For true learning to happen in any subject, it must be understood intuitively. Cooking is a great example of this concept. Cooking might begin as following the steps of the process to the letter but discovering the purpose of each part of the recipe is when the magic of learning starts to happen. Then, the cook can alter the recipe, making substitutions or additions, customizing the outcome of the dish. Cooking also requires a variety of tools – pans, forks, spatulas, etc. A person can follow a recipe that says, “use an 8” nonstick skillet”, but once a person can answer questions such as: “Can you use a different cooking instrument to achieve the same goal? Why?”, then the person is truly learning how to cook. But learning these things doesn’t happen all at once. You wouldn’t expect someone who has never cooked before to be able to pick up Gordon Ramsay’s cookbooks and be able to make one of his famous dishes on the fly. That person would need to make that dish dozens of times before they had truly mastered how to make that specific dish and the reason for each ingredient. They could then use those ingredients and techniques in other dishes and, as they began to grow the repertoire of ingredients and skills they had mastered, could start to combine them in other ways to make new dishes.

Math is the same way. Students need to spend time with each topic, digesting it, learning its details, and they need to do this regularly and often to truly learn a concept’s ins and outs. Much like cooking, learning math requires building a web of skills that support each other and very little of it has to do with the ability to memorize a recipe. For example, students can memorize “7 times 6 equals 42”, but this is just a string of words and numbers that when put together make what looks like the understanding of a math fact.

Students who rely on memorization often confuse operations because they are not comprehending what the operation is, they simply remember the words that make up the fact. To go back to our cooking example, confusing multiplication with addition is like confusing a knife with a spoon – they are tools with different purposes and understanding what they are used for allows for more versatile use of both. In math, skills and concepts that have been learned become tools, while facts and rote processes that have been memorized can often be misconstrued, like trying to eat spaghetti with a spoon.

Many students can “get an answer” in math by memorizing a series of steps and following those steps to get to an answer. Most of the time, this mechanical process would go unnoticed as the correct answer was reached and no one is the wiser. However, many students eventually reach a point where this memorization no longer works, and they cannot rely on their ability to simply memorize a series of steps and then execute those steps correctly. Then, when a new set of steps comes in, the previously memorized steps often get thrown out to make room for the new information. All too often, this goes on for a long time until someone notices that the student is no longer able to parse what is happening in the classroom. When this is happening, students are doing math, but they are not learning math and haven’t been for some time.

When students need to be reminded about or shown tools again, the tools have not been learned and are not always available as tools when needed. Students go through stages of learning where they may be aware of a tool’s existence but not really know its function, much the same way as a child might be aware of the existence of a cheese grater but might not know when it is appropriate to use or even what it is used for. Even after learning what a tool’s function is, students still need to use the tool to perform that function regularly to be proficient in its use.

When a student is truly learning math, they can effortlessly recall which tools to use to approach problems they have done in the past as well as use those tools in new scenarios or applications. Being able to solve a variety of word problems without examples months after the core skill has been learned is a strong indicator of true learning and retention. Similarly, being able to describe a strategy using simple terms is also a strong indicator of a student’s comprehension of a strategy.

Some important indicators at various levels are included below. Often when we encounter students who are struggling, they have not been learning math for some time. Math is one of the few subjects where every new skill builds on previous skills and identifying opportunities to help students truly understand the strategies and reasons behind them is crucial to long-term retention and success.

Doing Math | Learning Math | Perils of Only Doing |
---|---|---|

Doing “7 + 6 is 13” by counting on fingers | Being able to explain “I know 6 plus 6 is 12, and 7 is one more than 6, so 6 plus 7 is one more than 6 plus 6.” | Miscounts or includes 7 in start of count, getting 12; lacks efficient strategies for more difficult problems |

Knowing “4x8 is 32” only as a memorized fact | Being able to explain “I know 4 groups of 8 means I can add 4, eight times or I can add 8, four times” | Mixes up with another memorized fact and says “4x8=36” |

Rounding 3672 to the nearest hundred using a multi-step algorithm | Understanding that 3672 is between 3600 and 3700, and is closer to 3700 | Missing or mixing steps, getting “700” or “3772” |

Doing “3/5 + 2/5 is 5/5” by “adding the top numbers and keeping the bottom numbers” but not understanding why | Being able to explain “3/5 means 3 pieces out of 5 in the whole thing and 2/5 means 2 pieces out of 5 in the whole thing, so if I add them together, I have one whole because I have all 5 pieces” | Simply adding numbers seen, getting “⅗ + ⅖ = 5/10” |

Getting “Half of 36” by trying to find half of 3 and half of 6 | Being able to explain “If I take half of each piece of a whole, I will find half of the whole. 36 is 20 plus 6, half of 30 is 15, and half of 6 is 3, so half of 36 is 18.” | Being able to find half of 26, but not half of 36 |

Can get an answer to proportions using “Cross Multiplying” | Being able to explain what a “proportional relationship” is and why Cross Multiplication works | Conflating “Cross Multiplying” with Multiplying Fractions |

Completing today’s math worksheet | Being able to do that same worksheet a month later | Viewing math as simply “finding an answer”, often rushing or making careless mistakes |

Needs someone to check for correctness | Can explain why the answer is correct | Falls into a pattern of “learned helplessness”, making learning independently difficult |

Problems with multiple steps are difficult | Problems with multiple steps are solved fluidly, real world applications make sense | Math is relegated to something only for school, real world application potential is not valued |

Relies on a calculator for arithmetic | Uses efficient number sense strategies | Struggles with arithmetic fluency begin to affect algebraic foundations |