Learning math can be comfortable. Essentially, everyone can learn mathematics. When it’s presented well,
learning mathematics needs about the same ability level as learning to read. You may have heard of people who have gotten discouraged about learning math, but I believe that is because they had poor instruction.
I’ll discuss some of the ways mathematics can be presented well, but first I want to say what that presentation will accomplish, so it will be clearer what the good teaching does.
For learning a mathematical principle, a student needs several steps or stages.
A. Understand the statement of the principle.
B. Have a way of remembering it.
C. Recognize where to apply it.
D. Have enough practice using it to feel confident in its usage.
E. Get feedback and make corrections.
Here are some of the aspects of good teaching. Each of them helps one or more of the steps A through E.
1. Relate the new principle to something (mathematical or otherwise) that the
student already knows about. For example, basic algebra principles can be presented
as helping with arithmetic, like 2998 * 3002.
2. Sometimes draw a diagram or illustration of the principle, so it can be
visualized. Usually a larger diagram is better than a small one.
3. Sometimes make two statements of the principle, one easy to remember and one
precise. In any such statement, strive for clarity.
4. Show simple examples of the principle’s usage. This can even precede aspects 1 through 3,
depending on the complexity or abstractness of the principle.
5. Decide whether to present the proof. The proof can help with understanding
why the principle is true, which can help with A, B and C. Or it can be too
much of a sidetrack.
6. Observe the student using and stating the principle, and give feedback.
7. Spend enough time on the principle.
A student learns that he has a test coming up sometime next week; a week goes by, and the student just now starts studying for his exam: the day before it! We’ve all been there; sometimes, it’s unavoidable, but students who continuously fall into this trap of procrastinating need to improve their study methods.
One of the best scientifically proven ways to study is through spacing assignments, known as the spacing effect. Students divide their studying time up into multiple periods, instead of all at once, which is called cramming. Cramming will help students remember information in the short run; however, they will lose this knowledge over time. Spacing has the opposite effect: it is less stressful, and students will have more time to process information and store it into their long-term memories. According to numerous studies, students who study through spacing perform significantly higher on retests compared to students who crammed.
One major criticism from learners who are new to the spacing method is that they forget what they learn. They feel frustrated because the material they spent time practicing days before they’ve since forgotten and now need to relearn. Their perception is justifiable; people are bound to forget some information, but what they don’t realize is how crucial forgetting is in the learning process.
When students must relearn material, they strengthen their neural connections involving the subject, which helps them solidify the knowledge. Ultimately, most courses in school build on previous ones, so it’s imperative for students to understand and recall information from prior units. For example, when a student enters Algebra Two, it’s assumed that they know the material from Algebra One, and the course will build on those topics. However, if the student only crammed for the Algebra One tests, then most of that material has been lost, and there are many gaps. While initially, it may frustrate students to relearn material that they’ve just studied, it’s an essential part of the learning process. Finding these gaps also exposes what the student must review more, and by doing so, they will strengthen those connections and have greater ease remembering the material later.
The opposite effect occurs during cramming; students proceed past the unit without retaining much of what they’ve learned. Students can avoid this dilemma by working on their assignments diligently, quizzing themselves without aid from their notes, and acknowledging areas that require more attention. By implementing these strategies, they’ll create the habits to succeed in whatever field they’re pursuing, from sports to law enforcement to medical school.
“Show your work”, “Show your work”, “Show your work”, etc.! The all too often phrase said by math teachers to students. And the students reply is often like, “But, I got the right answer!”.
So, why isn’t the answer good enough for most math teachers? The answer to this question is that good mathematics is not about the answer, it is about reasoning, which will eventually lead to the correct answer.
Good mathematics is about reasoning. Solutions in mathematics should never involve “leaps of faith”, guessing, or the idea that it just seemed right. The beauty of mathematics is there is always a clear, logical argument as to why the solution is correct. The ideal should be that all students can learn how to present these arguments in such a way that anyone, anywhere can follow the solution. This is what I am calling a universal solution.
What constitutes a universal solution?
Well, first of all, it should be factually correct. All information contained in a solution should be true and accurate. Mathematics is built upon definitions and axioms. It is from there that we can launch forward to proving new ideas. In classrooms, most students are not focusing on the proof of new ideas, but learning to explain clearly the ones that already exist. Second, the solution should address the problem in a clear and coherent manner and not contain any irrelevant information. Finally, there should be a logical flow, from step to step, with sound reasoning as to why each claim can be made. In summary, always ask yourself, could my solution be understood by anyone, anywhere, anytime?
Example of a Universal Solution
Explain which value is greater, 0.003 or 0.0006691?
When comparing numbers, the digit to the left of any other digit will always represent a higher place value. For example, a 9 in the ones place will always be less than any digit that is in a higher place value, like even a 1 in the tens place. For 0.003, the first non-zero digit, 3, occupies the thousandths place while in 0.0006691, the digit in the thousandths place is a 0. Although there is a 6 in the ten-thousandths position and 6 is greater than 3, thousandths are greater than ten-thousandths, so any digit in the thousandths position will always be more than a higher digit in a lesser position. Therefore, 0.003 > 0.0006691.
When Should Students be Introduced to Calculators? I am sure that this topic could spark an interesting debate. Well, I certainly have no interest in debating, but, I would like to bring to light, my professional belief on this topic, after working with math students for almost 20 years.
First, I would like to explain theoretical mathematics as doing math independent of the world. It uses reasoning, proof, and abstract concepts to establish truth upon truth. While things going on in the world around us may inspire a theoretical mathematician with new ideas, the study of theoretical mathematics does not depend on the world around us. So, some might argue that if math does not have a direct “real-world application”, then it is useless. This is not true since even applied mathematicians draw on theoretical mathematics to solve problems related to the the world.
Research has shown that students receiving more instruction in theoretical math do better overall because they build a mathematical foundation that will allow them to extrapolate math to other real-world situations. If math is just taught for a specific application, then the student will not be able to transfer that knowledge to another context. Theoretical mathematics does not change. It is simply, truth.
Now, let’s get back to the calculators! What is the purpose/role of calculators in mathematics? To make it simple, calculators are used to speed up extensive calculations involved in real-world problems. The problem is that students who haven’t developed a strong math foundation are using these calculators for much smaller calculations that they should be doing by hand, to strengthen their grasp of numbers, or even in their head in some cases. When students do computations by hand, they develop a feel for number patterns and a respect for mathematics. They build a foundation of mathematics that will be evergreen, and thus withstand the test of time and any changing technology.
Consider this example: I watched an advanced 9th grader go to her calculator to compute 105-90. A person with a strong sense of numbers would likely “mentally” compute this by knowing that from 90, it’s 10 more to 100 and then 5 more past 100 so the difference is just 10+5 =15. Once this skill is developed, it is certainly quicker to state this difference without a calculator! Just in case you think this example is too “complicated”, what about watching a student perform 13-9 on a calculator? That is not what a calculator was invented to be used for!
So, what am I suggesting? I am suggesting that calculators aren’t really needed until a student has a solid grasp of number sense. So solid, they won’t “forget” how to do basic arithmetic on all numbers, including integers, fractions, and decimals. So, when is this? Well, clearly it will vary from student to student, but in general, I would go as far as to suggest not allowing calculators until at least precalculus. And even then, limiting their use. Even the AP Calculus exam and the SAT have “no calculator” sections!
I say to all math teachers, let’s put the pencils in student’s hands and give them lots of paper and let them DO MaTh! There is no shortcut or tricks to learning math. Each student has to walk the road and allow their brain to make the connections.
Summer Learning—An Enrichment Opportunity
Summer school! What student or parent wants to hear that? Not too many, and that is because the term “summer school” has a negative connotation associated with it. Usually it insinuates that the student fell short during the school year and has to use the summer to make up their loss. But what about “Summer Learning”? Does this sound more favorable?
What Do Studies Show?
According to a 2010 study by the Wallace Foundation, just 25 percent of school-age children participate in a summer learning program. Why should only failing students use the summer to play catch up? Why not all students use the summer to maintain and get ahead? I suggest that parents and students see the summer downtime as an opportunity for enrichment and learning for fun, if you will. Throughout the school year, our students experience a lot of pressures from grades and distractions from other students and activities and rarely get to learn for the sake of learning and to be enriched.
In addition, numerous studies have shown that students forget a portion of what they have learned during the school year over the summer. This causes many teachers to spend a lot of time reviewing skills and delaying lessons at the beginning of the year. Students just fall behind as they try to jump back into the school year and keep up.
Nature of Learning Math
In particular, math is one subject that requires consistent practice and repetition. Could you imagine if you interrupted your workout routine for 2 months and did nothing? Surely you might experience, weight gain, muscle loss, and a decrease in cardiovascular strength. It should make sense that the same thing happens to our math skills if we don’t stay in good practice.
There are many proven benefits to summer learning. These may include students’ grades upon their return to school, their attendance, and even classroom behavior. Summer enrichment and learning programs can be found in many places in the community. MaThCliX® offers summer math lab and several enrichment classes during the summer. Math enrichment can even be practiced at home! You can visit https://www.greatschools.org/gk/articles/build-math-skills/ to learn about some ways to build up math skills around the home.
Take a moment and think about your greatest talent or accomplishment. Did it come naturally to you? Regardless of your inherent skills and natural gifts, I’d wager there’s no fluke as to why you became proficient at it? You probably practiced it. Even those once-in-a-lifetime achievements are not flash-pan. It takes consistent rehearsal to master any skill, conventional wisdom pointing towards about 10,000 hours of active repetition to completely develop any proficiency, skill, or craft.
Cognitive sciences hold that the brain retains information through the creation of neural path via a process called “rehearsal”. Building a highway incorporates this rehearsal process into a simple analogy. The first time we see a new vocabulary word or equation, the brain begins the process of cataloging that information by making a very linear path to the idea. Prefatory manipulation of that same information soon develops that road into a two-lane highway, one could think of as an ellipse. Over time, the more we use a given fact or review a certain concept, we keep elaborating on this highway and eventually our brain adds passing lanes and short-cuts along that same highway; instead of one to and fro’ route created to reach the destination, the brain builds a myriad of possibilities to aide in our scholarly journey.
As soon as you find something you enjoy, or something that really sparks your interest, “go with it”. Do not hesitate to find sixth gear and “put the pedal to the metal”. As we break down individual pieces of something we begin to not simply understand what, but why. Several highly successful entrepreneurs and artists vigorously rehearse this truly simple process:
- Write. It. Down.
Get a small notebook, and keep it and a pen on you at all times. Pockets are for things, not your hands. Even a simple one-word note can lock it in your “thought processor”.
Did you capture everything the first time? What was it that you really wanted to say?
- Re-work. Re-build. Re-do.
Was that everything you wanted to know? Is that the only reason you wanted to know it? Is that the last time you want to visit this?
- HAVE FUN WITH THE PROCESS
If at first you don’t succeed, try, try again.
Many awesome things are nonsense the first time we experience them, right? At some point we all could not talk or form sentences, we could not write, we could not read… In grammar school we constantly rehearsed these skills, and continue to rehearse these techniques the rest of our lives. Mastering a skill takes no more effort than continuous performance and utilization of that skill. As the saying goes: “You know how you get to Carnegie Hall, don’t ya?” Practice. That first draft pales in comparison to that final, proofed, and re-worked draft. Your first solo will probably not be the best, but your last will certainly NOT be the worst.