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
Practice makes perfect, right? Kind of--but it takes the right kind of practice, something known as interleaving. Most of us are familiar with the usual "other" type: if we want to "master" the free throw, for example, we might take 300 shots in a row, until it gets dark, or our arms feel about to fall off.
Yet an 1978 experiment from the University of Ottawa involving 8-year-olds and bean bags demonstrated, practicing one thing successfully does not necessarily lead to transfer in a real situation. One group practiced throwing bean bags into a target from two and four feet. The other group practiced only at 3 feet. When the final test was conducted at three feet, the group that practiced only at two and four feet actually did much better.
Another study in 1986 from Louisiana State University studied how women players practiced three types of badminton serves. Group A performed blocked practice, rehearsing only one type of serve each session. Group B performed serial practice of the three serves. Group C did random practice of the three serves. After three weeks, group C (random) performed best, followed by the group B (serial), while group A (blocked) was the worst.
In 2007, at the University of South Florida, 24 grade 4 students had to calculate the number of faces, edges, corners and angles in a prism. Half the students learned in a blocked format, while the other half did the exact same problems but randomly. Results: the mixed-study group outperformed 77 to 38 percent.
Particularly in math word problems, the greatest challenge for students is in selecting the right strategy or algorithm needed. Therefore, mixed or random studying will give students the greatest amount of real practice necessary when they encounter a variety of questions and problems.
Repetition appears to show rapid improvement but then it plateaus. Varied practice, on the other hand, seems to show slower improvement but the results are a greater accumulation of skill and learning.
Here is where interleaving is so important to practice. It is the mixing related but distinct material during study. It works for pretty much all areas of learning: skateboarding, playing an instrument, cooking, algebra, art. Shorter breaks of different but related activities will achieve the most success, about 10-15 minutes each.
Real life is not so cut and dried, and you need to be able to recall a variety of facts, information and skills in order to solve real-life problems. Having experience in doing many different types of tasks will lead to the greatest amount of transfer of learning.
Source: How We Learn, Carey
We are using Khan Academy for independent skill learning and practice in math at certain times, or at home. For those who don't know, Khan Academy is an online one-on-one "tutor" for all math skills from probably kindergarten to university level. The great thing is that is keeps track of individual student progress, gives hints to problems, and videos to explain. It is all free and is a safe, secure website for students, parents and teachers.
I plan to use it during math for early finishers, as well as for others who need extra practice and assistance. However, I will also make sure that ALL students have some access during the week. We currently have 7 devices for them to use in class. Students, of course, can log in at home for extra practice.
There are badges and points for mastering skills, so there is motivation for kids, as well. From previous experience with students in previous classes, I've found that even those struggling find it often less stressful and more motivating to work with a computer and on their own, as opposed to sometimes working with peers or sharing in a large class. Also, using technology seems like fun to them, even when they're doing "work." (For example, kids love reading on Epic, a online book app, even more than actual physical books, it seems.) Also, in a short period of time (10-15 minutes), students are able to master skills quite easily.
The other great thing is that Khan Academy matches your child's skill level. Therefore, questions will simplify for those struggling or at a lower-grade level, or it will give higher-grade level questions for those who need a challenge. Just looking at our reading/writing levels in our class is a clear indication of the variance of levels in a single class.
Finally, if students can master all of the important skills and concepts quickly with the aid of Khan Academy, then we can devote more time in class for bigger picture ideas, such as problem solving real-life math issues, or working on math models or projects, for example.
Important note: Khan Academy should not be seen as a replacement for math study, but simply a powerful tool used in conjunction with math at school, home or elsewhere.
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.