Sometimes when an optimization model is formulated the model yields a lot of alternate optimal solutions meaning that for the same value of the objective function the model yields multiple value of the non basic or decision variables.
Alternate optimal solutions occur mainly due to some portion of the polyhedron being parallel to the objective function. In such cases all points along the segment of the portion that is parallel to the obj function will be affine transforms and would yield the same value of the obj function.
In a practical situation the implications of this would be that when one is trying to solve a problem of say trying to calculate the maximum profit given the effort to manufacture 10 different products and the total constraint on available labour in the plant. Supposing the problem has 10 decision variables and two constraints. Due to degeneracy explained above it may yield an optimal solution of maximum profit of USD 10000 for multiple combinations of the product mix required to be manufactured in the factory.
In such cases it is very difficult to figure out which production mix to choose as the optimal criterion as there are actually multiple values. The parallel portion of the either the edge of the polyhedron or the hyperplane that connects two planes of an n dimensional polyhedron can be disturbed slightly by tweaking the constraints a little bit.
The constraints in the linear programming model form the boundaries of the polyhedron or the hyper plane of the polyhedron. But just modifying the constraint say from 4*X + 5 * Y < 5 to 4*X + 5 * Y < 5.1 would result in altering the feasible region just little bit, but would cease to produce alternate optimal solutions.
In the same context we can also discuss what forms a feasible convex set and why linear programming problems require the set of constraints to be a convex. The optimal solution to a linear programming formulation is found out by traversing the set of constraints from vertex to vertex. So why does an optimal solution not fall somewhere on an edge that connects two vertices, but only on the vertex?. This is because the feasible set can be visualized as the boundary enforced by constraints. The constraints in a linear programming model would result in a polyhedron /polytope. When this is convex it means that any point connecting the two vertexes does not lie inside and so the extreme solution of the objective function will be necessarily found on the vertex.
In the above said study the researchers had offered students a math video game to try their hands on. When the children played the video game collaboratively or competitively with a fellow player they automatically adopted a mindset of mastery which is highly conducive to learning, as opposed to when they were playing the game alone. It was also noticed that keenness and enjoyment of students when they played children’s educational video games increased when they played with another student.
The finding of this study has been published in the Journal of Educational Psychology and elaborates how gaming consoles, computers and mobile-based education can yield increased learning benefits.
One of the lead authors of the study, Professor Jan Plass stated that they have found ample support for well-designed math video games that they can act as effective tools for teaching students subjects that are usually less popular amongst kids. All forms of game-based learning piques the students’ interest about that subject with broadening their interests beyond the aim of just collecting points or stars.
By putting interactive educational tools to use in classrooms such as a free educational game for children which can help to improve the plaguing problems inside a typical classroom environment, which puts students in a mindset for appearing smart rather than being interested in learning.
Two main types of motivational orientations were primarily identified among students when they were engrossed in playing educational video games – the mastery motivation of achieving a goal, where kids focus on learning and developing new skills and the second being, performance-based goal orientation in which the children focus on validating their skills. So, if we consider this scenario in a classroom then students may either be interested to get better at the game (better at math) while playing or may be interested in trying to prove that they are smart or trying to avoid looking incompetent from their peers, thus enhancing performance based motivation.
While the design and content of the video games are important factors that strongly impact on the beneficial outcomes from the game but the positive result from these studies definitely showcase that gamification of education can have a significant positive impact in learning.
Some forward looking thinkers even contend that education through games hold the key to the salvation of the concept of learning- from instruction to true education. But these are predictions only and it is time that will decide what we make out of this promising new medium.
When it comes to math, a lot of parents really notice that only a few kids enjoy it. If your child isn’t exactly like those math geniuses and is more likely to think up ways to not do school or any academic activity, don’t despair. With a dash of creativity and sass, your child can transform from being a math hater to being a math enthusiast. One of the things you can do, especially if your child is struggling with math, is to enrol him in an after school math tutoring program that’s known to utilize a big collection of teaching methods.
Oftentimes, children learn better when they’re not aware that they’re being taught. The best tutorial centers use games and other enjoyable activities effectively in teaching everything from fundamental mathematic principles to complex algebra. It’s best to go with these tutorial centers. Another smart trick used to promote fun math learning for kids is to create situations for practical application. Let the math lessons taught in school, and reinforced in tutorial lessons, come alive. For example, when you go grocery shopping, have your kid assist you with anything that involves numbers.
Of course, this may slow down the activity for you a bit, but practical application is known to create “true value” for math. Your child will have a great sense of accomplishment in helping you out with counting, simple addition and subtraction. Not only that, you take away the child’s fear of mathematical problems. Pretend play and role-playing games are other great math activities for kindergarten children. There are tutorial centers that set up mini markets and play classrooms for very young children to practice math.