Obtuse angle on the left (see upside down “T” figure) and perpendicular lines (right angle) on the right.
In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)
When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.
To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: “Ability” to execute or perform “tasks” – DOING.
Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.
61 cents per ounce is a rate of change. Graph the line modeled by this (y intercept is 0) and it becomes slope of the line. In referring to algebra we often hear, “when will I ever need this?” My response is “all the time!” Our job as teachers is to make this connection for students.
It is difficult to individualize instruction in a whole class or small group setting. I created and taught the curriculum for a Consumer Math course at the high school where I teach. For a class of 10-12 students, all with an IEP, I developed an approach that allowed me to individualize the instruction for each students.
In the photo below is an example of a folder set up I used with the students in Consumer Math. Each student would have a dedicated folder, kept in the room and updated daily. The smaller paper shows the individualized agenda. The other paper shows an example of how the folder can be used as a resource. Student computer login information, accommodations like a multiplication table or notes can be secured inside the folder. The agenda would be changed out each day. (In case you are wondering about the label in the agenda, “Math Group 4.” This particular folder was used in a special education training session for teacher candidates.)
Below is a screen shot of a video in a series of videos that provide instruction on algebra topics. The videos are designed to make algebra more accessible for almost all students.
The presentations include the following instructional strategies
The videos can be used in the following ways
The videos include a link to the handouts used in the presentation. Additional practice worksheets will be included as well.
NOTE: this is only a sample, with more samples to follow. Please share feedback or ask questions.
The topic of imaginary numbers is one of the most abstract and therefore difficult math topics to teach in algebra. Here is how I introduce it to students (emphasis that this is only an introduction).
I write 1, 2, 3… on the board (see photo above) and explain to the student “at some point in life you learned to count on your fingers, 1, 2, 3…” These are called the Natural numbers.
Then I explain, “later you were told that no cookies means ZERO cookies. Zero is a new type of number. We call 0, 1, 2, 3… the Whole numbers. You learned a new type of number.”
This continues, “A little later on you were told you could have half a cookie and so you learned about a new type of numbers called fractions.”
This continues with negatives. Then I explain that all these number types can be found on the number line. We call the set of all of these numbers Real Numbers.
I conclude with “Now we have a new type of numbers that are not found on the number line. These are called imaginary numbers. Just like before you had number types you had before and now you have a new one to learn.”
The point of this approach is to help the students understand that a new number set simply builds on previous number sets. Also, the students have encountered this situation before.