Numeracy
Mathematics is often an area of difficulty for individuals with learning challenges. A school psychology assessment will outline recommendations to teach to a student’s strengths within the cognitive profile, while also growing or supporting weaknesses.
Math can be challenging for people with any learning challenges because it involves all aspects of cognition; Verbal Comprehension (librarian), Fluid Reasoning (detective), Visual-Spatial (the architect), and the office staff (processing speed and working memory).
The suggestions below are general.
For individuals with a weaker librarian (Verbal Comprehension Index):
An individual with a weaker librarian (verbal comprehension), may be able to understand big picture mathematical concepts. However, he or she may get tangled in the language of word problems, may lack the abstract symbolic memory for algebra, or the ability to follow the step-by-step processes required in long division and other math processes.
Recommendations
Manipulatives, visuals, and guided discovery!
A Guided Discovery method might prove positive. This is a big picture/question down approach. It de-emphasizes the concept of ‘one right answer’ (also positive for anyone with math anxiety!).
New concepts and procedures should be introduced in, or with a connection to, the practical situations in which they will be applied.
Use Manipulatives whenever possible! For individuals with challenges within the Verbal Comprehension Index, the abstraction of language and symbols can prove challenging. Manipulatives help to make that connection to real life, making math concepts more understandable. Indeed, manipulatives are useful for all students (and grades). There are many commercial versions available but everyday items like egg cartons and sugar cubes can also be used.
Possible activities:
Pizza Fractions Create an instruction sheet with five different fractions on each. Students should create a pizza (using construction paper, or even the inside of an empty pizza box) and decorate the toppings to represent each fraction.
Weighing In Line up a variety of fruits and veggies, such as oranges, bananas, cucumbers, kiwis, tomatoes, and bell peppers. Ask students to predict the order of the foods from lightest to heaviest. Use a balance scale to test their predictions, then rearrange the foods according to their actual weights.
It is important to support the abstract with something concrete.
Concrete: students use three-dimensional objects to represent math problems.
Manipulative:
Pictorial: students use pictures to represent math problems
Abstract: students represent the problem using numerical symbols.
A good website to try: https://tapintoteenminds.com/concreteness-fading/
Systematic growth and support for math vocabulary and verbal reasoning!
Math requires connections within and between real quantities, math language, and written symbols. This is difficult for many children. It is particularly challenging for individuals with challenges with the Verbal Comprehension Index (that weaker librarian). Difficulty in language processing will make those connections hard to initiate and maintain. This also makes it hard for these individuals to retrieve facts from semantics-based, long-term memory.
Explicitly and repeatedly link math concepts and vocabulary.
Have the student develop a written file of math vocabulary. On a 3 x 5 index card, have them write the word, the definition, and make a picture to illustrate meaning. (Could be that real-life connection or photograph) Have the student file the words alphabetically. Provide opportunities for review.
As well as teaching specific terms like those above, students with language processing issues may know how to divide 15 by 3, but fail to connect it to the other phrases we use like ‘how many times does 3 go into 15’ etc. Make certain that all the terms and phrases used to mean a certain operation are explicitly taught.
Provide a brain-friendly cue to help students to remember math vocab. Difficulty in language processing will make those connections hard to initiate and maintain. This also makes it hard for these individuals to retrieve facts from semantics-based, long-term memory.
Examples:
The denominator is down.
Mnemonics My Dear Aunt Sally Says which helps us to remember the order of operations Multiply, Divide, Add, and Subtract.
4. Always provide an example (exemplar).
5. Using a Math Word Map which includes a visual and connects the concept to real-life applications might prove helpful. https://www.readingrockets.org/strategies/word_maps
6. Provide a hands-on method for reinforcing symbols and their meaning! For example, the illustrations below reinforce, the ‘greater than’, ‘less than’ and ‘equal’ signs.
8. Be aware of the linguistic complexity of all word problems that the student is expected to answer. Rephrase or rewrite the problems as needed.
Model self-talk and Verbal Reasoning!
In addition to visuals, explicit instruction is required. Talking ourselves through a problem can prove helpful for any of us. Children with language and verbal reasoning challenges will need to have this modeled.
Provide Strategy/Steps:
For example:
R Read the problem for understanding
I I know statement- list all the information given in the problem
D Draw a picture
G Goal statement declared in writing I want to know…
E Equation development
S Solve the equation
For students with a weaker detective (Fluid Reasoning)
These students may lack the ability to intuitively understand math but will be able to use verbal reasoning to work through the problems. If this individual has a strong working memory, they may do quite well in the early years and math challenges may not be suspected. However, as math becomes harder, or more abstract, challenges will occur.
Recommendations
Use fiction: A student with a weaker detective, may have a strong librarian and love stories and words. . Introducing a new math concept by using a book or story might be a great way to introduce vocabulary, connect math concepts to real life, and decrease anxiety.
Carol Fullerton has some excellent Math resources at: https://mindfull.wordpress.com
Carol emphasizes the idea of mathematical questions, processes, and discussions. Again this de-emphasizes that limiting concept of that ‘one right answer’.
Books for older students include:
The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger
The Man Who Counted: A Collection of Mathematical Adventures by Malba Taha
The Math Inspectors by Daniel Kennedy
Again manipulatives are very important and should be used even in more advanced grades. However, research says that manipulatives and visuals alone do not increase outcomes. Students need to be explicitly taught. This is particularly true for individuals lacking that intuitive math comprehension.
Provide explicit problem-solving in a step-by-step manner. Model teacher verbalization and encourage the student to also coach herself through the steps.
Write or state a verbal story from a math sentence in order to develop a better contextual understanding of numbers.
For students with challenges in the office staff? (processing speed and working memory)
Processing speed (the filing clerk):
Processing speed is like the filing clerk. It is involved in finding, storing, and processing information. Some of us are fast and some take longer to find and process information. Being slower does not mean that an individual has any lack of intelligence. Ellen Braaten’s book, Bright Kids Who Can’t Keep Up discusses students who have a slower processing speed.
Math in the early years can feel exhausting for children with a slower processing speed. Simple addition and subtraction are tiring and the individual may feel as though they are always falling behind their peers.
We used to be very derogatory about ‘drill and kill’ – the practice of constantly going through times tables and number bonds. However, we have learned that this helps automaticity and automaticity is important.
Indeed, having this automaticity of numeric facts is particularly important for individuals with slower processing. It will lift the cognitive load as they have to understand more complex mathematical concepts.
However, it will be even harder and more effortful to acquire.
I see it as a balance and every school team, student and parent will need to find the right fit. I always advocate is to find ways to make this practice fun. Card games, checkers, snakes and ladders, or any other board game can prove a fun way to develop automaticity. Rolling two dice and jumping is also an active, brain-break activity. There are also many online math games that might prove helpful.
Meanwhile, as the individual moves into higher grades with more complex math, ensure that the necessary supports, like calculators, are provided. It is vital that the student does not miss the major concepts in new material because all cognition is focused on addition etc.
Working Memory (the office manager)
Working memory is a form of memory but juggling is involved. The brain has to hold onto the information in order to make use of it (like remembering a phone number, before we had auto-dial) Working memory processes are involved in even the simplest mathematical tasks. For example, even to compare numbers, children must hold onto the meaning of the symbols (numbers) to identify the largest number.
More complex mathematical tasks take a greater toll on working memory.
Mathematical word problems are particularly arduous. Children have to read and integrate this new information with the information already present in memory. In addition, it is necessary to pay attention to and remember only the relevant information. Moreover, in order to reach the solution, it is necessary to create the correct mental representation of the problem, a process that requires visuospatial memory skills.
Recommendations:
Manipulatives! Moving and holding information can extend working memory sufficiently to allow information to reach long-term storage. Besides, they make math more meaningful and we all remember better when things are meaningful.
Get Physical with the information!. Let the students move around, use hands-on material, and put information on file cards so they can be manipulated. e.g. Chalk number line.
Develop routines! If something is routine, we don’t need to actively think about it! Routines decrease stress and lower cognitive load!
Make routines VISUAL.
Sticky notes can be a great way to quickly make instructions visual
Explicitly teach organizational skills
Brain breaks are important for everyone but doubly important for individuals with distractions and working memory issues. They help to focus and also provide time for working memory to get solidified into long-term memory.
Provide Example (exemplar)!
Use colour to highlight key facts: Ask students to highlight key mathematics operations or issues before beginning work on mathematics problems. For instance, highlight each time the signs change. This will help reduce careless errors.
Use numbered points for any sequence of instructions. Underline or highlight important information. Double-space written instructions. Write directions in a different colour on worksheets.
Keep oral language simple. If you repeat the instruction – use the same language.
Ask the student to paraphrase instruction – this checks for the accuracy of understanding and also provides verbal rehearsal.
Make instructions multisensory. Make certain instructions remain available in a visual form for students.
Formulas, definitions, and key mathematics facts should be available. Students can create one-page review summaries of notes, formulas, or definitions.
Additional Math Resources
Peter Liljedahl has a series of numeracy tasks that are high ceiling, low floor, and open-ended. These offer multiple points of entry for students and enable the teacher to have just one task that is accessible for all. Peter Liljedahl » Numeracy Tasks
Nikki Lineham has a website (Educating Now) that offers excellent teaching strategies that are inclusive, differentiated and work through concrete, pictorial, and abstract as concepts are being taught. s. Educating Now – The Art of Teaching Math
Dan Finkel’s site, Math for Love, has some great ideas.
Check out Jo Boeler and YouCubed.
Mathhelp.com has playlists by topic on youtube.
Carol Fullerton has resources for teachers, as well as, parents.
YummyMath.com has resources for everyone, students, teachers, and parents.
GetTheMath.org (uses short videos to show how algebra is used in the real world).
TedED (Math in Real Life) Check out all of TED-Ed's math videos
RealWorldMath.org
Eddie Woo, You can view his videos here and his printable worksheets here.
The Khan Academyacademy.org/.
The National Library of Virtual Manipulatives contains interactive, Web-based manipulatives and tutorials for mathematics instruction in Grades K-12. http://nlvm.usu.edu/en/nav/vlibrary.html
Just Math Tutorials is home to hundreds of math tutorial videos on a variety of topics ranging from algebra to calculus. http://patrickjmt.com/