Tag: conceptual understanding

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Speeding Up and Slowing Down on a Graph looking at Velocity and Acceleration

We all understand speed intuitively. Velocity and acceleration are more mathy and harder for many students to comprehend, especially on a graph. Even harder is the idea of when an object is slowing down or speeding up. In this post I attempt to unpack both.

Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling (think of backing up). Both cars below are traveling at equivalent speeds.

Velocity can be graphed (the red curve below). Here is information found in the graph.

  • 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).
  • The blue lines are tangent lines with slope equal to acceleration (the derivative of velocity). If the line is going down from left to right, the acceleration is negative. Up, positive.velocity graph with tangents

Below is an example of using prior knowledge and visuals to help make sense of speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. When the car is above the x-axis, it has positive velocity. Below, negative. When the car is headed down, the acceleration is negative. Up, positive.

Now for speeding up or slowing down.

  • Think of a bank account balance.
    • If you have a positive balance and a transaction that is negative, Your balance is moving towards 0.
    • Suppose you have a negative balance in your bank account and you have a positive transaction. Your balance is moving towards 0.
  • As the car moves towards a stop sign it will slow down. The speed is getting closer to 0.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0, like the bank balance analogy.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0.
  • As the car moves away from a stop sign it speeds up. The speed is moving away from 0.
    • Positive bank balance, positive transaction means balance is moving away from 0.
    • Negative bank balance, negative transaction means balance is moving away from 0.
speeding up and slowing down
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Lego Fractions

The photo below is courtesy of Robert Yu, Head of Lego Education China, as shared by Jonathan Rochelle, Director for Project Management at Google.

The use of Legos shown here is a classic (and wicked clever) example of manipulatives.

Lego fraction model

Before writing the actual fractions students can use drawings as shown below. The sequence of manipulatives, drawings then the actual “mathy” stuff constitutes a Concrete-Representational-Abstract (CRA) model. Concrete = manipulative, Representational = picture, abstract = symbolic or the “mathy” stuff.

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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 concept
definition 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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Busy Engagement vs Intellectual Engagement

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.

owl

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.

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Example of Using Support Class to Support Current Math Content

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.

examples of support class prerequisites

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.

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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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Scaffolding Higher Level Math Topics

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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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