**Book Review**

Vision in Elementary Mathematics is a well-developed book that works through many examples within the frame of Elementary mathematics. The author uses many pictures and diagrams to explain his thinking and stresses the importance that "a picture, is in fact, a most valuable way of reminding someone of a sustained chain of thought." This books walks you through 13 chapters that include multiple methods and ideas to work within the context of teaching Elementary Mathematics. The summary of each chapter is shown below.

__Chapter 1: Even and Odd__

The author works through many examples of even and odd problems using different shapes and methods to solve simple math problems.

__Chapter 2: Divisibility__

Here the author works through many divisibility tricks and why their methods work along with different ideas about our number system.

__Chapter 3: An Unorthodox Point of View__

The author says "a good teacher tries to find to exactly where the pupil failed to understand" and "a pupil can only learn good judgment by seeing the effects of bad judgment." This chapter talks about multiple different ways to get students thinking so that they can learn about different methods and when these methods are appropriate to use.

__Chapter 4: Tricks, Bags, and Machines__

This chapter the author says "early teaching is to enable the pupil to distinguish truth and falsehood, sense and nonsense for himself." There is also discussion about the four stages where students move from words to pictures, and then from simplified pictures to shorthand. This chapter has lots of different games that students can play to better understand certain mathematical content.

__Chapter 5: Words, Signs, and Pictures__

The author shared a story about a classroom that students had to write a story at the end of every sum they came upon in the class. This is a great idea to get students thinking about equations and their overall meaning. The author stresses that communication is the most important aspect in teaching mathematics and that "you cannot succeed unless you are willing to devote to it a considerable amount of time, thought, and activity."

__Chapter 6: Sudden Appearance of a Practical Problem__

The author says "in our teaching we should include elements that will appeal to the realistic side of every child, and assure the strongly practical child that we have not forgotten him." This chapter shows some different number tricks and relates mathematical ideas back to the proof of the Pythagorean Theorem.

__Chapter 7: A Miniature Problem in Design__

The author touches on different problems in mathematics dealing with algebra and subtraction. The author shows a card trick in this chapter that also relates to algebra.

__Chapter 8: Inversigations__

The author’s main idea in this chapter is that "the making of tables is an excellent way of improving accuracy in computation.

__Chapter 9: Routines of Algebra__

The author works through many different methods of algebra and says "we shall many times need to add, subtract, multiply, and divide."

__Chapter 10: The Routine of Algebra__

In this chapter the author stresses that "we need to be patient. We need to be careful not to present so many ideas at once that children will fail to grasp any of them." Also the author says "it is far better for them [students] to use any procedures which they have thought out for themselves, and which they can use correctly, than to attempt some streamlines proportion of incorrect answers."

__Chapter 11: Graphs__

This chapter talks about graphs and how "children should be encouraged to pose questions about graphs and asked to experiment freely."

__Chapter 12: Negative Numbers__

It is important to remember that "many children will have met negative numbers in connections with temperature below zero." The author shows different tables that can be used to represent negative numbers and show the multiplication of these numbers in relation to the number patterns. The author also stresses that "quite younger children can learn to appreciate that negative numbers do lead to results which can be seen to be satisfactory. They can become familiar with negative numbers as a working tool."

__Chapter 13: Fractions__

The author says that "pupils should spend sufficient time working at interesting and stimulating problems involving algebra for them to become thoroughly familiar with algebra and able to follow an algebraic argument in a textbook or scientific paper without strain or undue expenditure of time."

Recurring Big Ideas

Recurring Big Ideas

The big ideas that
the author is trying to make is that not all students learn the same. This
ideas that showing the students a correct answer without allowing the students
time to work through the problem will create problems for that student in the future.
It is also important to show solutions in multiple different ways so that
students will understand that there are multiple routes to solve a problem in
mathematics. The author touches on this idea that teachers strategically plan
their conversations with their students so that the student has the maximum
opportunity to discover the math on their own.

**Recommendation**

I would recommend this book to anyone pursuing for an Education degree. Whether that person is going for Elementary or Secondary Education, mathematics starts from the basics and without those basic understandings students will not be able to succeed. I feel this is a great book and was easy to read as a mathematics major, but I feel it could be read by any education major and would be a very useful tool in the classroom. This book includes many pictorial ideas for showing mathematical problems and it also describes different way to work through problems that might arise within the classroom.

Critique

Critique

The one and only critique I have about this book is that the author is very repetitive and there are multiple times throughout the book that the author talks about previous work on a problem. There is a sequential flow throughout the book that makes sense in the long run, however the author seems to repeat many ideas within the frame of the book. Overall it is a great read with many ideas that can be used for classroom teaching.

**Use in My Future Classroom**

In my future classroom I will use many of the techniques
mentioned in the book. I will allow for maximum student involvement in the discovery
of the materials. I will allow an opportunity for student to make mistakes in
order to learn what not to do in the future. I will focus my teaching on the
student’s needs and provide as many visual representations as necessary. In the
long run these changes will provide a major difference in our community.
Without these changes students would continue to be given repetitive book problems
to work on during class using a strategy that is given in the book. This change
will allow students an opportunity to explore mathematical problems and learn
to their fullest potential. My goal as an education is that I want all my
students to realize that math is interesting and fun, so that they will further
their involvement in mathematics in later grades and learn to love math.

Handy guide to the book, and a good recommendation. I'd love to see what you say those recurring big ideas are, though. Can you see yourself using these visualizations in the classroom? What difference would they make?

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