# Graph paper is beneficial for all math students!

When you are shopping for school supplies, think about an extra journal or package of three ring binder paper: quad lined paper is the first and most economical help for your struggling Math student. A small minority of students gets visually confused by the squares on the page. So it is always best to search for paper that is very lightly printed with ‘unobtrusive’ squares: the emphasis should be on what your student is writing and drawing not on the grid.

Neatly lining up calculations prevents errors. Quad paper can also help illustrate many concepts that are more complicated to explain in words but are easily seen on paper, such as multiplication, area and perimeter, for making graphs, and drawing congruent shapes, when showing transformations in size, slides, flips, and rotations and many more.

# Instant Fix

Graph paper works wonders both for calculations and understanding concepts

Although there is no instant fix for dyscalculia, there is an instant fix to maximize your credits for the math you can do! Using graph paper (quad paper, quadrille paper) will help you both with calculations and understanding concepts! It can benefit applying concepts as well as communicating your thoughts about the math problem.

Making your own drawing to represent the question makes it more fun and clarifies all types of math work, you can also add colors to your work!

Initially using graph paper might take a few minutes more, but it will save you time in the long run as well as increase your grade.

Starting in KG

Having a square to write in will make learning to form the numerals easier. KinderGartners and 1st graders need larger squares and should be taught to leave an empty square between each numeral and between lines to prevent the numerals ‘bleed into each other’.

Elementary

Depending upon fine motor skills / penmanship and the progress in math, somewhat smaller size squares will come in handy now.

Using graph paper makes it quick and easy to draw a number-line and to show your additions and subtractions as ‘jumps’, including the multiples of ten numbers as ‘stepping stones’. It can also easily illustrate multiplication as repeated addition and division as repeated subtraction. Extending the number-line to the left from zero later explains negative numbers.

Using the sides of the squares as a guide for drawing will enable you to easily visualize and compare areas and perimeters and help you solve problems about area and perimeter.

Writing each numeral in it’s own square will help you to keep the    digits in multiplications and long divisions aligned, so you do not mix up your units, tens, and hundreds, etc.

The grid will help you find lines of symmetry, complete symmetry drawings (like our free download activity) and make artistic patterns. When you cut out shapes or fold and cut the sides of a large square to make a star or snowflake, the printed squares will help you make strait cuts or folds. These activities are enhancing visual-spatial abilities, one of the components that contribute to being a mathematician.

Grid paper helps you to make the link between manipulatives and drawing models. By drawing a model the student shows he has internalized the concept that was illustrated by using manipulatives. The model shows his/her thinking and helps the teacher/tutor/parent to see if there are any remaining misconceptions, and if so which help is needed.

Middle and High school

Together with switching from wide ruled to college ruled paper you will now probably go for the small squares, allowing for more complex algebraic equations and working with geometric shapes.

In algebra it will keep those little ‘devilish’ minuses and other small math signs like powers etc. securely locked in your equations.

When you use the grid to draw a few points using an x- and y- axis on graph paper it is much easier to immediately ‘see’ the whole line, calculate the slope, and find the x and y intercepts.

Drawing your own graph before using your graphic calculator engages the brain more and forms a stronger memory than only looking at a ready made textbook or calculator graph.

Translations, rotations, reflections and dilations are easily understood making a drawing on graph paper as well as vectors.

Dysgraphia

Just getting into the habit of using a separate square for each numeral or sign will land you those extra points you deserve in algebra, in particular when you also have dysgraphia.

Do not back off because using graph paper in class or for homework means you need to copy the question on your paper before starting to solve it. The copying (or errors in it) shows your teacher or parent that you have (or have not) read the question correctly.

In a nutshell

Graph paper can be used in multiple ways, such as for many arithmetic, measurement, algebraic, geometric, and recreational math purposes. It is also beneficial to students without Math problems and should be available in each math class, it should be dubbed ‘Math paper’.

# Online Learning

Students are different in interest and abilities, some need more repetition and some need hardly any. An interactive, self paced option can offer advantages to all types of learners.

New technology is here to help us. As dedicated teachers and tutors, we would love to be the proverbial octopus that can tailor instruction to 8 different learners at the same time, but in reality it’s not working and our national math achievement is not something to be proud of. I am ready to accept the helping hand of technology as long as it is truly research based and the student is not left on his own when he stumbles, but gets immediate constructive feedback.

Internet programs are very convenient, can be used anytime in school and at home and will help bridge the long summer holiday, when many newly learned math abilities are forgotten.

Interactive Math learning software should have attractive, but not overpowering graphics: the attention is still on the mathematical content instead of on the comic or movie part. The progression from one concept to another should be clear and logical so students do not get frustrated with questions that are still too hard for them, and teachers/parents/tutors should be able monitor progress and still be able to step in where needed and last but not least: cheer the successes!

There are numerous good websites for classroom and individual use:

For example, the Illuminations website by the National Council of Teachers of Mathematics (NCTM) provides a great number of tried and proven lesson plans and online activities. For several of the activities you will also find a link to the activity on iPad. You can search on level and type of activity. An invaluable resource for each math teacher and tutor!

The NASA website has a list of engaging math activities and lesson plans per grade band. The website can be used both in the HS classroom and by individual students.

Have a look at the SHODOR.org website, a National Resource for Computational Science Education. They maintain a great website, under Activities and Lessons you will find Interactivate’showing students over 100 math concepts in a novel interactive way.

The additional advantage of websites for individual use is differentiation. Many programs just provide drills for arithmetic. We are now aware that speed drills for math facts can cause math anxiety. Luckily there is a growing number of websites that presents concepts in a novel and interactive way that is hard to do with text books alone.

A few suggestions from the long list of math oriented websites that can be used for individual learning for younger students with dyscalculia are the Number Race and the Number Sense website. Websites that offer all levels are the Khan Academy and the BBC ‘Bitesize’ website. Middle and High school students can use Connected Math and SchoolYourself.org. Also check out the Mind Research Institute (you can try several activities for free).

The Number Race was developed for ages 4 – 8 developed by Dr. A.J. Wilson in collaboration with Dr. S. Dehaene from the Unicog Institute (INSERM) in Paris and is available for free from SourceForge in several languages. It uses recent research in neuroscience to specifically target youngsters with dyscalculia.

The London Knowledge Lab developed the Number-Sense website, with five highly acclaimed activities for students with dyscalculia: Learn Time, Number-bonds, Dots2Track, Number-line, and Chicks Test. (This website uses Flash, so not suitable for Apple Pc’s). The Number-bonds is a tetris like game where you combine the well known Cuisenaire rods to make ten. Different levels ultimately leading to the abstract notation. You will also find the contact to the Low Numeracy Network and can sign up for regular updates and new information on the topic.

The Khan Academy has super clear lessons presented on a ‘blackboard’ with oral explanations in all levels, from simple addition to multivariate calculus. There is a built-in progression, that takes students on a journey through the areas of Math and a progress monitoring system. Teachers can connect with their students using the Coach function. Unlike many educational computer programs that span only elementary, middle school or high school level, this one spans many levels, so the student on the verge of one of these school borders does not need to wait till graduation day…

Word problems area a huge stumble block for many students with dyscalculia, in particular those who also have dyslexia. Thinking Blocks, by Math Playground, is based on the Singapore way of using strips to represent numbers. Students are guided step by step through the process of problem solving. You can choose from word problems focusing on Addition-Subtraction, Multiplication-Division, Fractions, and Ratio-Proportion. It is available as PC game and on iPad.

The BBC Bitesize website has a wealth of interactive and fun Math games and learning activities. The games and activities are grouped according to level: KS stands for Key Stage. KS 1 and 2 are for elementary, KS 3 and 4 for middle and high school, and GCSE stands for General Certificate of Secondary Education (UK). Most of the lessons have three parts 1. Revise: the explanation how it works,  2. Activity: using novel ways to learn the content, with immediate corrections, and 3. Test: after doing the test, you can print your answers with guidance for correcting answers.

The Connected Mathematics Project for Middle School at the University of Michigan offers a lot of information, not only math content for students, but also a section with in depth explanations for parents, who like to help their teenagers and interactive resources, including online math games using Java: Connected Math

A newer website for Middle and High School is SchoolYourself.org It offers great oral explanations and clear visuals. Students are asked to participate in various points during the explanations and hints are provided to go to videos where you can brush up the required basic knowledge.

Check out the Mind Research Institute, a neuroscience and education-based non profit corporation: www.mindresearch.net. They developed a math game where students help penguin Jiji to bridge a gap using mathematical concepts: the kids love it and it is effective too!!

Happy calculations!