Encouraging Mathematical Thinking

Parents, do you want to encourage your young people to think mathematically? Here are some ways to accomplish that.

Preschoolers

Nurturing Mathematically Talented Preschoolers—While it can be difficult to find programs for three- to five-year-old mathematically precocious children, there are things that you can do at home to encourage and feed their talents. Click on the link above and find suggestions that include

• Specific building materials, including LEGOs, Tinker Toys, Lincoln Logs, K’nex, Magna-Tiles, tangrams, and blocks of all shapes and sizes. Ideas are also provided for using these building sets.
• Ways to use mathematical logic in everyday conversations
• Ideas for working with fractions

There is no need for formal lessons. All of the suggestions provided are applied through play activities.

Elementary School Students

10 Practically Fun Math Games and Activities for Your Preteen—While the title of this article suggests that young people be close in age to teenagers, many of these activities are appropriate for much younger children. Author JC Ryan lists eight indoor activities and two outdoor activities that parents may not automatically think of as building math skills.

Kindergarten through High School

Have You Seen These 8 High Quality, Free Maths Websites?—Activities here cater to an enormous range of abilities and offer math related subjects from basic counting through calculus, current economic theory, and puzzles.

What Should We Be Teaching Kids for the 21st Century?

Dr. Judy Willis is an authority on brain research. She has a unique background, having been both a neurologist and a classroom teacher. She has written several books and writes a blog for Psychology Today. One of her blog entries, Whose Children Will Get the Best Jobs in the 21st Century? http://www.psychologytoday.com/blog/radical-teaching/201104/whose-children-will-get-the-best-jobs-in-the-21st-century , offers an interesting perspective on what we should be doing to prepare students for today’s world.

According to Willis, the best jobs in the future will go to applicants who have the

• skillsets to analyze information as it becomes available
• flexibility to adapt traditionally believed facts are revised
• ability to collaborate with others
• ability to articulate one's ideas

Rather than just learn a lot of information, students need opportunities to discover the connections between isolated facts, build networks of concepts, and apply what they learn in new contexts. Critical analysis, judgment, creative problem solving, and the ability to evaluate and apply data to new situations are all vital.

Parents can prepare students by

• helping children develop personal responsibility
• explicitly teaching how to focus attention, study, organize, prioritize, plan, and set goals
• teaching how to make the switch from memorization to mental manipulation by comparing and contrasting concepts and applying big ideas to solve new types of problems
• teaching how to evaluate sources of accurate information and then to use critical analysis to assess the veracity/bias and current/potential uses of new information
• finding out the topics children will study in the coming school months and then promoting interest by introducing things that relate to the topic, providing background knowledge and interest

Willis provides many concrete ideas for parents to teach these skills at home. I strongly urge you to read the article. The ideas provided would make a great beginning for discussion in a parent support group.

Using Primary Sources

Most students study history using only secondary sources—articles, reference books, and textbooks—all written at some point after the actual event. Secondary sources tend to interpret or analyze historical events.

Primary sources, on the other hand, were created during the time period being studied. They reflect the individual viewpoint of a participant or observer. Primary sources include autobiographies, diaries, e-mails, interviews, letters, minutes, news film footage, official records, photographs, raw research data, speeches, art, drama, music, novels, poetry, buildings, clothing, DNA, furniture, jewelry, pottery, etc. These sources enable the researcher to get as close as possible to what actually happened during an historical event or time period.

Today, the Internet provides access to a wealth of primary resources. In earlier years, one would have had to travel great distances to various libraries and museums to gain access to this information.

The American Library Association’s Reference and User Services Association has posted an article titled Using Primary Sources on the Web, which can be used as an exercise in critical thinking. It provides information on

• Finding primary sources
• Evaluating primary sources (including, among other things, understanding the purpose of the website where the primary sources are displayed and the credentials of the person who created the website)
• Citing websites appropriately

Repositories of Primary Resources contains links to Internet sites for primary sources all over the world. Want to find a digitized photo of a street scene in Colorado in the mid-late 1800s? Do you want to find crime reports for the United States in 1935? Do you want to see an original score written by Beethoven? Do enough searching on this site and you will find such information.

The Library of Congress is in the process of digitizing many of the important documents in American history. As of the writing of this blog entry, they have posted documents from 1763-1877.

These are just some of the many sources for primary resources on the Internet. For a particular topic of interest, do an Internet search using the subject of your search (e.g., Civil War women) plus the words “primary source.”

Exercising Minds through Questioning

Bright students often come to class thinking they must know all the right answers. What they (and many adults) may not realize is that thinking is not driven by answers, but by questions. It is the sense of wonder and curiosity that drives understanding. As the old saying goes…The more you know, the more you realize you don’t know. The more you learn about a subject, the more you realize there is to know. Perhaps having students list thoughtful questions at the conclusion of a unit would be a better determiner of knowledge gained than taking a test.

Young people learn to develop inquiring minds when they hear their parents and teachers ask thoughtful questions of themselves and others. One way to do this is to use Socratic Questioning. Socratic questions help to

• Clarify issues
• Question assumptions
• Justify statements
• Understand the ideas of others
• Imagine consequences
• Relate different issues 

Divergent questions are also useful. They usually begin with words or phrases such as

• Imagine…
• Suppose…
• Predict…
• If…, then…
• How might…
• Can you create…
• What are some possible consequences…

Here is one example of questioning for young children in kindergarten or first grade after reading and discussing Jack and the Beanstalk.

Q. What did Jack do when he got to the giant's castle?"

A. Jack hid from the giant, found the goose that lays the golden eggs, was discovered by the giant, fled, reached the bottom of the vine, and then chopped it down. The giant, of course, tumbles down, breaks his neck, and Jack lives happily ever after with his mother and his newly found wealth.

Q. Did Jack trespass illegally? (In kindergarten terms, "Did Jack go into someone's house where he did not belong?")

A. Yes!

Q. Did Jack steal the goose that lays golden eggs?"

A. Yes!

Q. Did Jack, then, refuse to give back what did not belong to him?

A. Yes!

Q. Then did Jack escape down the bean vine and cause the giant to be killed?"

A. Yes!

Q. If Jack trespassed, stole, and murdered the giant, why is the giant the villain of this story?