The idea is that the student will have to count squares and eventually skip count by how many rows. By doing so the student is more engaged (or less passive) in determining the product byy engaging the visual representation. I am interested in feedback and will revise if this could be useful.
Human senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
We have a filter that allows only some of these stimuli in. We focus on the ones that are most interesting or relevant to us.
Our working memory works to make sense of the stimuli and to package it for storage. It is like a computer, if there is too much going on, working memory will buffer.
The information will be stored in long term memory.
Either it will be dropped off in some random location and our brain will forget the location (like losing our keys)
Or it will be stored in a file cabinet in a drawer with other information just like it. This information is easier to find.
Analogy to Classroom Learning
Here is an analogy to what happens during school instruction. You are driving down the street, like the one shown below.
There is a lot of visual stimuli. The priority is for you to pay attention to the arrows for the lanes, the red light and the cars in front of you. You have to process your intended direction and choose the lane.
Other present stimuli may be filtered out because it is not pertinent to your task: a car parked off to the right, the herbie curbies (trash bins), the little white arrows at the bottom of the photo. There is extraneous info you may allow to pass through your filter because it catches your eye: the ladder on the right or the cloud formation in the middle.
Maybe you are anxious because you are running late or had a bad experience that you are mulling over. This is using up band width in your working memory. Maybe you are a relatively new driver and simple driving tasks eat up the bandwidth as well.
Impact on Students
For students with a disability that impacts processing or attention, the task demands described above are even more challenging. A student with ADHD has a filter that is less effective. One with autism (a rule follower type) may not understand social settings such as a driver that will run a red light that just turned red. Another with visual processing issues may struggle with picking out the turn arrows. Their brain may start to buffer, like a computer.
Here’s a common word problem used for linear functions and equations (y=mx+b):
There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6.
Often the word problems like this are presented on a sheet of paper in isolation as an attempt to make the math relevant and to develop conceptual understanding. For students who have trouble with conceptual understanding, words on paper are likely too abstract or symbolic to allow applications like the one above to be meaningful.
The real life application is useful if presented more effectively. Here’s an approach to use the same scenario but in a more relevant and meaningful presentation. The photo above shows the current amount of snow – call it 6 inches. Students can be shown the photo to allow for a discussion about accumulation and for their estimates of the amount of snow shown. The photo below shows an excerpt from a storm warning. Showing this warning and a snow fall video can allow for a discussion about rate of snow fall and the purpose for storm warnings. Combined, this approach can lead into the above word problem.
Once the application is presented students can be asked to compute snow levels after 1 hour, 2 hours etc. Then they can be asked to determine how long it would take for the accumulation to reach 18 inches (the prediction for the day this post was composed). After computing the answers WITHOUT the equation the students can be shown how to use the equation – the “mathy way.”
Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.
In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.
Here’s a typical exchange – working through problems 11, 12:
Me: “Look at the table, what’s changing?”
Student: “the cost”
Me: “How much is it changing?”
Me: “20 what?”
Student: “20 cost”
Me: “What are you counting when you talk about cost?”
Me: “So the price is going up 20 what?”
Me: Show me this on the yellow” (student knows from before to write +$20)
Me: “What else is changing?”
Me: “By how much”
Student: “1 people…person”
Me: “write that on the green”
Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
Me: “What do you do next?” (they typically know to move over and up)
Me: “Use green to highlight the over” (they highlight)
Me: “How much did you go over?”
Student: “1…1 person”
Me: “Now what?” (Student goes up.)
Me: “Highlight that in yellow.” (They highlight.)
Me: “How much did it go up?”
Student: “2…20…20 dollars”
Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
Me: “So what is the rate of change?”
Student: “$20 and 1 person”
Me: “Look at the problem at the top. What is the 20?”
Student: “$20 per person.”
I point out that you can find this rate or slope in the equation, the table and in the graph.
This is a screen shot from explorelearning’s Gizmos. This site has various simulations related to science and math. This one shows multiple reprsentations for a system of equations. The site has 5 minute test trials which can be used to present a topic in class. I used the one for photosynthesis for a 7th grader with asperger’s who was collecting data for his science fair project.
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.