Multiplying by Adding
Multiplication simply is not repeated addition, and telling young pupils it is it makes little difference whether you can do one abstract thing with numbers or a. Teach your students how repeated addition relates to multiplication. This lesson builds number sense and supports a conceptual. They are ready to make the connection between repeated addition, groups, and I use this project when I start my multiplication unit to teach repeated addition.
I remember that my mind was clearly messed up about math in high school. While I knew the problems involved logarithms or roots at that level are supposed to be very easy, I could not understand what I'm really doing.Multiplication and Repeated Addition
I guess somehow in my mind, I made axioms in arithmetic that worked well for integers and rationals but I could not extend and generalize them to real numbers. As a results, I never learned trigonometry in high school. I could not understand why everybody in the class could so easily just add or multiply trigonometric functions of x and mix them with all other functions while I could not understand what even adding or multiplying a number by a irrational number means. This lack of understanding of math hunted me all the years in the university.
I passed my math courses with B- and Cs. I could not understand anything as I would always get stuck in the basic definitions that did not work out. I did not get to clear my mind about it till years later during my PhD in engineering when I had the opportunity to study, on my own, and finally get to learn axiomatic set theory, peano numbers and such.
I feel all my years at the high school and university was ruined because I never learned the basics of mathematics from the beginning. Maybe it was my teachers' fault, but probably they didn't know much about math themselves.
Maybe it was my father's fault that kept repeating not to move to next step until understanding everything correctly and truly.
How Are Addition and Multiplication Related?
I've started to learn math on my own, from scratch. I am 36 now. I hope this year I can finally finish basic calculus and understand it. Log in to post comments By Anonymous not verified on 25 Jul permalink Nobody abandons a paradigm merely because it has contradictions and anomalies. A scientist who leaves her paradigm is no longer a scientist. A teacher who abandons a paradigm of pedagogy has to jump onto some new bandwagon. The new paradigm may or may not solve all the problems, but it has to offer hope for people in crisis.
The problem in meta-theory comes before ecponentiation, or multiplication, or even addition. It comes at counting.
If the students have not been properly taught what an integer is, they'll never get anything else right. I gave an example in "week73". There is a category FinSet whose objects are finite sets and whose morphisms are functions. If we decategorify this, we get the set of natural numbers!
Teaching Multiplication: Is it repeated addition? | ScienceBlogs
Well, two finite sets are isomorphic if they have the same number of elements. I like to think of it in terms of the following fairy tale. Long ago, if you were a shepherd and wanted to see if two finite sets of sheep were isomorphic, the most obvious way would be to look for an isomorphism. In other words, you would try to match each sheep in herd A with a sheep in herd B. But one day, along came a shepherd who invented decategorification.
This person realized you could take each set and "count" it, setting up an isomorphism between it and some set of "numbers", which were nonsense words like "one, two, three, four, By comparing the resulting numbers, you could see if two herds were isomorphic without explicitly establishing an isomorphism!
According to this fairy tale, decategorification started out as the ultimate stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome through the process of "categorification".
Okay, so what does this have to do with quantum mechanics? Well, a Hilbert space is a set with extra bells and whistles, so maybe there is some gadget called a "2-Hilbert space" which is a category with analogous extra bells and whistles. And maybe if we figure out this notion we will learn something about quantum mechanics. I always quote "God created the integers, all the rest is the work of man" and cite the grman author Google it and that I am not violating separation of Church and State.
How Are Addition and Multiplication Related?
Of course I am not teaching Category Theory to year-olds. But, as you have shown, Mark, it is good for us to know, to get clear on the basics of Math. Now, I have had an on again, off again, tentative relationship with Category Theory since I was a teenager.
There is a gradual and accelerating takeover of parts of Math, including Foundational, by means of Category Theory. This has a bright side and a dark side.
Category Theorists have beliefs in things even sillier than the infinities of Set Theory, about which I need not believe in completed infinity, can use the Cantor stuff, have seen the morass deeper in theory, and don't matter for Science anyway except as to whether or not space or time are continuous or discrete, which don't use either sets or categories anyway.
Category Theory is more "gestalt" and less "analytical" in terms of which half of your brain is engaged. This is a paradigm shift, socially, I can say without accepting or denying the claims on either side of the battle. In any case, Engineering and Science can pretty much watch with detached amusement, or ignore the fight, until the winners start new invasions. The Categorists tend to see Biology and Sociology and Economics as ripe for conquest, due to "networks" and their uses.
Decategorification and recategorification are not merely clumsy big words. They code for an agenda. First, that the real world has structures which set theory throws away. So how do we put the real world structure back into Math?
How Are Addition and Multiplication Related? Have you heard anytime that addition and multiplication are related? If no, then you students can use this skill to improve your math multiplication. Many students find it difficult to multiply the numbers, but if you practice this skill to hone the multiplication skills, you will find it easy to solve various multiplication problems.
Multiplication is also known as repeated addition. This explains everything about how addition and multiplication are related. However, to understand it in a better way, we will consider few instances.
It is described in next paragraph. Understanding the Relation in a More Lucid Manner: For example, if you rolled 2 and 6, you would lay out two plates and place six linker cubes on each plate.
Then, on a sheet of paper write a repeated addition equation and a multiplication equation to go with the model you built i. Write the rules of the game on the board for student reference: Instruct students to build four or more problems with their partner.
Have students number their problems on their paper as they record so that you can see how many models they have built.
Repeated Addition and Multiplication
Independent working time Hand out the Repeated Addition worksheet. Instruct students to complete the worksheet independently.
Circulate and offer support as needed. Review the worksheet as a class. Provide pre-drawn models and ask students to write a repeated addition problem to go with the model. Provide a repeated addition problem and ask students to write it as a multiplication problem. During the dice game On a Roll with Multiplicationhave students roll two dice to determine how many linker cubes to place inside each group.