Teaching Language & Methods

The ability to work independently; a willingness to try even if they aren’t sure they’re right.

An understanding of what numbers mean and their relationships to each other; the ability to work with numbers fluently using efficient strategies, not just memorized processes.

An awareness of their own thought process; the ability to not just get a correct answer, but explain how they got it; a deep understanding of why strategies work.

...from any number, to any number, by any number, forwards and backwards.

Counting starts at the beginning of math development and is the key to developing Numerical Fluency – smart addition and subtraction skills. Our goal with Counting is for students to move beyond counting by 1s to counting by larger numbers, fractional numbers, and in patterns.

The whole is equal to the sum of its parts; any one part is equal to the whole minus all of the other parts.

Understanding Wholes and Parts is critical to the development of Number Sense. A strong understanding of Wholes and Parts and their relationships allows students to break down complex problems and is a key framework for solving word problems.

Reasoning in groups.

Proportional Thinking is introduced early in the Mathnasium curriculum. Students practice “reasoning in groups” of physical objects like lemons and glasses of lemonade, aided by visual clues. Grasping this fundamental concept leads to a stronger understanding of more abstract topics such as ratios, proportions, direct and indirect variation, and algebraic reasoning.

How many? Of what?

Quantity and Denomination examines two aspects of numerical value. Quantity asks “how many?” or “how much?” Denomination asks “of what?” These aspects are obvious in objects (4 apples) or measurements (10 miles), but also apply to more abstract concepts such as place value (49 = 4 tens and 9 ones), fractions (2/3 = 2 thirds), and algebra (4x^2 = 4 x-squareds).

We can only add and subtract things that have the same denomination (name).

We naturally apply the Law of Sameness without thinking about it. We can’t add quantities of apples and bananas unless we give them the same name (fruit). The Law of Sameness can be applied to many mathematical concepts, including addition and subtraction of whole numbers, decimals, fractions, units of measure, and combining like terms in algebraic expressions.

How much altogether?

Example: For 7 + 4, we would say “How much are 7 and 4 altogether?”

How far apart? How much is left?

Example: For 9 - 7, we would say “How far apart is 7 from 9?” For 9 - 2, we would say “If you take 2 away from 9, how much is left?”

This number, that many times.

Example: For 9 x 3, we would say “How much is 9, 3 times?” or “How much is 3, 9 times?”

How many of these are inside of that?

Example: For 54 ÷ 6, we would say “How many 6s are in 54?” For 54 ÷ 7, we would say “How many whole 7s are in 54? How much is left over?”

Asking guided questions to lead students to come up with the answer.

Example: The instructor asks the student, “So what does it mean to cut something in half?”

Presenting concepts as statements rather than questions.

Example: The instructor says, “In the order of operations, multiplication and division are always done in order from left to right.”

Anticipating student needs and preemptively preparing them for success.

Example: The instructor asks the student to explain how they are going to solve the first problem on a page, rather than waiting for the student to ask a question.

Not too many words (but not too few); Keep It Short & Simple.

Example: The simplest explanation for what “half” means is: “two equal parts” or “two parts, the same.”

Reinforcing concepts without pressure to “get it” immediately.

Example: After walking the student through the first few problems on a page, the instructor asks them to try the next one on their own and moves on. When they come back, the student says “I forgot what to do.” The instructor does not say “Why don’t you remember? We just did this!” but instead guides them to recall how they solved the previous problems.

Teaching strategies to enhance numerical fluency and limit reliance on algorithms.

Example: When the instructor sees the student writing down “99 x 3” to solve with the multiplication algorithm, they introduce the idea of thinking “99 + 99 + 99 is really close to 100 + 100 + 100.”

Introducing, explaining, or reinforcing concepts with meaningful pictures, diagrams, or objects.

Example: A student is having trouble understanding the concept of borrowing, so the instructor uses base 10 blocks to illustrate the idea of breaking a group of ten into ten ones.

Asking additional questions beyond what is printed on the page.

Example: A student breezes through a page on finding half of even numbers up to 24, so at the end of the page the instructor asks the student, “So how would we find half of 3? Of 44?”