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.

I've found that a multiplication table can be a great tool to help students visualize the relationship between squares, their sides and areas, square numbers, and square roots.

Here are some activities to consider:

Here are some activities to consider:

- As an instructional lesson, demonstrate how to draw a square on a blank multiplication table. Articulate and show the relationships between the sides and area of the square, how those values can be represented on the multiplication table, and then how that visual information can be used to find the square root of the square number. Use this visual representation to show the relationship between the square root of the number and the answer (e.g., "the square root of 16" and "4"; the square root of the area of a square gives you the length of the side...). There are lots of ways to play with this -- have fun!
- Have students create their own square root multiplication tables using a blank multiplication table (these are available to download on my
). Several of my students have really enjoyed doing this, both because they can be creative, and because it makes the concepts super concrete and accessible, which makes them feel knowledgeable and in charge.**website**

- For students who have a very significant difficulty processing visual information and may become overwhelmed by too much visual information, consider having them use separate multiplication tables for each square root instead of drawing them all on one page.
- Once students can see the basic relationship, this table can also be used to approximate and visualize square roots of non-square numbers (e.g., the square root of 20 is between 4 and 5) by creating or visualizing squares in between the perfect squares and/or using the line created by the perfect square (roots) as a diagonal number line. Note that some students may want to find the square root of 20 by finding the number 20 on the multiplication table. This becomes a great teaching opportunity -- if the numbers on the multiplication table can be used to find the exact square root of 25, why isn't the same true for finding the square root of 20?
- Give students time to practice using their multiplication table to find square roots as an independent exercise to build skill and understanding, then have them practice using their tables while completing math problems that require them to find square roots (applications!).
- Consider allowing students to use their visual aid during subsequent assignments and assessments pertaining to square roots, especially if they have known difficulties processing visual information. Some of my students have put their square root tables in their math notebooks in addition to a regular multiplication table for easy access during assignments and assessments.

Enjoy!

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.

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.

Expanded function tables are a great tool for scaffolding and supporting students' thinking as they plug-in different values of "x" to solve for "y." An expanded table gives students a space to write down their thinking for each step of the process. This is beneficial because it takes away the demand of having to hold information in their working memory, which decreases their cognitive load and gives their brain more space to think. It also helps students to keep track of where they are in the problem solving process. For students who have a hard time remembering the sequence of steps, an expanded table also provides a guide to what step comes next. And for students with calculation difficulties, a 2-step expanded table provides space to write out any calculations that they need to solve on paper (i.e., "Side Math"), and it allows students to focus their whole attention on doing this calculation because the table is acting as their working memory and is also saving their place in the problem solving process for them.

Here are some ways to integrate expanded function tables into your teaching:

- Introduce and model using an expanded function table during direct instruction.
- Teach students how to determine when it is helpful to use an expanded table.
- Teach students how to draw expanded function tables, and teach them where they should do this if their classwork/homework is in a workbook or on a worksheet with limited space.
- Make hard copies of expanded function tables available for your students to use as needed. You can download expanded function tables from the
page of my website.**Math Aids**

Students with math fact difficulties cannot easily bring to mind the multiples of 6 and multiples of 8 and then hold them in their brain while comparing them in order to identify the lowest common multiple (24). Using a multiplication table, students can identify the multiples of 6 in one column and the multiples of 8 in another column, and then look for common multiples. Students can use two parallel columns, or one horizontal column and one vertical column, as shown below. It sometimes helps to use a straight edge to help students track the columns for easier visual scanning. |

Students with math fact difficulties cannot easily bring to mind the factors of 9 and the factors of 24 and then hold them in their brain while comparing them in order to identify common factors and then determine which one is the greatest (in this case, 3 is the only one). Using a multiplication table, students can look for numbers (i.e., factors) that have both 9 and 24 as their multiples. |

Students with math fact difficulties cannot easily bring to mind the multiples of 3, determine which of those multiples is closet to but still less than 17 (i.e., 15), and then determine what the quotient is when you divide that multiple by 3 (or in my students’ heads, how many 3’s does it take to get to 15). Using a multiplication table, students can easily identify the multiples of 3 and quickly see that 15 is the greatest multiple that is smaller than 17, which means that 3 can go into 17 5 times. |

Students with math fact difficulties cannot easily bring to mind the factors of 18 and hold them in their mind while determining which factor pairs add to 11. Using a multiplication table, students can easily identify the different factor pairs that multiply to 18, then add each of these factor pairs together to determine which ones add to 11. |

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:

1

Students with math learning disabilities often lack automaticity of basic math facts that makes it harder for them to do more complicated math.

2

Students with math learning disabilities almost always have brain-based difficulties processing visual information, which makes it harder for them to “see,” remember, and “do” math as other students do.

Students with math learning disabilities almost always have brain-based difficulties processing visual information, which makes it harder for them to “see,” remember, and “do” math as other students do.

3

Students with math learning disabilities often have strong language and verbal reasoning skills and may excel in other academic subjects, which may make their difficulties in math look like a lack of effort rather than a brain-based difficulty.

Students with math learning disabilities often have strong language and verbal reasoning skills and may excel in other academic subjects, which may make their difficulties in math look like a lack of effort rather than a brain-based difficulty.

4

Students with math learning disabilities make mistakes that look like “careless errors” that can easily be corrected, when instead they are a manifestation of the disability and represent a significant area of difficulty.

Students with math learning disabilities make mistakes that look like “careless errors” that can easily be corrected, when instead they are a manifestation of the disability and represent a significant area of difficulty.

5

Students with math learning disabilities tend to have difficulty thinking flexibly about math problems, and may struggle to know what to do when they are taught multiple ways to solve a problem.

Students with math learning disabilities tend to have difficulty thinking flexibly about math problems, and may struggle to know what to do when they are taught multiple ways to solve a problem.

6

Students with math learning disabilities may demonstrate mastery of math concepts or skills in isolation with repeated practice, but will often have difficulty using and applying concepts, facts, and procedures when the problems are out of context or are more complex.

Students with math learning disabilities may demonstrate mastery of math concepts or skills in isolation with repeated practice, but will often have difficulty using and applying concepts, facts, and procedures when the problems are out of context or are more complex.

7

Students with math learning disabilities tend to spend and exert more time, focus, and effort to learn or do the same amount of math as other students.

Students with math learning disabilities tend to spend and exert more time, focus, and effort to learn or do the same amount of math as other students.

8

Students with math learning disabilities have to exert more brain power in order to do the same math as other students, and can very quickly “max out” their cognitive load and become overwhelmed.

Students with math learning disabilities have to exert more brain power in order to do the same math as other students, and can very quickly “max out” their cognitive load and become overwhelmed.

9

Students with math learning disabilities may get good math test scores and grades that suggest they are doing just fine in class, when really, they are often working extremely hard and sometimes at a pace that is excessive and unsustainable in order to achieve and learn.

Students with math learning disabilities may get good math test scores and grades that suggest they are doing just fine in class, when really, they are often working extremely hard and sometimes at a pace that is excessive and unsustainable in order to achieve and learn.

10

Students with math learning disabilities often experience distressing emotions about their math learning experiences, including sadness, anxiety, fear, confusion, anger, frustration, despair, embarrassment, and lack of self-confidence.

Students with math learning disabilities often experience distressing emotions about their math learning experiences, including sadness, anxiety, fear, confusion, anger, frustration, despair, embarrassment, and lack of self-confidence.

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.

Students have different internal experiences and thoughts that can lead them to do math too quickly. I encourage you to think about your specific students and what their reasons might be. Here are some of the things that might be going on inside your students' heads and bodies:

*The faster I do this, the faster I can be done!**Smart kids do math quickly, so I should do math quickly too – that makes me smart, right?**My hand impulsively goes faster than my brain can think and I can’t slow it down*(this may be especially challenging for students with executive-functioning-related difficulties, such as students with ADHD or Autism Spectrum Disorder).*I’m anxious about doing this math, and this speed matches the pace of my nervous energy.**This is how fast the teacher did the problem in class, so that is the speed I need to go.*

The best way to help a struggling math student is to help them to develop skills to address their difficulties. In the case of the speedy math student, the skill we want to teach is:

Or, as I say lovingly to my students once they understand the skill we are trying to build:

- Teach students to
**stop and think**before they begin a problem. Some students will need to be invited to put down their pencil so that they are not tempted to write before they are ready. Some students will need guidance on what to think about as they begin to solve a problem (more on this in a future blog!). - Teach students to
**say their thinking aloud**as they solve math problems, and allow their words to guide the pace of their problem solving. This may be talking at a conversational voice if the student is alone or working one-on-one with a teacher, or whispering or talking softly if the student is in a classroom setting. Some of my students have arranged with their teachers to take their tests in a corner of the room where they can talk aloud without drawing attention to themselves or disturbing other students. - Help students to
**get centered**before they begin doing math. This might be explicitly inviting students to take a minute to breathe or guiding students through a short mindfulness practice, or using your own energy, voice, and teaching environment to guide students into a calm, focused space (both physically and mentally) to do math. - Teach students
**appropriate expectations**for doing math. Many students have learned that they should be able to solve math problems quickly and easily in order to be good at math. Students need to learn that solving math problems takes time (depending on the person and the type and level of the problem, it can take math thinkers hours, days, or even years to solve a problem!), and that being fast at math does not equal being good at math and vice versa (ala the symmetric property :) ).

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.

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:

Developing early math skills does not mean giving preschoolers math workbooks or written arithmetic problems to solve (although some children may enjoy these!). In preschool, opportunities to build pre-math understandings are everywhere, and they can be fun, creative, and exploratory, and and they can be built around children's own interests, insights, and experiences.

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:

Our discussion highlighted an important distinctions in the ways that mathematics learning is viewed. On the one hand (and the left side of the white board above) is the notion that learning mathematics is about memorizing and applying procedures and formulas to calculate and solve math problems. From this perspective, students do math as a means to an ends. Their goals are to get the correct answers and to get a good grade. The math teachers associated this way of thinking about learning math with many negative emotions and feelings such as anxiety and torture.

On the other hand (and the right side of the white board above), the math teachers also described a view of mathematics learning that is expansive and meaningful. Learning math can be about engaging in mathematical thinking, making connections, and using logic, symbols, and a mathematical language to solve nontrivial problems. Several teachers shared personal reasons why learning mathematics can be interesting, fun, and important. One teacher shared her appreciation for the beauty in math that she wants all of her students to see and experience. Another teacher shared his love for engaging in complex problems with required him to struggle and think creatively over several days before triumphantly uncovering a solution. The teachers shared many positive emotions and feelings associated with this view of mathematics learning such as excitement and appreciation.

As one of my graduate school mentors, Dr. Alan Schoenfeld, wrote,

On the other hand (and the right side of the white board above), the math teachers also described a view of mathematics learning that is expansive and meaningful. Learning math can be about engaging in mathematical thinking, making connections, and using logic, symbols, and a mathematical language to solve nontrivial problems. Several teachers shared personal reasons why learning mathematics can be interesting, fun, and important. One teacher shared her appreciation for the beauty in math that she wants all of her students to see and experience. Another teacher shared his love for engaging in complex problems with required him to struggle and think creatively over several days before triumphantly uncovering a solution. The teachers shared many positive emotions and feelings associated with this view of mathematics learning such as excitement and appreciation.

As one of my graduate school mentors, Dr. Alan Schoenfeld, wrote,

"Learning to think mathematically means developing a mathematical point of view and competence with the 'tools of the trade,' using the tools for mathematical sense-making."(Schoenfeld, 1992, p. 334) |

For parents and educators, it is important to recognize our personal views about mathematics learning and what it means to learn math, as these guide the ways that we work with children. Our own views and beliefs about math come out in the ways that we structure math lessons and activities, the ways that we model mathematical thinking and problem solving, the ways that we help with math homework, the ways that we respond when children are having difficulty with math. I would encourage anyone who works with a child learning math to consider what it means to learn math and what that child needs to continue developing as a mathematical thinker.

Reference: Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning. New York: Macmillan.

I'm very excited to be working with students around these areas this summer. If you have any questions or you think your student might benefit from Summer Therapeutic Math Tutoring, please do not hesitate to contact me!

• Accommodate different learning styles;

• Create a variety of testing environments;

• Design positive experiences in math classes;

• Avoid tying self-esteem to success with math;

• Use student surveys to determine attitudes toward math;

• Emphasize the fact that everyone makes mistakes in math;

• Make math relevant to dally life;

• Let students have input Into their own evaluations; and

• Emphasize the importance of original thinking.

l. Let your children know that you believe they can succeed at math.]]>

2. Be ready to talk to your children about math and to listen to what they are saying.

3. Be more concerned with the process of doing math homework than getting the correct answer.

4. Don't tell children how to solve math problems; ask questions and guide them through the process.

5. Practice estimation with children whenever possible, e.g. ,while shopping or on a trip.

6. Provide a special place for study that accommodates the child's learning style.

7. Encourage group study.

8. Expect homework to be completed.

9. Don't expect all homework to be easy, and don't rush your child.

10. Seek positive ways to support your child's teacher and school.

11. Ask the teacher for a mathematics course outline for the year.

12. Find time to occasionally sit in on your child's math class.

13. Try to better understand standardized test results and placement decisions.

14. Avoid drilling your child on math content or using drills as a punishment.

15. Model persistence and pleasure in demonstrating everyday use of 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:

**Processing skills**-- Sometimes when children have difficulty learning, there is an underlying deficit in some type of psychological processing such as auditory or visual processing. When a processing delay is present, it may be indicative of a learning disability. Processing-related mathematics difficulties are low in prevalance (about 5-10% of the school-aged population has a learning disability, and only 20% of these students have difficulties related to math). Most students exhibit sufficient processing abilities to learn math.**Math Skills**- One of the fundamental types of mathematical thinking involves being able to execute basic mathematical procedures. These skills range across developmental levels from completing math facts and solving long division problems in Elementary School, to simplifying equations and graphing data in Middle School, to applying algorithms and solving complex equations in High School and beyond.**Conceptual Understanding**-- Children's mathematics learning depends on their understanding of mathematical concepts. The National Council for Teachers in Mathematics outlines five content areas, which provide a nice framework for thinking about the types of concepts children work with in mathematics: number and operations, algebra, geometry, measurement, and data analysis and probability.**Metacognition and Problem Solving -**Metacognition refers to children's abilities to regulate their own thinking and learning. Students who are very successful in mathematics are able to think flexibly about math, and to combine their math skills and conceptual understandings to think critically and solve math problems. Metacognition allows children to understand what a problem is asking, to identify the types of concepts and procedures they should try to to solve a problem, and to think of what to do when they become stuck.

**Beliefs -**Research has shown that children hold a wide variety of beliefs about mathematics. These beliefs include beliefs about the subject of math (e.g., math has nothing to do with the real world), beliefs about math learning and problem solving (e.g., doing math means memorizing facts, there is only one right way to solve a math problem), and beliefs about oneself and others in relation to math (e.g., I'm not a math person, girls aren't good at math). Children's maladaptive beliefs about math and themselves as math students can greatly impede their ability to be successful in math.**Confidence -**One type of belief that is particularly important to children's success in mathematics is their self-efficacy, or self-confidence in their math abilities. Many children learn through school experiences and feedback from others that they are not good at math, and as a result they lose confidence in their abilities and begin to believe that they cannot do it. The good news is that through positive experiences with math, positive math self-efficacy and self-confidence can often be restored.**Anxiety**- Anxiety is often the emotional reaction that we see when working with children who are having extreme difficulty with math. Due to anxiety, children may avoid doing math homework, dislike going to school, or exhibit other symptoms of anxiety especially before or after math tests.

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.