A student reported to our schools math lab where I reside. He had a handout on proportions shown in the photo below and stated that he didn’t know what to do.
I have algebra students, in high school and college, who struggle with evaluating expressions like 2 – 5. This is a ubiquitous problem.
I have tried several strategies and the one that is easily the most effective is shown below. When a student is stuck on 2 – 5 the following routine plays out like this consistently.
I implemented this approach because it ties into their prior knowledge of 5 – 2. It also prompts them to analyze the situation – do some thinking.
As Piaget highlighted, our brains make connections between new information and previous information (prior knowledge). I introduce the concept of congruent triangles by connecting it to prior knowledge of identical twins (photo above).
This connection is carried throughout the chapter. For example, to show triangles are congruent we look at parts of the triangle, just as we can look at shoe size, pants size and height of 2 people to determine if they are twins (see photo below).
The use of the “-” symbol is challenging for many students. They don’t understand the difference between the use of the symbol in -3 vs 5 – 3. To address this I use a real life example of multiple uses of the same symbol (1st 2 photos below) then break down the “-” symbol (photo below at bottom). I suggest this be introduced immediately prior to the introduction of negative numbers.
The photo above shows an excerpt from the presentation of notes in an algebra class using an UDL approach. The following strategies are implemented:
This allows for Multiple Means of Representation as found in the UDL Guidelines.
The graph shown involves derivatives – a calculus level topic. Before getting into this heavier mathy stuff, consider the title of this post and the other content presented on this blog. Making math accessible to all students is not a special ed or a low level math thing. It is a learning thing. This artifact is what I drew to explain the math concept to a student in calculus to help her grasp the concept as well as the steps. The following are strategies used.
As for the mathy stuff, the concept of velocity was address by its two parts: speed (increasing or decreasing) and direction (positive or negative). The graph was broken into the following parts: decreasing positive, decreasing negative, increasing negative and increasing positive. Each part was presented with possible y-values (data) and the sign. The most intuitive part is increasing positive which is a car going forward and speeding up.
I find that when I provide intervention, this approach especially by addressing conceptual understanding is effective as the students respond well.
This is a portion of scaffolded notes I provided for a lesson on functions. This shows two key strategies I often employ: scaffolding and connections to prior knowledge.
The scaffolding is seen in how blanks are provided for students to fill in key information. This saves time on copying notes while still engages students in note taking. In the notes handout I include photos that enrich the notes.
The prior knowledge is the photos. The guys are Kanye West and Chris Humphries (basketball player) with Kim Kardashian (dark hair) and Amber Rose. Kanye was dating Amber and Chris was married to Kim. Kanye then cheated on Amber with Kim. Most students fully understood this situation which allowed for carry over into the concept of functions. While this connection is not concrete as in CRA representations, it does make the concept more concrete for the students.
Note: a visitor asked if this presentation sent a message to females. That’s a fair question. My response is that the natural follow up is to show Kim matched with Kanye and Chris and ask if that is a function.
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:
Most if not all educational strategies would address some aspect of this model.