Aversive stimulus. Positive reinforcement. Is positive feedback a forgotten classroom practice? Findings and implications for at-risk students. Working memory and reinforcement schedule jointly determine reinforcement learning in children: Potential implications for behavioral parent training. Front Psychol. Introduction to Psychology: Gateways to Mind and Behavior. Domjan, MP. Your Privacy Rights.
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How Punishment Influences Behavior. How Stimulus Generalization Influences Learning. Both the number of groups and the direction of each group are to the right. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number. In one sense though, this visual argument is just mathematical consistency represented using a number line.
If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.
Karen Lew has this analogy. Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, we are getting less negative. This analogy between multiplication and addition and subtraction helps students nicely connect the two concepts. Joseph Rourke shared this context. How much more money did they have 5 days ago? Here, the loss per day is one negative and going backwards in time is another.
This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers. Prerequisite knowledge: All contexts that build new understanding require students to understand the pieces of the context fairly well, so it is especially important to probe how students understand an idea when it is presented contextually.
From Dr. Alex Eustis , we have this algebraic proof that a negative times a negative is a positive. First, he states a set of axioms that apply to any ring with unity. A ring is basically a number system with two operations.
Each operation is closed, which means that using these operations such as addition and multiplication on the real numbers leads to another number within the number system. Each operation also has an identity element or an element that does not change another element in the system when applied to it.
For example, under addition, 0 is the additive identity. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below. From these axioms, we can prove that a negative times a negative is a positive. Prerequisite knowledge : While I went through and added the justification for each step of the proof that was missing, I needed a fair bit of fluency with the original set of axioms.
I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms.
This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic. From this, we can show that ab and — ab have opposite signs and therefore that a positive times a negative is a negative.
Using the fact multiplication is commutative, a negative times a positive is also negative. In the early curriculum multiplication is introduced in the context of whole counting numbers and is appropriately defined there as repeated addition. The computation is quite different with the numbers 5 and 4 serving switched roles. Question: How would you convince a young student that groups of is sure to have the same value as groups of ?
HINT: Put dots in rectangular arrays. Repeated addition allows us to multiply a positive number and a negative number. Using piles and holes this looks like:. Interpreting negative times a positive and negative times negative through repeated addition, however, is problematic. The truth is that multiplication has no meaning here in context of repeated addition. We have entered new territory and if we want to open up our world to new types of numbers it is not surprising that previously concrete, literal definitions begin to flail.
So we have to engage in a sophisticated shift of thinking, letting go of the question What is multiplication? How would we like multiplication to behave? Comment: Let me stress this point.
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