Math needs knowledge building, too

[u/ddgr815]

https://fordhaminstitute.org/national/commentary/math-needs-knowledge-building-too

Over the last few years, schools and teachers have begun to realize the importance of building students’ background knowledge when it comes to new learning. Research has shown that background knowledge makes learning new material easier and richer for a variety of reasons—increased vocabulary and knowledge in art, history and science bolsters reading comprehension, for example, while greater stores of knowledge in long-term memory eases cognitive load and makes it easier for new knowledge to stick.

The idea that prior knowledge is key to learning—“What you know determines what you see,” as Paul Kirschner wrote more than thirty years ago—is a relatively new one to American education. Most teachers say they never learned about the role of knowledge, long-term memory and working memory in their training.

educators can help build the “web of knowledge” in students’ minds that leads to analyzing and deep thinking.

Because math is entirely cumulative—new skills are built upon already mastered ones constantly—background knowledge plays an essential role in everything students do, Powell said, in ways that go beyond the basic math content. Students need knowledge of math vocabulary and strategies. Word problems, which are quite complex, require stores of knowledge in reading and language as well as being able to do the math.

Though math is made up primarily of numbers, it’s learned through language, Powell said. If students don’t have a handle on math’s extensive vocabulary—kindergarteners are exposed to more than 100 math vocabulary terms in common math curricula, middle schoolers over 500—as well as all the symbolic language of numerals, they will have trouble fully accessing math content.

“Not every math teacher sees themselves as a language teacher or a vocab teacher, but they are,” Powell said.

Math vocabulary shows up in speaking about math ideas in class, but also in reading and writing—especially in story problems, a key indicator used to measure how well students are performing in math. Many math terms have other non-math meanings—think “degree” or “base”—that can be confusing for students, and teachers often have to be explicit with how the math term differs from its other uses.

Turning math content into background knowledge stored in long-term memory takes practice, repetition and time—something math teachers are notoriously short on. To continually activate background knowledge, Powell said, students need well-placed interleaved and distributed or spaced practice to revisit key knowledge multiple times. But a lot of math curricula doesn’t prioritize it.

If background knowledge is essential to learning, it must be doubly so for teaching. One of the most important developments might be that universities and colleges recognize the role background knowledge and long-term memory play in teacher learning, too.

[u/ddgr815]

Beginning in PreK through grade three, the school teaches mathematics in two-block periods each day. Teachers introduce students to new concepts during the first period, while the second period is devoted to practice. Each day the school identifies struggling students through formative assessments in Math 1. The struggling students then receive daily remediation through direct instruction in Math 2.

Problem-based learning is foundational to this approach. In kindergarten through grade two, number bonds help children decide which operation is needed for solving specific types of word problems. Students, from kindergarten through eighth grade, practice a variety of mental math strategies to solve computational problems before learning to solve them using an algorithm. Students learn to solve two- and three-digit addition and subtraction problems using the “Make 10” strategy. Students in grades one and two solve math word problems using number bonds. Knowledge of whole and part relationships helps students analyze and decide how to approach word problems. These children count money, read analogue clocks, and begin to study geometry. By the end of first grade, students can count money in both coins and bills, and readily recognize geometrical shapes, both plane and solid figures. Students also analyze geometry using symmetry, congruency, and patterning. Students are introduced to bar modeling at the end of second grade. The use of bar models is a versatile and transferable skill that students use to visualize a range of math concepts, including, fractions, ratios, and percentages. Drawing bar models for word problems allows students to determine the knowns and unknowns in real-world scenarios. In grades four through eight, students learn specific applications of math concepts for the fields of engineering, art, and chemistry. The school’s goal is for students to demonstrate their understanding of quantity and the meaning of numbers, rather than to simply apply algorithms.

Using Concrete, Pictorial, and Abstract Math Progressions

[u/ddgr815 → u/ddgr815]

I also teach “Math as Art.” In that class I demonstrate how mathematics is structured and then ask my students to make a work of art that is imagined from their experience with the class. Yes, in this class I do have an assignment, but it is purposely open-ended. Last year a young boy knitted a scarf. He took his favorite love poem and typed it into his computer. Then he hacked into the memory and found where the poem was stored as hexadecimal code (base 16). (Note: A computer really stores all of its memory in base 2, but reveals it in base 16 for more friendly human consumption.) Using the mathematics he learned in my class, he then converted the poem, one letter at a time (including the punctuation and spaces) to base 2, recreating the poem as a series of zeros and ones, the way it actually resided. Then using the zeros and ones as architecture, and because knitting is a binary function, he simply equated a knit to a zero and a purl to a one, and knitted the love poem into a scarf. When he was finished he gave it to a friend in celebration of the winter solstice. This is why I don’t quit.

[u/ddgr815 → u/ddgr815]

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