A common method to learn multiplication facts is through skip counting. In turn, this is a means of learning division facts (see next paragraph). The challenge for many students is they struggle to learn the skip count routines or cannot engage brute force memorization effectively (e.g., have a working memory deficit).
The challenge with multiplication by skip counting is keeping track of two sets of numbers while memorizing the order of the skip counting. That is another example of the rubbing belly and patting head phenomena in math where one extra task demand undermines the process.
A hack I use to scaffold this process to reduce the task demand during the learning process is to provide rows from a multiplication chart (below) for the facts of focus (3s and 4s in this example). The same approach can be used for division facts, e.g., in the image below right I have the student choose the row of the divisor (3) and then skip count to until reaching the dividend (12). The idea is the student has less task demands while learning the process and seeing the number pattern. This allows for more repetitions or rehearsal.
For students more severely impacted by a disability or who simply struggle with the patting head and rubbing belly of skip counting, the appropriate times table row can be provided for each problem to allow the student to circle (below). This allows for a hands on approach with even less task demand. You could also laminate the rows to make them reusable in lieu of several consumable ones requiring more paper. I like the consumable as I use that for data collection.
Below is a model for information processing (retention and retrieval). Here are a couple key points I want to highlight:
A lot of information is filtered out so what gets through? Information that is interesting or relevant.
Information that is connected to prior knowledge, is relevant or that is organized has a better change of being stored effectively for retrieval.
Working memory has a limited capacity. Consider what happens to your computer when you have a lot of apps open. Your computer may start to buffer which is basically what happens to our kiddos if instruction involves opening too many apps in their brains.
Long term memory is basically retrieval of information. Think a student’s book bag with a ton of papers crammed in it. How well can he or she find homework? Compare this to a well maintained file cabinet that has a folder labeled homework with the homework assignment in question stored in this folder. That paper is much easier to retrieve. This is analogous to long-term memory. If the information is relevant or meaningful it will be stored in the file cabinet folder and more easily retrieved. In contrast, rote memorization like the rules teachers present students are papers crammed into an overflowing bookbag.
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.
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.
Chunky peanut butter means you there are pieces or chunks of peanuts. The peanut butter is broken into distinct parts. Chunking in teaching means breaking a concept or lesson into distinct, smaller parts. This makes it easier for students to process the lesson and the concept. They focus on less content at one time which is important given that working memory has a limited capacity, especially for many students with disabilities.
Here is an example of how it can play out. On a lesson on slope (compute slope of a line given two points from the line) the steps are as follows for (1, 5) and (4, 11):
write the formula y2-y1/x2-x1 (rise over run)
plug in the ordered pairs (11-5)/(4-1)
compute 6/3 = 2
But there is also the conceptual understanding that is often lost on the students. These steps do little to help with conceptual understanding.
In the photo below the teacher is presenting a conceptual piece as a chunk of her lesson before she gets to the steps listed above. She has drawn a triangle to represent going downhill, another going uphill and a horizontal line segment representing no hill. The rise and run are listed for each (the horizontal line segment has no rise). The students wrote a rise over run ratio showing the slope for each then practiced this before the teacher moved onto to the steps listed above.
Upon completion of this chunk the teacher can give a practice or pop quiz to help students fill in the gaps individual students may have before moving on to the next chunk.
The next chunk would not yet involve the formula. A slope triangle would be drawn under the line to visually represent the rise and run. Students would practice finding slope using this approach with the next step bringing in ordered pairs and the formula listed above.