Hi math teachers,

As you are continuing to adjust to teaching online and figuring out how to create effective learning experiences for all of your students, one concern I see coming up is how to support students with learning needs including those with __math learning disabilities__. When it comes to teaching students with math learning disabilities online, here is some good news: **virtual tools work extremely well to support students with visual processing challenges! **Here are 5 ways that you can support students with math learning disabilities while teaching online.

The first strategy I use when working with any student with a math learning challenge is to make everything bigger. Whether a student is having difficulty seeing and perceiving the visual information due to a developmental delay or their brain just isn't cooperating at the moment, they will have an easier time perceiving, processing, and manipulating visual information when it is larger.

Students with math learning disabilities have to work harder to process visual information than their peers, which means they take on a higher cognitive load. When there is extraneous visual material on the page or screen, their brains have to work harder to select the relevant visual information while filtering out the irrelevant material. This leaves little brain space to learn and do math. Excluding non-essential visual stimuli will lessen students' cognitive load and make it easier for them to perceive and focus on the relevant visual information and solve the math problem.

It can be extremely difficult for students with math learning disabilities to perceive the pieces within visual representations and to visualize their relationships. These students benefit from direct instruction using a combination of drawing and descriptive language to help them to see what they cannot see on their own and make the connection between these visual pieces and the mathematical concepts.

Whether solving a multi-step algebra problem or transforming a graph from a function, students with math learning disabilities struggle to fluidly manipulate visual information to obtain a desired result. This process requires them to not only perceive visual information, but to work with it flexibly while also keeping track of multiple steps. Students will do best when they are able to focus on one step or visual attribute at a time and have easy access to the visual information and tools they need as they move forward to solve the problem.

Students with visual processing challenges may experience difficulty organizing visual information. For these students, not only is the input of visual information challenging, but so is the output. This leads to visually unorganized papers, which contributes to more visual processing challenges. These students will benefit from direction instruction on what to write and where to write it.

RELATED RESOURCES:

Click __here__ to download centimeter graph paper and other free printable math tools.

Check out my post __"10 Things to know about students with math learning disabilities"__ to learn more about students with math learning disabilities and download my __free fact sheet__.

]]>To my math teacher friends and colleagues,

Happy (almost) spring break and congratulations on making it through your first few weeks of distance learning! I know it's been a challenging transition for many as we deal with the implications of the coronavirus pandemic both personally and professionally, and that most of us are still adjusting to teaching remotely.

Happy (almost) spring break and congratulations on making it through your first few weeks of distance learning! I know it's been a challenging transition for many as we deal with the implications of the coronavirus pandemic both personally and professionally, and that most of us are still adjusting to teaching remotely.

During this transition, the big questions that I hear coming up for math teachers are:

"What do I teach?"

"How do I teach?"

and

"How do I stay sane and healthy in this moment?"

"How do I teach?"

and

"How do I stay sane and healthy in this moment?"

In this post, I wanted to share 3 tips to help you address these questions and offer a bit of mental health support as you continue to navigate distance learning. I've also created a free Mental Health Resource Kit that includes reflection questions and a self-care guide to help you incorporate these tips into your teaching practices.

If you find any of these tips helpful, I'd love for you to share in the comments below what one thing you are interested in exploring or trying in your own practice. And if you know of others who might benefit from these tips, please pass this on to them!

The math teaching landscape has changed vastly over the last month. Just a few weeks ago, you were teaching in your classrooms where there were familiar structures and routines in place to support your teaching practices. You had lesson plans and worksheets and activities and assessments planned out. Your classroom was setup intentionally to support student learning. You had a vision of what you were teaching, where you need to get to by the end of the school year, and how you were going to get there.

Fast-forward to now (has it really only been a few weeks?!) and we are teaching in a an unfamiliar landscape filled with all kinds of constraints and unknowns. Teachers are juggling many new factors including changes in schedules and "class" lengths, alternative methods of delivering instruction, and the implications of not being physically in the same space as their students, just to name a few.

So, what do you teach and how do you teach it?

The natural reflexive strategy that I've seen many math teachers make is to take their existing classroom practices and to smoosh them into an online format. They are converting their in-person lessons to pre-recorded videos or syncronous online lessons, sending students their usual handouts and assignments as printable PDFs, having students use breakout rooms to engage in group activities, and even giving students real-time tests to complete and submit online.

If this smooshing method is working for you, awesome. However, if you have been attempting to squish your classroom teaching methods into an online space and it's feeling not quite right, that's completely understandable, too. For those of you who are feeling like squishing and smooshing isn't working for you, I want to offer you another way to think about how you design online learning experiences for your students.

When I help teachers to make instructional decisions, I often go back to this rule: *Form follows function**. *What you do or create depends on what you want to achieve. Your actions depend on your goals. In teaching terms, this means **what*** we teach and *how* we teach depends on what we want students to learn and experience*. Following this mindset allows us to think about teaching in a very different way. For any decision you are trying to make about your teaching, if you go back and ask yourself what goal you are trying to achieve, it will help you to determine the best action to meet your teaching goals and your students' needs.

Using this reframing, one of my math teachers figured out how to structure her twice a week 90-minute synchronous sessions to meet her goals of serving low, medium, and high achieving students and getting her finger on the pulse of how each kid is doing. Another teacher figured out how to balance teaching skills with having fun math learning experiences by stacking two instructional units that she would typically teach sequentially during the school year. Pre-pandemic, another math teacher re-structured the second half of her Algebra I curriculum (polynomials & quadratics) for a class of students with significant learning differences by considering her students challenges and needs, her content learning goals, and the constraints of her classroom. If this way of thinking appeals to you, it can be a practice that you take with you when you return to the classroom.

Whether you are wanting to make big changes to the way you are teaching remotely or just looking for small adjustments to make your teaching to feel more connected, here are three questions to consider that may help you to figure out how and what to teach as you move forward with your distance learning plans:

- What are my teaching goals?
- What are the limitations and affordances of my current teaching environment and situation?
- What are ways I can address my teaching goals given my remote teaching reality?

These questions can help you to answer big picture questions (e.g., how do I want to assess my students?) as well as day-to-day questions (e.g., how do I want to structure my 30 minute synchronous lesson with students today?) and week-to-week (e.g., how do I divide up this unit into multiple days/sessions?)

With at least another month of remote teaching in our future, now is a great time to give yourself the opportunity to be thoughtful and creative about your teaching, and to think differently about how you engage with distance learning.

So, seriously, this transition to distance learning happened so quickly! One day schools were open, and then all of a sudden, they were closed and teachers were expected to be up and running and serving students within a matter of days! I'm sure you did A LOT in very short amount of time, and I hope that you can take a moment to appreciate yourself for all that you have done to get to where you are right now!

In this fast shuffle, what I am seeing among math teachers who care about being good teachers and providing the best learning experiences they can for their students (I'm guessing since you're reading this that you likely fall into this category) is an understandable spike in stress and uncertainty over how they are teaching remotely, and a continued scramble to figure out how to be the best online math teacher they can be. Many of you are feeling pressure, whether it's coming from outside or within yourselves, to figure this distance learning thing out STAT!

To address this pressure, I'd like to get logical for a moment and refer back to some basic learning theory that we all know as teachers and that Helen Hayes put so succinctly:

The path from novice to expert is filled with learning experiences that help us to gain insights, skills, and mastery along the way. Just as we would never expect our math students to become experts in any new concept or skill right away, it would be unfair to expect ourselves to become masters of distance learning in such a short amount of time. So, following this model, the following has to be true: **you don't have to have it all figured out right now.**

An important way to support yourself in this moment is to give yourself the gift of time. Instead of trying to bypass the inevitable novice to expert path, I encourage you to focus on the next step on your path to developing your distance learning practice. Just work on that one step that is right in front of you. Of course you can think about what will follow in ways that will help you plan, but it may not be necessary to think too deeply about things that are a few steps down the road. Here are some questions you may want to consider:

- What is the one thing I need to do or figure out today?
- What is the most important thing for me to do right now?
- If I do _________ today, that will be enough.
- What can wait until later?

For those of us who are planners, I know there's a common fear that we don't have it all figured out, bad things will happen. Here are two things I try to remember when this doubt or uncertainty pops up. First, even if you don't have a perfect plan, you very likely have the skills and insight to react in the moment and make things work. Trust that you'll know what to do. Second, the magic of taking things one step at a time is that you could end up creating or discovering something amazing that you wouldn't have thought of if you tried to figure it out in your head all at once. When nothing is known, anything is possible!

Be patient. Be creative. Be open. You've got this. And if you're still feeling uncertain, Tip #3 will help you even more...

With everything going on in the world and in our personal and professional lives, now is the time to double down on self-care. In my work with math teachers over the past few weeks, three practices have stood out as being especially important right now. Whether you have an existing mindfulness or self-care practice or are interested in trying some things out, here are three things you can do to support yourself in your transition to distance learning:

DEVELOP A DAILY GROUNDING ROUTINE

When we are stressed, anxious, or scared, it's natural to look for things around us to make us feel calm and safe. We take actions to try to ensure our security, tell ourselves stories to convince ourselves that we are or will be okay, and use outside distractions to block our emotions from our conscious experiences. Sometimes these strategies works, but sometimes we still feel unsettled.

When we are stressed, anxious, or scared, it's natural to look for things around us to make us feel calm and safe. We take actions to try to ensure our security, tell ourselves stories to convince ourselves that we are or will be okay, and use outside distractions to block our emotions from our conscious experiences. Sometimes these strategies works, but sometimes we still feel unsettled.

In lieu of looking to external forces to give us a sense of safety and security, we can develop personal grounding practices that help us feel centered and grounded from within ourselves. Grounding practices bring our attention from the outside world into our internal physical and energetic experiences. They connect us to our breath and our bodies, and allow us to release the thoughts, emotions, and physical tension that we are holding. With intention and attention, we can connect in with images and sensations that help us to connect our bodies to the earth, to allow us to feel held and supported by the earth, and to help us find our internal sense of stability.

There are many ways you can find grounding through personal practice, including meditation, qigong, and yoga practices. Some of my favorites right now are root chakra meditations and warrior poses. I encourage you to experiment and find a practice that feels right to you. If you are interested in learning about the centering practices I do with teachers, please let me know in the comments below! Maybe we could do some Math Teacher Meditation!

CREATE AND MAINTAIN BOUNDARIES

Boundaries are so healthy for us, yet they can be really difficult to create and maintain. As you continue to practice social distancing and provide distance learning from your home, clear boundaries will help you to create the space and spaciousness you need as a teacher to serve your students, to protect you from outside stress and negative energies, and to maintain separation between your teacher life and your home life.

Boundaries are so healthy for us, yet they can be really difficult to create and maintain. As you continue to practice social distancing and provide distance learning from your home, clear boundaries will help you to create the space and spaciousness you need as a teacher to serve your students, to protect you from outside stress and negative energies, and to maintain separation between your teacher life and your home life.

I recommend considering three types of boundaries: physical boundaries, mental boundaries, and emotional boundaries. Physical boundaries include maintaining a dedicated space for distance learning in your home and being mindful about how you move in and out of your home and work spaces. Mental boundaries include being intentional about when your mind is in work-mode versus life-mode and creating practices that support this transition. And emotional boundaries include distancing yourself from triggers that agitate or upset you either, whether through personal choices or energetic work. I like to include boundary work in my guided meditations with teachers to help them establish an energetic separation between themselves and the people and thoughts that challenge them.

CONNECT WITH WHAT MATTERS

Everything we've talked about in this post has been about ways that you can be good to yourself and support yourself during times of transition. My final recommendation for you is to take time to connect in with your heart and the positive emotional experiences you have each day. Acknowledge your successes. Meditate on gratitude. Reflect on moments of joy and laughter. As you focus on what is going well, try not to over-intellectualize it and instead see if you can experience the feeling sensations of gratitude, compassion, and joy in your body.

As you consider developing a distance learning self-care practice, know that it doesn't have to be bi and you don't have to do everything at once. Similar to developing your online teaching practices one step at a time, a daily self-care practice can start small and grow over time (or stay small). Consider choosing one thing (three at most!) that you'd like to try, and give it a go for a week. When you are just beginning, 3 to 5 minutes of practice is a great start. I'd suggest choosing a time of day that works well for you (e.g., first thing in the morning, right before or after you teach, before you go to sleep), consciously setting an intention to practice, marking this time in your calendar, and committing your intention to a colleague or loved one. If you'd like to share your self-care practice intention in the comments below, I'd be honored to be a witness. :)

Everything we've talked about in this post has been about ways that you can be good to yourself and support yourself during times of transition. My final recommendation for you is to take time to connect in with your heart and the positive emotional experiences you have each day. Acknowledge your successes. Meditate on gratitude. Reflect on moments of joy and laughter. As you focus on what is going well, try not to over-intellectualize it and instead see if you can experience the feeling sensations of gratitude, compassion, and joy in your body.

As you consider developing a distance learning self-care practice, know that it doesn't have to be bi and you don't have to do everything at once. Similar to developing your online teaching practices one step at a time, a daily self-care practice can start small and grow over time (or stay small). Consider choosing one thing (three at most!) that you'd like to try, and give it a go for a week. When you are just beginning, 3 to 5 minutes of practice is a great start. I'd suggest choosing a time of day that works well for you (e.g., first thing in the morning, right before or after you teach, before you go to sleep), consciously setting an intention to practice, marking this time in your calendar, and committing your intention to a colleague or loved one. If you'd like to share your self-care practice intention in the comments below, I'd be honored to be a witness. :)

I hope that you've found an idea or two that you can take with you, reflect upon, and perhaps try. Please feel free to leave a comment below -- I'd love to know what your big takeaways are and/or what intentions you'd like to set for yourself. And, in case you still need it, here's one more chance to download my free Mental Health Resource Kit containing Reflection Questions and a Self-Care Guide to support your practice off-line.

Best regards,

Adena

]]>Adena

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**

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.

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. |

10_things_to_know_about_students_with_mld.docx.pdfThere 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 (__click here__ to download this fact sheet):

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.

Update: 4/2/2020

Here is a PDF version to print or share with other math teachers, students, parents, and educators!

Here is a PDF version to print or share with other math teachers, students, parents, and educators!

Best, Adena

]]>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 :) ).

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:

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.

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 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 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:

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!

In 1999, the National Association of Elementary School Principals (NAESP) published an article on **Addressing Math Anxiety**. The article defined mathematics anxiety as an **"irrational dread of mathematics that interferes with manipulating numbers and solving mathematical problems within a variety of everyday life and academic situations."** The NAESP article shared some wonderful recommendations for addressing math anxiety, which were originally written by the National Council of Teachers of Mathematics and other mathematics education scholars:

• 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.

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:

**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.