Although intuition can and should be used in math--particularly for estimation--learners still need to reason to make sense of some of the more challenging and counterintuitive concepts in math, such as zero, negative numbers, irrational numbers and infinity, to name a few.
In elementary school, it begins with mathematical reasoning, "an evolving process of conjecturing, generalizing, investigating why, and developing and evaluating arguments" (Lannin, Ellis, Elliot, 2011)
Math is often considered the science of patterns, whether in the form of numbers, shapes, operations, and relationships. Seeking patterns is what children do naturally as they experience the world around them.
Conjecturing and Generalizing
After a pattern is discovered, mathematicians develop hypotheses to test if they are true. Hypotheses in math is called conjectures, basically working theories. Often conjectures lead into generalization, to see if there's a universal pattern. It goes from "it's true to this case" to "it's true in all cases!"
Zager, Tracy Johnston, Becoming the math teacher you wish you'd had, 2017
Lannin, Ellis, Elliot, 2011. Developing Essential understanding of Mathematical Reasoning for Teaching Mathematics in Pre-K-Grade 8.
Too often in math class, mathematically interested students are shut down in class, often by well-meaning teachers who either don't know the underlying mathematical concept, feel publicly challenged, see math as a set of rules and formulas, or perhaps are on a tight schedule to "cover" the curriculum.
Most mathematicians feel mathematics is not so much being obedient to a set of fixed rules and regulations, but rather using creativity and risk taking to solve elegant problems. Paul Lockhart, mathematician, says, "Math is not about following directions, it's about making new directions." They are most intrigued and challenged by unsolved or open problems. This requires taking risks, something you might not attribute to learning/teaching math. Mathematician James Tanton says, "Math is being able to engage in joyful intellectual play--and being willing to flail (even fail!)."
Zager talks about the joy children experience when discovering or realizing new mathematical understanding in their everyday lives. It doesn't matter to them if a mathematical law or property already exists in some math textbook.
In a grade two class, Zager observes a math lesson and notices several things. First, the teacher uses wait time, wanting deep thinking and thoughtfulness, not simple ideas and speed. Also, she keeps students focussed on mathematical thinking, not unrelated matters. The teacher makes sure they are using specific and clear mathematical language. Finally, she teaches students to challenge themselves and take risks.
PROMOTING OBEDIENCE VS. ENCOURAGING RISK
Obedience: Memorizing algorithms is necessary. "What's the rule about adding fractions?"
Risk: Different ways to solve problems exist; try to figure out which work best. "That's an interesting approach. Will it work all the time? Show your work in a way people can follow."
Obedience: Smart and easy are common words. Speed is key. "Wow, that was fast! You must be smart!"
Risk: Challenge and try are common words. "This problem is really interesting. Nice job, you really tried hard! I have a challenge for you today."
Obedience: Students speak up when they know the answer.
Risk: Students speak when they have a question, notice something, have an idea, build on a student's thinking, agree or disagree, or have an answer.
Obedience: Students are passive and do what they're supposed to.
Risk: Students are encouraged to takes risks and expect them to come up with novel ideas. "You might be inspired by Alvin's example. Is there a volunteer to try? We will help."
Source: Zager, Tracy Johnston, Becoming the math teacher you wish you'd had, 2017
Willingham's cognitive principle is children differ in intelligence, but the good news is intelligence can be improved through persistent hard work. This has been the Asian educational view for a long time, although with Dr. Dweck's growth mindset ideas, Western thought is changing in that direction. Intelligence is essentially how "people reason well and catch on to new ideas quickly." The current view of intelligence is that there is a general intelligence (g), which contributes to verbal and mathematical intelligence. Therefore, verbal scores are related to math scores, although individual verbal scores relate closer to each other. The "g" is not clearly known, but could be related to the speed or capacity of working memory.
What Makes People Intelligent?
It's the classic nature vs. nature debate; is it genetics or the environment that makes people intelligent? Through many twin studies, genes are responsible for about 50 percent of our smartness. What's interesting is that is starts off young, about 20 percent, then increases to about 60 percent in later life. The bottom line: genetic effects can make people seek out or select different environments. For example, imagine you start off life with a little better memory, more persistence, or simply more curiosity. Your parents pick up on this trait subtly, and begin to use a larger vocabulary or discuss deeper-thinking ideas. This leads you to spend more time with "smarter" kids, and grades become a natural focus. On the other hand, genetically you may not have the physical abilities, which leads you to avoid many sports and instead pick up a book and read instead.
Though genetics plays a large role, intelligence is malleable and can be improved.
Implications for the Classroom
Praise Effort, Not Ability
You want kids to understand they are in control of their intelligence. Praise effort, persistence, and taking responsibility for the work. Be careful of insincere praise, as kids are not easily fooled.
Hard Work Pays Off
Remind students that it takes hard work to be smart, just like it takes hard work and practice to be a successful athlete; natural talent can only take you so far.
Failure leads to Learning
Again, the most successful people (think entrepreneurs, inventors, athletes) take risks and fail in order to succeed. Michael Jordan talks about all his mistakes and failures on the court, which ultimately led to his greatest successes. Remind students that failure is not necessarily embarrassing or negative; it's an opportunity to learn something new.
Study Skills are Necessary
Help struggling students with techniques and methods of effective studying, memorizing, and organizing their time. They need to be self-disciplined and resourceful, as well.
Catching Up is the Goal
In order to catch up with the brighter students, they will need to work even harder than them. There is no easy solution or magic pill. They may need to revamp their entire schedule and drop activities that do not contribute to their educational goals.
Show Confidence in Them
As a teacher, set high standards and expect students to meet them. If they do an substandard job, simply state what they have done and give them feedback for improvement. Do not overpraise them for a mediocre job.
Source: Willingham, Daniel T., Why Don’t Students Like School? (2009)
Teachers nowadays are being asked to differentiate learning by meeting students’ individual learning styles, differing cognitive abilities and multiple intelligences. Is this possible? And how effective is it? Willingham, a cognitive scientist, turns that notion on its head. He states that children are more alike than different in terms of how they think and learn.
COGNITIVE STYLES VS. ABILITIES
First, let’s differentiate between cognitive styles and cognitive abilities. Cognitive ability is the capacity for success in certain types of thought; for example, mathematical concepts. Abilities are how we deal with content and how well we think. Cognitive styles are biases or tendencies to think in a certain way, such as thinking sequentially or holistically. Styles are how we prefer to think and learn. Of course, more ability is better than less, but one style is not better than another.
COGNITIVE STYLES (a sample list)
Three characteristics of cognitive styles: 1) stable within an individual during different situations and times; 2) consequential: has implications for future actions; 3) not an ability measure
There are people who have very good visual or auditory memories. However, Willingham explains why teaching different modalities to learners with a prefered style is ineffective. He gives the example of a visual learner and an auditory learner learning vocabulary words. In theory, showing the words with pictures to the visual learner while playing a tape with words for the auditory learner should be most helpful. Yet studies show this is not the case. Why not? Because it is not the auditory or visual information that is being tested--it is the meaning of the words. Generally in schools, students need to remember what things mean, not what they look or sound like. So, if this theory is wrong, why do 90% of teachers (and students) believe it to be true? Willingham chalks it up to several reasons, the first being accepted wisdom: it must be right because everyone believes it. Another reason is because a similar fact is true: kids are different in their visual and auditory memories. Learners may have good visual and auditory memories, but this not being a “visual or auditory learner.” Lastly, the psychological phenomenon known as confirmation bias comes into play here. Once people believe something to be true, then all future ambiguous events are seen through that viewpoint. For example, people believe crazier things happen during a full moon, and, in fact, crime and births increase during a full moon. However, when there’s an uptick in crime and babies on non-full moon nights, no one bats an eyelash. In conclusion, Willingham says that all cognitive styles, not just visual-auditory-kinesthetic, suffer from the same issues; at best, the evidence is mixed.
ABILITIES AND MULTIPLE INTELLIGENCES
Over the years, studies and experiments have shown that some kids are good at math, some are musical, others athletic, but not necessarily the same kids. This must indicate there are different mental processes at work here. In the mid-1980s, Howard Gardner, a Harvard professor, proposed his theory of multiple intelligences: linguistic, logical-mathematical, bodily-kinesthetic, interpersonal, intrapersonal, musical, naturalist, and spatial. At the time, many psychologists felt contention to Gardner’s theory. However, educators were (and are) interested in the three claims of his theory: 1) they are intelligences, not abilities or talents; 2) all eight intelligences should be taught in school; 3) many or all of these intelligences should be used to teach, matching the different intelligences of students. Gardner made the first claim, while the other two were made by others, although Gardner disagrees with them. Gardner argues that some abilities, in particular logical-mathematical and linguistic, have greater status in education than say, musical ability. He questioned why one was called “intelligence” while the other was a “talent.” Claim 2 is made on the basis of equity and fairness, that all intelligences should be acknowledged and celebrated. However, Gardner feels that curricular decisions should be made by the values of community, and his theory should only be a guide. Cognitive scientists believe Gardner has simply relabelled talents as intelligences, rather than “discovering” musical or spatial intelligence. The third claim is to use multiple intelligence modalities to introduce new knowledge. For example, when learning how to use commas, students could write a song about commas (musical), search the woods for things that look like commas (naturalist), and create sentences with their bodies (bodily-kinesthetic). So, in theory, students would come to an understanding of commas easier if taught with a particular intelligence in mind. Gardner wholeheartedly disagrees with this notion. The different abilities are not interchangeable; mathematical concepts need to be learned mathematically, and skill in music will not help. Writing a poem about your bat swing will not make you a better batter. These abilities are separate enough that one strong skill can’t compensate for a weaker one.
CONTENT VS. STUDENTS
Since catering to cognitive styles have been shown to be essentially ineffective, think in terms of curricular content. For example, in socials, a country’s geography should be seen, an anthem should be heard, and a traditional meal should be made and eaten.
CHANGE PROMOTES ATTENTION
Variety is the spice of life and the surge in energy during lessons. Switch between talking and listening to something visual; go from deductive thinking to free associative thinking; quick brainstorming could lead into thoughtful, reflective responses. Give all students practice in these different mental processes.
VALUE IN EVERY CHILD
Every child is unique and valuable, regardless of their intelligence. Trying to be equitable and egalitarian and have everyone possess “multiple intelligences” may be misleading. Also, determining who is “smart” depends on which intelligences you consider and at what level; is it top 10 percent or top 50 percent? In reality, there will be many students who are not especially gifted in any of the intelligences. Telling a child they are smart or have a skill in an area they don’t rarely works. In fact, telling a child they are smart actually backfires in reality.
Source: Willingham, Daniel T., Why Don’t Students Like School? (2009)
Daniel H. Lee
This blog will be dedicated to sharing in three areas: happenings in my classroom and school; analysis and distillation of other educators' wealth of knowledge in various texts; insights from other disciplines and areas of expertise that relate and connect with educational practices.