In math instruction, everyone wants a “magic bullet” strategy that will help all students understand math immediately–and, I’ve found it!

Many times I’ve had students in my office struggling with a difficult concept we’ve discussed in class or, unfortunately, upset about a recent bad grade on a quiz. After going over the problem and doing some examples, I hear a common refrain: “I understand the problem when we do it together but when I sit down to take the test I forget everything and my mind is a blank. I read the book and the examples and I know I that know the material. What’s wrong with me?”

I can see the frustration in their face and hear it in their voice. And I completely sympathize. Luckily there is a helpful solution. It’s a common phrase among mathematicians: Mathematics is not a spectator sport. To really learn something, to see the pattern of a solution and to internalize it, you have to get your hands ‘dirty’ and do it yourself. I realize that can be a tall order so I follow the advice I was given when I struggled with the same thing as an undergraduate student.

When I work through an example with a student, I’m always careful to make sure they are contributing to the development of the solution. I can see the big picture but in the beginning they can often only help with the small steps. Once we’ve finished with that problem and before we move on to the next question, I put away our previous work and have the student work through the problem again, step by step, completely on their own. Surprisingly, even just recreating a solution they just saw minutes ago can be a challenge. Sometimes I’ll push a bit more and ask how things would have changed had this part been negative, or what if that part had a difficult denominator. By playing around with the material on their own, the student internalizes the procedure and masters the concept.

It may take some time but the good news is that it works! And it works really well! The student doesn’t just walk away ‘feeling’ like they know something or understand what they saw me do. They know it! Because they just did it themselves and they made the process their own.

By Michael Munn, Private Tutor

If I were allowed to instantaneously eradicate one bad habit among physics students, I would choose their addiction to numbers. That is, the propensity of many students to plug numbers into an equation at the first opportunity. I think that this addiction stems from several factors: fear of algebra, the comforting familiarity of arithmetical operations and teachers who didn’t sufficiently emphasize abstract reasoning. Since it prevents students from gaining a deeper understanding of physics, I try hard to help students break this addiction.

It isn’t easy. The allure of numbers is powerful: a two is always a two, and three is greater than two and five times four is twenty—what’s d minus h? Or v squared divided by g? But if v is 5 and g is 10 then…For some students, numbers call like sirens and it takes a great effort to resist their call. “But I got the right answer,” is a common reply when I exhort students to leave the numbers until the end of the problem. As I’ve written elsewhere, solving physics problems isn’t just about getting the right answer. So I continue to confront students who reach for numbers like candies in the hope that the benefits of fewer numbers and more algebra become clear.

What are the benefits of going number-free? As I tell students, there are three immediate ones:

1. Checking your work becomes easier. If you didn’t get the right answer, it’s hard to figure out where you went astray if all you have in front of you are numbers. But with equations it’s much easier to see where you dropped a minus sign or forgot to square a factor.

2. Variations on a theme are no big deal. Say you’ve been asked to find out how fast a car can go around a rough banked circular track without skidding off course. You’ve gone through the calculations, plugging in numbers all the way, and you get an answer. You checked in the back of the book (or online) and it’s right. Now you turn to the second part of the question, which asks you to explain what happens when the angle of banking is changed. With only numbers on the page (or the screen)—oops! But if you have an equation for the speed in terms of the angle and other variables, there is much less work to do.

3. Physical meaning is more readily seen. To derive that the braking distance of a car is proportional to the square of the speed is not just about getting a numerical answer. The relationship between the two quantities describes a physical process—the energy due to the motion of the car (kinetic energy) is converted by braking (through friction) into heat.

And then there is a long-term benefit. If the student and teacher persevere, eventually the addiction is broken. The student starts to see things intuitively and has a feel for what it means to do physics. In the words of the Nobel-prize winning physicist Richard Feynman, “We have to learn…how to apply the mathematics to the physics in order to understand the world.”

By Neer Asherie, Private Tutor

A significant portion of the 2nd, 3rd, 4th, and 5th grade Common Core Standards Math are based on fluency with multiplication facts. Smaller numbers are introduced in second grade while all the basic facts are reinforced in 3rd. Once your child reaches 4th and 5th grade it is expected that they have the multiplication facts 0-12 memorized in order to move on to much larger numbers. Besides knowing this fluency, students must understand when to use these strategies as well as explain how they arrived at their answer.

In my years of teaching experience, multiplication facts are my absolute favorite thing to teach. Multiplication facts are great to teach because they have multiple entry points depending on the students prior knowledge, learning style, and grade level. I use the entry points of skip counting and building arrays in order to make the academic content accessible to all of my students.

Skip counting
Your young learner will benefit most from multiplication practice if they are able to skip count. This is best done by practicing the easy multiples first, of 1, 2, 5, and 10. Once those basic skip counting strategies are mastered up to 100, it is important to then begin skip counting by more challenging numbers such as 4, 6, 7, 8, and 9. Fluency in this basic skip counting will allow for automaticity and fluency with multiplication problems between 0-10 which are the basic building blocks of higher multiplication problems. This can be practiced around the house when counting, why not skip count instead of simply counting?

Arrays
If a student is lacking background knowledge in skip counting, a great entry point for multiplication is making arrays. This allows for all students, regardless of prior knowledge and ability access to multiplication tables. In my classroom it has made my students feel successful year after year. You simply show the young learner how to draw one. Let’s say, for example, the student must find out what 3 x 6 is. Show the student how to draw the number three six times, in six rows:

Then, we count all of the marks to find the answer. This also provides entry to the Commutative Property of Multiplication. From here, you can show the student how the array of 3 x 6 and 6 x 3 will always give you the same answer. Students will gain a deeper knowledge of how multiplication works and how to solve any problem they are confronted with. Also, introduction to the names of the mathematical properties are vital for success in standardized testing.

Reinforcement with games and “multiplication master” challenge
Once multiplication has been taught through the 0-12 times tables, it is vital that you continue reinforcing with your child. This reinforcement is crucial to automaticity with multiplication in the higher grades. Combined with plenty of positive reinforcement, I love to challenge my students to become “Multiplication Masters.” Students become multiplication masters by using skip counting to write out all of the multiplication facts between 0-12 and passing an oral quiz on the facts they just wrote down. This way, students are not only reinforcing multiplication, but also how to write equations and skip counting. Once my students pass all the way up to 12, I reward them as I see fit and also encourage them to help other students practice.

This is something that can be done anywhere. Anywhere with your child you can practice the facts and high-five them for correct answers. Oral word problems that include multiplication such as, “If there are 3 girls and each one has 5 apples, how many apples are in all?” are a great way to keep the multiplication fresh.

Besides practicing the facts, you can play multiplication “Top-it.” This is a very simple game used with only number cards, where each player must lay down two cards. Each player multiplies the two numbers they have laid down, states the answer, and whoever has the higher number gets the cards. After finishing the deck, the player with more cards wins.

It is important to not underestimate the importance of automaticity with basic multiplication facts. And, fortunately, it is something that is extremely rewarding to teach with multiple entry points for diverse learners.

Written by: Editorial Team, My Learning Springboard, Inc.