For years over many settings from middle school to college I have witnessed students struggle to make sense out of inequalities like x < 4. Not only is the concept of an inequality of an inequality challenging, merely reading the symbol is problematic. This post shows an concept based approach to making sense of inequalities.
First, I start with a topic of interest and possibly prior knowledge for the students (age to get a drivers license – below). I present the idea of an inequality in context before I show any symbols. In this case, students identify ages that “work”.
Then I introduce the symbol (below). In this case, I include equal to for the inequality (x > 16 vs x > 16). The students plot the same points then we discuss that there are many other ages that work. These ages are called solutions. We put a closed circle on all of the solutions. Then discuss that ages are not exactly whole numbers so we can plot points on all the decimals. Then we discuss that the solutions keep going to the right so we keep drawing dots to the right. There are so many dots we draw a “line” instead of all the dots.
Then we do the same steps for a situation in which the number listed (52 in the case below) is NOT a solution. The students put dots closer and closer to the number but cannot put a closed circle on 52 as a solution (top photo below). Then we present the symbols and talk about the number as a cutoff point that we get really close to but cannot touch. Therefore we use an open circle to show the number is NOT a solution.
This introduction can be followed by problems on a handout, ideally with context then without.
Math is a language with words and other symbols that also makes sense of the world around us. We consume and know more math than we realize or allow ourselves credit for.
When buying the latest iteration of an iPhone, we may call forth algebra. How much will you pay if you buy an iPhone for $1000 and pay $80 a month for service? Well, that depends on how many months you will use this iteration before moving on to the next iPhone. The number of months is unknown so algebra gives us a symbol to represent this unknown number of months, x (or n or whichever letter you want).
Just as there is formal and informal English (or other language), we can engage algebra formally or informally. You don’t need to write an equation such as y = 1000 + 80x to figure out how much you will pay. You can do this informally, compute 80 times 10 months + 1000 on the calculator. Then try 80 times 12 months etc.
Math provides us a means of organizing and communicating ideas that involve quantities like the total cost for buying an iPhone.
The difficulty in learning math is that it is often taught out of context, like a secret code. In contrast, a major emphasis in reading is comprehension through meaning, such as activating prior knowledge (see below).
In fact, math absolutely can be taught by activating prior knowledge. An approach is to work from where the student is and move towards the “mathy” way of doing a problem.
Without meaning, students are mindlessly following steps, not closer to making sense of the aspects of the world that involve numbers.
Educators teaching math typically start with the “mathy” stuff first. For example, for finding the sales price teachers may start with showing students the steps to calculate (photo below).
I start with the concept, either with a pictorial representation or actual objects to represent the underlying concept. In the photo above, I show an object (related to the student’s interest – this student is into weight training) on sale. The $50 circled in yellow represent the original price. I explain the concept of being on sale and discount and show that 20% is $10 to take away (marked out). This leaves $40 (in green) which is the sales price. This allows for conceptual understanding before showing him the “mathy” way of doing the problem.
All too often math topics are introduced first with the skills and steps. This is backwards. The photo above shows how I introduced solving equations a high school student with autism using the concept as an entry point.
We discussed what was involved in buying a car, including payments (no interest) then I posed the problem seen at the top. I asked him to figure out the monthly payment. He worked out the problem, overlooking the down payment. With a minimal prompt he self corrected. I followed this by “showing him the mathy way of doing the problem.” (Seen in the bottom half of the photo). He conceptually understood why the -1,000 was the first step and x had meaning.
Math is an esoteric subject for most people. Good instruction makes information meaningful. One method for making information meaningful is to connect new information to prior experience.
In this situation the new information involves determining whether shapes are similar (see photo below). One example of student prior experience with this topic would be shrinking people down. In the photo above I use Mini Me and Dr. Evil and their respective (and fabricated) weights and shoe sizes as measures that will eventually give way to measures of sides of a polygon (below). When working on the problem below the students can be prompted by recalling the analogy of Mini Me and Dr. Evil.
Solving equations with a variable on both sides proves to be exceedingly tricky for many students. My approach is to focus on the individual expressions taken from both sides of the equation and to present them in the context of a relevant real life situation. The photo shows a snippet of the handout I use. The table is scaffolded to help students compute costs based on number of toppings. The pizza places charge the same at 3 toppings. Domino’s charges more for 0-2 toppings and Pizza Hut charges more for 4 or more toppings. The color coding fleshes this out.
Overall the kids are actively engaged and the variable, expressions and the overall equation has meaning.
One model for memory is called the Information Processing Model or Dual Storage Model.
Here’s the suggested process in this model in a class instruction context:
Our senses receive stimuli. In the classroom students hear the teacher or a classmate talking, see the teacher’s notes or the note being passed to them, smell various things in class, taste their gum etc.
The sensory register filters out most stimuli which means the teacher’s lesson is competing with all the other stimuli for attention. Most students are either visual or hands on learners yet the majority of instruction is conducted through auditory means. Information in a lesson that is meaningful or interesting is more likely to make it through the register.
The information that makes it to the working memory is processed. Working memory has a limited capacity. Like a computer, if it is attempting to process a lot at one time it slows down. It is hard for some students to process a lot of auditory information if they are a visual learner so as they are attempting to process they may be missing other parts of instruction. This is why scaffolding and other strategies are important. They help reduce the amount of information the student has to process. The working memory also attempts to organize and make sense of the information -Gestalt Theory. In the photo below are some examples. When I present the image below under “closure” and ask people what they see, the response is almost always “a triangle.” The really is no triangle there but the brain fills in the gaps. The brain wants to make the visual information meaningful.
The information that makes it to long term memory is filed away. Effective learning means the stored information can be readily retrieved. Think of computer files or files in a file cabinet. I have a file for Gabriel’s IEPs so I can easily retrieve them. Contrast that with how a student may stuff his homework assignment into his bookbag but later cannot find it. Effective storage is enhanced when the information is organized and makes sense. This is helped by making the information meaningful or by addressing prior knowledge (e.g. new IEPs filed with old IEPs).
Most if not all educational strategies would address some aspect of this model.