When my son was in preschool I asked him who was in his class. He replied, ” Natalie, she’s the yellow heart.” Children learn color before they learn words because it is easier to process.
This is found in children’s toys with color used to guide use of toys.
The obvious use of color in real life in traffic lights. The colors represent different concepts with red being used universally in the U.S. as representing stop. Color is used to partition an object into sections, as often seen in maps of areas with different sections. Think of how many highlighters are sold to college students to help them highlight key passages in textbooks.
The use of color help convey information, especially sections of a whole is an effective and easy to use instructional or support strategy.
The top two images below show my earliest attempts to use color. The student for whom this was used was a 7th grade student with asperger’s who tested in math and reading at a 1st grade level.
In lieu of referring to the “horizontal line” I can refer to the “yellow line” as in “find the yellow 3” for plotting the point (3, -2). Color, as in the aforementioned yellow heart, is much more intuitive for students, especially those with a disability.
Color was used for the same student to represent positive and negative numbers, first with concrete tokens then with colored numbers on paper.
More examples are shown below. Color helps a student focus on the different parts of an equation or different parts of a ruler.
Color can also help organize a room into different parts. Each color represented different courses I taught, e.g. green was used for algebra 2. The room is more organized because of the sections outlined in color. Consider how this can help a student with ADHD, autism or an executive functioning disorder.
This is an example of color coding (highlighting) to help make a calculus problem accessible. You don’t have to know calculus to see that the yellow sections (left and right of the 0) are going up while the green section is going down. Color coding breaks a whole into parts that are easier to see and understand – works in preschool all through calculus!
This example involves adding integers which is a major challenge for many students. There are two strategies present in the photo.
Color coding is an effective way to break down a concept into parts. Here red is used for negative numbers and yellow for positive. The numbers are written in red and yellow with colored pencils.
The chips are a concrete representation. Typically integers are only presented in number form and often with a rule similar to the one below. The strategy is to count out the appropriate number of red chips for the negative number and yellow chips for positive number. Each yellow chip cancels a red chip and what remains is the final answer. If there are two negative numbers then there is no canceling and the total number of red chips is computed (same with positive and yellow).
+ + = +
– – = –
+ – use the bigger number…
Rules are easy to forget or mix up especially when students learn the rules for multiplying integers. Concrete allows students to internalize the concept as opposed to memorize some abstract rule in isolation.
There are two strategies used in this example for rounding.
The number line is a different representation for a rounding situation. In CRA this is the representational or pictorial level. Typically students are taught to round by looking only at the numbers which is purely symbolic.
The color coding helps the students discriminate between the number being rounded and the choices for rounding. As I’ve written previously, color coding helps a student discern different parts of a concept.
I had a 7th grader who could not plot points. He has asperger’s and tested at a 1st grade level in math. Color coding the coordinate plane worked well for him.
As I have written previously color coding is an effective method to break a concept into smaller parts. Finding 5 on the yellow line is an easier direction to follow than finding 5 on the horizontal or x axis for many students.
The numbers in the ordered pairs are color coordinated with the axes colors. Students learn inductively that the first number in the ordered pair relates to the x or horizontal axis (yellow goes with yellow). Identifying the x-axis can be a subsequent step as the act of plotting the point is the immediate goal. In the photo you can see that the first few problems were color coded but eventually this support was faded and he continued to plot the points correctly.
Color coding is a form of scaffolding. It can be used to highlight specific parts of a diagram or problem or to help differentiate between different parts. Above is my first attempt at using color coding.
The 8th grade student simply could not interpret the table to answer the question – “explain the trend…”. I originally attempted to draw arrows from number to number to no avail. When I colored the two columns and asked him to tell me about the pink numbers then the yellow numbers he was able to interpret then answer the question.
The working memory for many students can be quite limited. Teachers often include many little details that are easy for us to process but can take much more effort by the student. It’s like a computer that has too many applications running at one time and slows down. Asking a student to look at the pink numbers can be much easier to comprehend than asking him to look at the column for year.
In PART 1 of Word Problems I went over my approach to teaching 1 and 2 step word problems involving addition and subtraction. In this post I cover multiplication, which is exponentially harder (pun intended – lay people see photo below).
As seen in PART 1, I color code the different parts of the problem:blue for the multiplication or division (rate), yellow for stand alone numbers, green for addition or subtraction, and orange for the unknown quantity.
To identify the multiplication or division parts, I focus on situations that involve groups of items, e.g. two cupcakes in every package or $5 in every lawn (for every lawn) as opposed to key words (as explained in PART 1). The students focus not on a single term such as “each” but on the situation. I use the groups of items as the structure for the equation, e.g., 5 x # lawns. The additional step in a two-step word problem can be connected within this structure, e.g., 12-9 in the top photo below.
Before working on the actual word problem handouts, I present the problems with a Google Jamboard to help flesh out the concept of multiplication as groups of items. Here is a link to the scaffolded handout.
After the Jamboard, I will use a scaffolded handout to help them unpack the structure. This is a scaffolded handout I use for 1 step multiplication word problems and the additional step, and show the additional step off to the side. This would be followed by problems on a typical worksheet as shown in excerpts above.
The problem below is a division problem. For division problems, I like to continue the focus on groups of items, in this case groups of wings. The difference is the number of items in a group is not given. This is a prompt for students to divide (which is how they will compute unit rate in the future). The division provides the main structure of the problem and the additional step can be attached, as is the case with 34+ 11 shown below. This way division is built on their prior knowledge of how to do word problems and they learn one additional step.
Several special education teachers responded in a poll indicating that the most difficult math topic to teach in elementary school is solving multi-step word problems. This happens to be a topic that is massively important and the first of several dominoes that will fall all the way through high school and beyond. One and two-step word problems are cited in the Common Core domain of Operations and Algebraic Thinking (images below) and the CCSS Coherence Map shows how these two standards lead to future algebraic thinking and algebra topics.
Before I get into what I call a conceptual approach to word problems, which I recommend, I will share that I am not in favor of the key word approach (image below). The major flaw, as I see it, involves how the brain stores or memorizes information. The key word approach is based on rote memorization. For many of the students with special needs, this is exactly what they do NOT need, more taxes on their working memory.
Here is the approach I use, with a focus on addition and subtraction first (followed by a forthcoming PART 2 blog post on multiplication and division). The handout used is from Math-aids.com.
I train students to highlight the quantities cited, along with any verbs. In the top photo is a legend for the elements of the word problem I highlight.
The yellow is used for quantities given in isolation. For example, Jason found 7 seashells but this was not presented after an preceding number.
In contrast, Fred found 6 seashells, which was in addition to what Jason found. Hence, Fred added to the number found and the green highlighting indicates this. Also note that the “+” is highlighted to indicate the adding on context.
Orange is used to indicate the quantity that is unknown. This helps focus their attention on the number they are looking for and is an ancestor to the eventually use of a variable.
The blue will be addressed in the PART 2 blog post. (Note: I do not use the term rate but wrote it for emphasis for the blog posts.)
The two-step addition and subtraction problems follow the same structure and involve minimal additional processing.
Additional notes about the process I follow.
I use a chunking approach in which I present the students several problems and have them practice 1 step at a time.
highlight just the unknowns (orange)
highlight the given values (yellow)
highlight the additional values (green)
then have new problems where they highlight all 3
then I would have them write the equation for the the previously highlighted problems
finally, they would attempt all the steps on a 3rd set of problems
There are additional types of problems such as Billy and 5 more tokens than Joey. If Joey has 8 topics, how many does Billy have? I would address these after the students show fluency with the process and the concept of using an equation to model a word problem. They do not follow the same type of structure I present above.
Our students may need help developing a conceptual understanding of addition and subtraction as well as the concepts underlying word problems. In my work with students I often find this to be a major obstacle in student progress with word problems. Hammering out conceptual understanding is likely to be a highly effective investment with a long range effect. It is not as easy to implement as the keyword strategy but we get what we pay for.
Several special ed teachers identified solving multi-step equations as the most challenging math topic to teach in middle school math. Here is my approach to teaching multi-step equations like 3m + 4m + 1 = 15. .
First, I use a task analysis approach to break down the math topic like we cut up a hotdog for a baby in a high chair. MOST of the steps involved are prior knowledge or prerequisites skills. I present these in a Do Now (warm up, bell ringer, initiation) – see image below. This allows me to fill in the gaps and to lay the foundation for the lesson. The prerequisite skills include simplifying expressions and solving 2 step equations. I also present meaning for the equation with a relevant real life problem that is modeled by this equation. By attempting the walkathon problem without the “mathy” approach, the students will more likely understand the equation and why they add 3m and 4m.
After reviewing the Do Now I use Graspable Math, which is a free online application that allows users to enter their own expressions and equations. These can be manually simplified and solved by moving parts around. Here is a tutorial on how to do this. This allows them to manually work with the simplifying and the equation before working on the handout, in a concrete-representational-abstract approach.
This is followed by a scaffolded handout with the use of color coding. I have student work on the first step in isolation as that is the new step (the other steps are prior knowledge and were addressed in the Do Now). This avoids all the work on the other steps that can result in sensory overload and allows me to address mistakes in the new content immediately.
This handout can have the equations removed and be used as a blank template to follow. In turn this would be followed with regular solving worksheets.