## Velocity, Acceleration, Speeding Up and Slowing Down

**Tagged**acceleration, car, concept, concepts, conceptual understanding, prior knowledge, slowing down, speeding up, stop, stop sign, velocity

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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.

Owls are symbols of intelligence but the purported reasons are based on the appearance of awareness and the deft hunting skills. It is claimed that the appearance and skill sets are confused with actual wisdom.

I find a parallel between the perceived wisdom of the owls and the perceived learning of students. Through my years in education I have seen teachers praised for their student centered activities. The students may be energetic and on task by an activity which is often considered a touchstone for learning. What is often missing is independent assessment to determine actual learning.

Once I was covering a class for a teacher widely praised for his activities and multimedia activities. In the class I covered the students were taking a test. It was clear that the majority of the students were hesitant about their performance. Several were looking around, one pulled out a phone and a couple looked at other people’s paper. Very few were locked in on completing their test.

I am not suggesting that multimedia or student centered activities are ineffective. My point is that there is a perception that such activities are inherently effective and reflective of actual learning. There is a difference between being intellectually engaged and being busy. The owl deftly executes action and skill but that does not indicate higher level functioning. Conceptual understanding requires more than simply being engaged by activity. Hopefully this is food for thought.

I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.

For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math.

This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.

Here is a link to a Dropbox folder with the documents I will address in my presentation. (Note: documents will not be uploaded until Jan 19, 2017.)

In a previous post I presented an approach to teach and assess the concept of addition. This document shows all the steps I use including the one shown in the photo above.

Imagine being asked to explain the climate in Spain and the photo above is the resource you had to use. If you didn’t speak Spanish this would be a challenging task for two reasons. First, you may not know the climate so this is new learning. Second, you don’t know the language used to explain the content – a double whammy!

The photo below shows a resource you are more likely to encounter. The language (if you are an English speaker) is natural for you which leaves you to focus on the content alone. Language is a barrier to learning math.

The photo below is an excerpt from “*Why Do Some Children Have Difficulty Learning Mathematics? Looking at Language for Answers*” by Joseph E. Morin and David J. Franks. It shows another element to the language barrier in learning math. In this example the term *over* is used to describe the location of the the white square (bottom frame) but the students understand *over* as a term used in 3-dimensional space (top frame). The misunderstanding of a single term can throw a student totally off in learning a new math topic.

Below is an excerpt from Malcom Gladwell’s book Outliers. He explains that the languages of Math and English do not get along very well. “Thirty” has to be translated into the concept of 3 TENS. Compare this to Chinese and Math. The problem in Chinese is read as “three-tens-seven” which is already in Math terms. This extra step of translating is often a problem for our students, especially those with special needs.

In teaching math the issue of the language of Math is an additional issue to address separately. I teach students to learn math in their own language (informal English, using manipulatives etc.) and after the concept is learned I show students the “mathy” way of talking about it. This follows the Concrete-Representation-Abstract (CRA) approach to presenting math.

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.

- color coding – each of the 4 sections written in different color
- connecting to prior knowledge – the concept of velocity was presented in terms of a car’s speed and direction (forward or backing up)
- chunking – the problem was broken into parts and presented as parts before exploring the whole
- multiple representations – the function was represented with a graph, data (1, 2, 3, 4, 5) and a picture (at the bottom)

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.

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