Tag Archives: concepts

Velocity, Acceleration, Speeding Up and Slowing Down

DISCLAIMER: This is a very mathy, math geek post but it also has value in demonstrating instructional strategies and multiple representations.

We all understand speed intuitively. Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling. Both cars below are traveling at equivalent speeds.

velocity vs speed

The velocity can be graphed (the red curve below). Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left. The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed). (I will address the blue line at the end of this post as to not clutter the essence of what is shown here for the lay people who are not math geeks.)velocity graph with tangents

Below is an example of using instructional strategies to help make sense of the graph and of velocity, acceleration, speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. As the car moves towards a stop sign it will slow down. When a car moves away from a stop sign it speeds up.

The concept and the graph analysis are challenging for many if not most students taking higher level math. This example shows how instructional strategies are not simply for students who struggle with math. Good instruction works for ALL students!

speeding up and slowing down

It is counterintuitive that when acceleration is negative the car can be speeding up. The rule of thumb is when acceleration and velocity share the same sign (+ or -) the object is speeding up. When the signs are different the object is slowing down. This rule is shown in the graph but the stop sign makes this more intuitive.

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Concepts vs Skills – Need Both

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.

calvin hobbs toast

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.

definition conceptdefinition skill

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.

solving equation with conceptual understanding first

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Documents for Task Analysis Presentation – DADD 2017 Conference

dadd-conference

task-analysis-overview

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.)

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A Meaning Making Approach to Word Problems

Here is a typical story or word problem.

Dakota helped her father bake cookies. They baked 9 sugar cookies and 3 chocolate chip cookies. How many cookies did they bake total?

When solving word problem the focus is often on following steps, e.g. read the problem and identify important information. There is also a focus on identifying key words, e.g. “total.” The problem with both is they rely on rote memorization. How do we identify “important” information? Focusing on the word such as total does not address the concept of total but is more of a signaled command like “sit.” Students see “total” and they know they are supposed to add. The problem is they often don’t understand why.

The entry point to word problems should be a focus on the underlying concepts. For example, present the word problem with cutouts of the actual cookies and physically demonstrate “total” by pulling all the cookies together. Similarly, you can have cutouts of the tadpoles and demonstrate the concept of how many are left.

word problems focus on concept first concept first approach

Words are symbolic representations of ideas. Same with math symbols (below). Addressing the concepts, vocabulary and the process with this approach is a concrete-representational-approach (CRA). The equations below would not be addressed until the conceptual understanding was developed. When word problems presented do not include the term “total” the student can process the context as opposed to being reliant on the signal.

word problems focus on concept first traditional approach

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Math is a Uniquely Challenging Subject

Slide1

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.

Slide2

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.

Slide3

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.

Slide4

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.

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