read Anne Marie's article about our talk, please visit the Bay Tree Blog.
I wanted to share a slide I made earlier this year for a lecture at Cal, which I am re-titling: "All the things that are going on inside your brain when you do math." This framework helps me to understand my students as math learners, to identify factors that may be making it difficult for them to learn math, and to develop ways to support their learning and development in math.
Last November, I gave a presentation to the parents at the Creative Play Center in Pleasant Hill, CA on how to support their children's early math development. In the talk, I drew from professional recommendations from the National Association for the Education of Young Children (NAEYC), the National Council of Teachers of Mathematics (NCTM), and my own experience and expertise as a math educator and former preschool teacher. Here are some of the key ideas we discussed.
When thinking about children's early mathematical development, parents and teachers should consider three areas of development: Conceptual Understanding, Mathematical Thinking, and Psychosocial Development. Here are are some descriptions about what each area looks like in early childhood:
This month I had the privilege of leading a presentation and discussion with the El Cerrito High School Mathematics Department on Creating Mathematics Learning Environments for Developing Mathematical Thinkers. I began the presentation by posing a question to the math teachers: What does it mean to learn math?
I asked the teachers to consider their own professional and personal thoughts and beliefs, as well as how they might have answered the question when they themselves were students. I also asked them to think about how their math students and their students' parents might answer the question. Take a look at what the teachers had to say:
I am please to announce that I will be offering Summer Therapeutic Math Tutoring for students who have had a hard time in the math course this past school year and would benefit from individual sessions this summer to address their academic and emotional challenges with math and gear up for their next math course this fall. Here are some things that students and I will tackle together this summer:
Re-Learning the Hard Stuff -- Students will identify the math areas that were confusing and/or did not make sense to them in their last math course, and have opportunities to understand and master these skills and concepts. Not only does this boost the knowledge and skill base they will have for their next math class, but it helps to teach students to process of identifying what and how things in math are difficult for them and what they can do to address those difficulties.
In 1999, the National Association of Elementary School Principals (NAESP) published an article on
Recommendations for Teachers
• Accommodate different learning styles;
The Complexities of Being a Math Student: Understanding How Students Learn Math and Why Some Struggle So Much
As a school psychologist and former mathematics instructor and department chair, I know that math can be hard for some students. When I talk with parents, they share their concerns about their children's failing math grades, declining self-confidence, and increasing anxiety. They share their struggles with helping their children at home, especially when the math material goes beyond what they remember from school, when a topic is being taught in a different way from what they learned, or when their attempts to help with homework turns into a fight. What I have consistently seen is that all parents want to know how to help their children become happy and successful in mathematics.
When a student is struggling in math, the first step to figuring out how to help him/her become a successful math student is to identify the reason or reasons why he/she is struggling. In other words, what is preventing your student from learning? Learning and doing math is an intricate process, and there are number of areas which could contribute to difficulties in math. Some of these areas are outlined below:
There are many ways to gather information and identify the nature of a child's difficulties in math and the interventions that may help to address the difficulties. Your child's teacher is often a good place to start. Teachers are often able to provide insights about a child's math skills and learning behaviors, and can often make recommendations about the areas in which a child needs extra practice or instruction. Also within the school setting, a Student Success Team or Student Study Team (SST) can further help to identify causes of learning difficulties and develop an action plan. If there is suspicion of a learning disability, parents have a right to request an assessment from their school district at no cost, or they may also choose to seek an assessment from an outside provider. Finally, some children may benefit from additional academic and/or psychological support from a private practitioner to identify and address their difficulties and help them to develop their mathematical thinking skills and self-confidence.