Grades 1-3
Why We Teach
Laying a Foundation of Conceptual Understanding and Remembered Skills
We teach mathematics because it is a fundamental skill our students need to function in life. We emphasize it in Lower Elementary (Grades 1-3) because children at this age are ready to develop the mental rigor and discipline mathematics makes possible.
We have three main goals for our Lower Elementary math program:
To build math confidence. We want to enable all of our students to view math as a fun process of solving quantitative problems. We want them to graduate with an earned confidence in their abilities to handle mathematical challenges.
- To habituate the right approach for tackling math problems. We teach our students not by rote memorization and drill, but by guiding them to develop a deep, conceptual understanding of the meaning and use-value of mathematics. We also emphasize the importance of paying close attention, of double-checking results, and understanding errors, so we can avoid repeating them. Once introduced this way, our students are less likely to fall into the "memorize and forget" trap later in their schooling, and will approach math problems with the attention to detail they demand.
- To build a remembered foundation of math facts. In her early elementary years, each student has to develop automaticity in a series of basic math skills and facts. As teachers in the upper grades well know, if a student does not learn skip counting or fails to memorize her multiplication tables early, she risks falling behind later in other areas of math. Our students learn their addition, subtraction, multiplication and division tables—and experience how knowing math facts enables them to be faster, more accurate workers, thus motivating their further efforts. They come to crave, rather than fear, the challenge of a sophisticated mathematical problem, implicitly knowing that they've developed the capacity to meet such challenges.
How We Teach
Conceptual Understanding By Moving From Concretes To Abstractions—And Retention Through Daily Practice
The Montessori mathematics materials enable our students to obtain an unparalleled conceptual understanding of mathematics. For example, instead of memorizing rules about place value or carrying, our students actually experience these mathematical concepts simply and concretely, with basic bead materials. They deepen their understanding through a variety of sequentially organized activities—and gradually develop a more abstract understanding as they graduate to pencil and paper. They also progress from simple problems to more complex ones within each level of abstraction.
Once students have learned a new skill, we ensure they attain proficiency at it through deliberate practice. We include systematic, daily math facts practice. This practice consists of problems for keeping skills fresh, and "story problems" that draw on the Singapore Math program we use in the upper grades, which enables our students to apply their skills to real, and increasingly complex, problems.
This combination—a deep conceptual understanding, consistent facts practice, and application opportunities—makes for an excellent math foundation, and leads seamlessly into our Upper Elementary & Junior High/Middle School math program.
How Montessori Math moves from concretes to abstractions: the example of Long Multiplication
Children first encounter multiplication in our Montessori primary program. They learn that it is a special form of addition—that is, putting the same quantity together multiple times. They use the Colored Bead Bars for this: these bars are made of different colored beads according to the numerical value of the bar. The 10-bead bars consist of 10 golden beads strung together; the 9-bead bars have 9 dark blue beads; the 8-bead bars are brown, and so on. To do a multiplication problem, let's say 7 x 4, the student would take four of the white 7 bead bars, and count all the beads to get the result, 28. He would then convert the result into two golden ten bars, and a brown eight bar to symbolize 28.
Gradually, in late primary and into elementary, students are introduced to more advanced multiplication problems and strategies to solve them more quickly. They also move from very concrete presentations to more abstract ones, and finally graduate to solving long multiplication problems with pencil and paper alone. Throughout, our teachers have a wide variety of materials at their disposal—materials which build upon each other, and are integrated by a consistent, systematic use of colors (which helps as a memory aid).
- Students learn about the decimal system (place value) with the Golden Bead materials—units of individual beads, bars of ten beads, squares of 100 beads and cubes of 1,000 beads. Our elementary students also get the unique experience of working with the Wooden Hierarchical Material, which demonstrates, in concrete terms, the proportionate difference in size between a single unit and a million!
- They learn skip-counting with the Bead Chains that repeat the colors of the smaller bead bars: for example, a short 5-chain has five light blue 5 bead bars hooked together, and will make a square of 25 when folded together.
- They use the multiplication board to understand and begin to memorize the times table. On this board, children set up and develop their own multiplication tables, which they often bind into little booklets and use to memorize their multiplication facts.
- They are introduced to multiplying larger numbers using the Golden Beads—exchanging units of beads to tens for carrying, and tens to hundreds. (Of course, they have first learned to add, and are now simply adding the same quantity several times.) The photo above shows the quantity of 6,425 set up with the Golden Beads, a set-up that makes it very clear what large numbers the children are dealing with, as there are thousands of beads in this set-up.
- They learn to multiply more abstractly with the Stamp Game, where units, tens, hundreds and thousands are represented by color-coded number squares, instead of beads. The photo show 2,321 x 3 set up with the Stamp Game.
- They are introduced to long multiplication (where the multiplier has two or more digits) with Montessori's unique Checkerboard material. Here the place value is indicated by the bead's position on the board, and partial products are made visible. After setting the problem up with numerical tiles placed around the checker board, the child places the designated number of colored bead bars in the correct spots. For our example, he would take three light blue 5 bead bars, and place them in the unit square; three green 2 bead bars in the tens square, three yellow 4 bead bars in the hundreds square, and three lavender 6 bead bars in the thousands square.
This YouTube Video provides an excellent demonstration of using the checkerboard to solve a single-digit multiplication problem, a point of familiarity for students just being introduced to the Checkerboard:
- They work with the Small Bead Frame, and then the Large Bead Frame, the Montessori version of an abacus, where place value is indicated by bead position, and where students need to apply math facts to move the right number of units, tens, hundreds and so on.
Throughout, we introduce our students to increasingly more complex multiplication problems and ever larger numbers; we also guide them to apply math facts to work faster:
- More complex problems. The multiplicand will grow to two digits, then three. The materials help to visualize what that means. For example, the differently colored squares in the rows of the checkerboard indicate the decimal places for the results.
- Larger numbers. Our students are fascinated by and eagerly do problems into the millions and beyond. With the Large Bead Frame, students can do math into the millions—and the Checkerboard can generate results up into the billion range. Not only do these large quantities challenge our students' skills, they are inherently motivating to youngsters who are enthusiastic about digging into big work.
- Using memorized math facts. Instead of counting out multiple bead bars, then exchanging with the checkerboard, we guide our students to do the math facts in their heads. For instance, to solve 6 x 8, instead of putting eight six bead bars on the checkerboard, they arrive at 48 in their heads, and then place an 8-bead bar in the units, and a 4 bead bar in the tens. This shows students how knowing the facts makes them more efficient, and provides motivation to learn the facts. It's also necessary to solve problems on the Bead Frames—an example of how mastery at one stage in the sequence opens the door to the next stage.
- Writing the problem on graph paper. We teach them how to write down the problems on paper using the correct place values, and how to document partial products. This facilitates cross-checking and identifying the source of errors.
What We Teach
An Ambitious Curriculum of Arithmetic and Geometry
The Montessori Lower Elementary (Grades 1-3) mathematics curriculum is quite ambitious, especially compared with the California math standards. We teach arithmetic, and by 3rd grade our students complete the four basic operations (addition, subtraction, multiplication, and division) into the millions. We teach geometry, and by 3rd grade our students can describe the major geometric shapes and solids, and measure their areas and volumes.
Because many of our students join us from Montessori preschool (also called "Primary"), they can build upon their earlier math training and progress rapidly. For those students who join us from a non-Montessori background, we initially work with them through the more advanced primary materials to get them caught up and provide them with the same unparalleled basis of numeracy.
Our program covers the following areas:
Basic numeracy. This is a short review for our Montessori primary students, who have learned these skills in the mathematics area of preschool. Students new to our program will use advanced primary materials—from the Bead Cabinet to the Golden Bead Materials—to learn counting, skip counting, place value (the decimal system), and the basic four operations of arithmetic. With this exposure to the primary materials, our outside students also acquire the skills of using concrete materials to set up and solve mathematical problems—a skill they will use throughout their Lower Elementary years.- The four operations of arithmetic into the millions. By 3rd grade, our students will have mastered the four operations into the millions—including long multiplication with multi-digit multipliers, and long division with three-digit divisors. We also teach them the basic laws of arithmetic (e.g., associative, commutative and distributive properties) as well as multiples and factors.
- Decimals and fractions. Our students gain a solid understanding of decimals and fractions: they learn what each is by using the materials; then, they learn to work arithmetical problems with decimals and fractions.
Geometry. A deep geometry curriculum is a unique strength of Montessori elementary mathematics. Building upon their introduction to geometry in the Sensorial area in preschool, we introduce our students to congruency, similarity and equivalence. Our students learn to draw, define and describe lines, polygons, angles, circles and geometric solids. They learn how to measure length and area—and apply those skills to the geometric shapes and solids. They acquire basic skills, such as using a ruler and a compass, which are prerequisites for the later study of geometry.
What We Deliver
Confident Young Mathematicians—With Conceptual Understanding And Automaticity With Math Facts
By the end of 3rd grade, our students have gained an unparalleled understanding and comfort with numbers large and small—and, most importantly, they have come to view themselves as capable of dealing with mathematical problems. They have experienced, over and over again, how their minds can tackle these challenges and successfully solve them. They understand what they are doing, and they know it; they don't just repeat a process memorized by rote or a meaningless pattern, but instead self-consciously apply a deliberate method. What we see here is nothing less than the birth of mathematics as a systematic science.
Our students:
- Have mastered advanced math concepts and really understand them. As the table below indicates, our students often learn and practice math far more advanced than the state standards. For example, by 3rd grade, they multiply by 3- and 4-digit multipliers, whereas state standards stop in 4th grade with 2-digit long multiplication. (Presumably, calculators are used for anything more complicated.)
- Have mastered their math facts. They need to free up space in their minds for the more advanced math concepts they will encounter next.
- Have learned that mathematics is a useful tool of measurement that can help them solve a wide range of problems. Our students consistently apply their understanding in word problems, which help make the utility of math real to them.
- Have developed a confidence in their ability to do mathematics—and, for the most part, a liking for mathematical problems. Because they experience success in math every day, because they are not discouraged by being placed in a slow math group or by receiving repeated bad grades, because doing math with the materials is a fun, stimulating and developmentally appropriate activity, our students tend to like math! And they are motivated to learn more as they progress in school. Since our students learn math by progressing from concrete materials to abstract operations, math is meaningful, fun and rewarding. That's why our students excel.
| Example skills of a typical LePort 3rd grade graduate in mathematics | Grade this skill is expected from public school students in California |
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