Math Professional Development for Teachers

There is a hidden assumption about number—we are always counting something. In this video, we help you transform your understanding of the concept of number. This should be the first video you watch.

In the video Different Forms, Still Counting, you can see how fractions and decimals can also be understood as counts. From there, you might be curious how we think about conducting a count, something like 17(5) which we read as "17 of 5" or " 17 fives." Either way, it is count of fives, and we have 17 of them. You may want to watch the video of Eric investigating ways to conduct that count of 17(5).

As teachers, we've tended to focus our assessment on what learners are able to do mathematically. What Kevin has come to recognize is how important it is to attend to the ways learners think and communicate. In this video, we hear Kevin describe how he is shifting his assessments to include those behaviors that support a young person in learning.

Consider this. Even if a learner communicates a misconception, it is important to recognize that they communicated an idea. And, what if they were able to listen to someone else who offered a more accurate version of the idea and were also able to recount what that person said? That learner is exhibiting high quality learning even though their initial mathematical understanding was weak. They showed us that they have a capacity to learn the mathematics. Let's not miss this by being overly focused on their initial misconception.

So, beyond assessing mathematical knowledge, Kevin finds himself more and more interested in assessing and supporting the behaviors young people exhibit as they learn. For him, this is the Long-View. Children are always learning, with or without us. Kevin looks to optimize their willingness to ask questions, to listen critically, to learn so that they can learn at the highest level possible even when he is not there.

What feels like a good day is when all learners are participating in high level learning behaviors as they grapple with math concepts. He is watching for them to develop their understanding as they develop their ability to learn and support others in learning.

If you want a place to start, this graphic captures our thinking of what we might see when learning is happening.

We think of addition as a concept, an idea to be understood, rather than a mere operation or action to be done. Though many wait to introduce the idea of combining like terms until high school level math courses, at Long-View we start with this conception of addition from the very beginning. In this video, Kaylie guides you to read and consider additive expressions with this notion of addition in mind. She starts simply by reading each expression and then asking, "How many terms are there?" and "What is each term counting?"

The question of what is being counted is fundamental to how we reason with mathematical expressions. If you missed it, take a few minutes to watch as Kaylie introduces the idea that number inherently expresses a count.

Again, we access and strengthen our conceptual understanding through language, including reading the plus sign (+) as "and." When you think of and talk about addition as the combining of like terms, you nurture an understanding than can grow, develop, and transfer to numbers in any form (think fractions, decimals, and even irrational numbers!) as well as algebraic expressions. As you construct your understanding of addition in this way, you are Learning for the Long-View.

If you are ready to see where reasoning this way can take you, then we suggest watching Eric and Kaylie think together as they add mixed numbers.

And, yes, you can get one of those shirts.

If you have considered the ideas we shared in the videos Divide into Groups and Fraction Division Two Ways, then you have begun to construct an understanding of division as a multiplicative idea. You also may have come to recognize that the same conception of division can be applied to dividing fractions. We don't need a new procedure; we need only read and reason with the division as dividing the dividend into groups of the divisor. (You may remember that 8 ÷ 4 can be read as "eight divided into groups of 4." Then, the quotient (2) is the number of groups.) In this video, Eric shares one way of reasoning with fraction division that offers a conceptual foundation for understanding the commonly taught algorithm.

Here, we see one way to reason with fraction division is to have the fractions be counts of the same size pieces. That is, it works to have like fractions, with common denominators. Consider dividing ⅘ into groups of ⅖. Each group has ⅖ in it, so there would be 2 groups. But, the fractions in this video aren't like fractions. Eric creates like fractions much like Kaylie reasoned with fifths and tenths in this video on addition. Kaylie also created a great video on expressing wholes as counts of one-fourths.

With the first problem, Eric engages in relational thinking starting with the fact that 2(½) = 1 to arrive at the equivalence of ½ and 4(⅛). He could have also started with 8(⅛) = 1 and halved the number of eighths to yield 4(⅛) = ½. When you are Learning for the Long-View, you want to explore mathematically sound alternatives before arriving at the most efficient route. This allows you to grow your strength, stamina, and flexibility in reasoning with numbers. We are shooting for high quality thinking.

As Eric considers the second problem, he reasons with creating like fractions with thirds and fourths. Again, rather than invoke a memorized procedure, Eric guides you to consider how you can reason that ¼ · ⅓ = ¹⁄₁₂ in which case 4(¼) · ⅓, which is one whole ⅓, is equivalent to ⁴⁄₁₂. He uses an area model along with intentional language to help guide the reasoning and reassures us that in time we will internalize the area model. Then, the reasoning will be guided by intentional language.

We recognize that language is key to constructing knowledge. At Long-View, we make a consistent effort to use language that supports reasoning. This article is helpful as you begin to practice using language that supports mathematical reasoning. You might also find this quick reference on language helpful.

While strategies played an important role in math education by disrupting the over-reliance on a single procedure for an operation. Strategies helped us to think outside the box. However, teaching strategies doesn't honor intellectual freedom, nor does it promote the quality of thinking we are shooting for when we are Learning for the Long-View. We want to lead learners past procedures, past strategies to the kind of creative thinking that is possible within the beautiful discipline of mathematics.

The Long-View Math Block is designed to make this kind of thinking possible. Rather than a class, the Long-View Math Block is an experience, an experience in the discipline of mathematics, in which young people construct understanding by engaging as a community in practices authentic to the discipline. We encourage you to watch the video overviewing the Math Block. Or, if you prefer a good read, we have an article for you that captures how we are thinking about learning in a Long-View Math Block.

Studio is the portion of the Math Block when learners work in partnerships of 2 or 3 at whiteboards. Collaboration and visibility are crucial. Partners explore problems that require a variety of mathematical understandings. Studio is not “worksheet mastery practice” but rather an opportunity to try out new ideas, rehearse thinking out loud, with support from a partner.

The work that is presented on the whiteboard is reflective of the thinking of both mathematicians. They work to develop their skills as communicators and collaborators. Studio can be loud, messy, and full of life as the learners discuss and debate their ideas. We put together a graphic you and your learners could use as an anchor for building a conversation.

Learners are presented with a variety of problems during Studio. The mathematical footprint is large on purpose. We want learners to make connections between different problems and ideas. We want them deepening their understanding of familiar ideas, and we want them making attempts to transfer understanding to new problems. We have a sampler of Studio sets along with artifacts of student work available.

Studio often wraps up with a Public Share or two during which partners share their thinking on a selected problem. They may have excellent thinking or they may have come across a common misconception. The point of the Public Share is to have knowledge shared within the community. To get a sense of a public share, this video shows two young learners sharing their thinking following Studio.

A Long-View Math Block typically consists of two Thought Exercises, a Concept Study, and Studio. If you missed it, watch the short video overviewing the elements of the Long-View Math Block. You may also want to read an article we wrote that has great information on the Long-View Math Block.

Like all portions of the block, Studio is designed for learners to engage in learning. We have a graphic that may help prepare your learners to get the most out of their learning experiences.

Statements of equality often are misunderstood as a description of an action or operation. The equals sign is not an arrow. Rather, it indicates the equivalence of two expressions. Equivalence is a state or condition of being equal in value. Cathy explores several equations and describes how learners develop misconceptions if we overly emphasize a left to right interpretation of the equation, as if what is on the right of the equals sign is "the answer." Instead, lead them to understand equations as statements of equivalence.

By relating to equations, even simple equations such as 8 = 5 + 3, as a statement of equivalence, young learners establish an understand that will transfer to reasoning with algebraic equations. Yes, we made that video. If you are curious about how equivalence plays a role in solving equations, check it out.

A mixed number is inherently an additive expression. When you read it, you hear the terms, one of which is a count of wholes, or ones. For example, if you read "3 ½" as "three AND one-half," the "and" implies addition. That is, 3 ½ = 3 + ½. So, when asked to add mixed numbers, whether they are expressed using common fraction or standard decimal forms, you already have some like terms: the counts of wholes. In this video, Kaylie carefully reads the mixed numbers in an expression. Then, she combines the wholes and goes on to create like terms with the fractional parts of the mixed numbers. Keep in mind we are thinking of addition as the combination of like terms.

Kaylie explains a few ways to think about the relationship between fifths and tenths so as to express two-fifths as a count of tenths. First, she uses an area to model to consider the value of ½(⅕) and to establish the equivalence of two-tenths and one-fifth. She also offers a line of reasoning that starts with thinking about ten-tenths (which equals one whole) to arrive at the same equivalence. From there, Kaylie shows you how you can reason that since ⅖, or 2(⅕), is twice as many fifths as ⅕, it will be twice as many tenths.

Reasoning in these ways to create like terms with fractional counts is built on experience with relational thinking: considering the value of an expression by leveraging relationships such as doubles and halves. For example, since 1 whole is equivalent to 4 fourths, then 2 wholes is equivalent to 8 fourths (twice as many wholes means twice as many fourths) and 10 wholes is equivalent to 40 fourths. Watch this video to see Kaylie reasoning with counts of fourths using relational thinking.

With an understanding of a mixed number as an additive expression, you might be curious to see how you can reason with a product of mixed numbers. We have a video with Eric showing how both the distributive property and the associative property can be used to multiply mixed numbers.

In addition to commutation and association, multiplication allows for distribution. In this video, Eric shows how the distributive property can be helpful when multiplying mixed numbers. He also shows how you can express a mixed number as an improper fraction and set up for associating. That is, by using association, you can change what you are counting and look for opportunities to conduct your count (i.e., multiply) more efficiently.

It is also worth noting that by reading mathematical expressions, we access ways to reason with those expressions. At Long-View, we read addition and multiplication intentionally as "and" and "of" respectively. We find this choice of language supports reasoning. You can read more on our use of language. Also, you can see how we might reason with adding mixed numbers Kaylie's video Five-Tenths AND Two-Fifths.

The goal is thinking. Mathematical reasoning often gets undermined by following a set procedure. If you leverage what you know about multiplication—that it distributes over addition and is both associative and commutative—you can open up multiple lines of reasoning and be creative as a mathematical thinker. Hear what Lisa says about high-quality, creative thinking.

***NOTE: Eric misspeaks at 7:03 when he reads 1/5 as "five".