This lesson begins the first algebra chapter for grade 8. The lesson is taught in Swiss German and is 45 minutes long. There are 25 students in the class. Of the three possible tracks (with basic, extended, or highest demands), it is a “track two” class with extended demands.
| 00:31–05:41 | (Whole class) The teacher begins the lesson by discussing the meanings of “terms” and “variable.” He then shows how the statement “5 – 3 = 2” can represent the difference between the lengths of two directed line segments of lengths 5 and 3, using a compass to copy each length from two drawn previously. |
| 05:41–14:22 | Students work individually to represent 3 • 5 – 2 • 3 = 9 with directed segments of lengths 5 and 3. The teacher asks them to use compasses to draw the lengths. |
| 14:22–23:06 | (Whole class) With directions from a student, the teacher draws a representation of 3 • 5 – 2 • 3 = 9. The teacher asks the class how to represent 17 and 26 using strips of paper. Yellow strips represent length 3, and blue strips represent length 5. Students come to the board and use strips of paper to show their representations:
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| 23:06–27:35 | The teacher circulates as students work individually using the blue and yellow strips to represent 15, 12, 13, and 3. |
| 27:35–36:27 | (Whole class) Students come to the board to present their solutions:
The teacher assigns students to work in pairs on one representation for the following problem: Using directed line segments labeled x, y, and z, represent:
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| 36:27–42:40 | (Whole class) Pairs of students come to the board to present their solutions. |
| 42:40–46:32 | The teacher presents a problem of calculating with variables: an expression written in terms of variables that represent “real-world” costs. He asks students to explain what cost the expression represents. |
(20:00–23:02) The whole class works as a group on new material. Rather than offering his opinion of a student response the teacher encourages other students to give their reactions. (Compare 20:00–23:02 with segments of US1 where students work in small groups [1:13–1:41, 2:54–3:25, 6:39–7:20, 11:37–13:12, 30:42–31:26 ]. The students in this U.S. lesson had worked in pairs before, but received no explicit instruction about how to work in groups of four.
The strips and line segments the class uses to represent quantities are similar to the “bar models” commonly used in Asian lessons (see 36 student problem solutions). A difference may be that the strips and line segments appear to be new to the students in this lesson; in Singapore students are introduced to bar models in the third grade.
(14:30–16:44) The teacher introduces the strips. (18:41) The teacher says, “If you calculate 3 × 5 you have to take the three 5 times and not one sheet of 5 and one sheet of 3.” The teacher makes an important connection and differentiation of how to represent additive and multiplicative expressions.
However, a possible drawback of drawing the line segment using a compass is that the class devotes a lot of time and mental energy to an activity that is not the focus of the exercise.
The entire lesson seems to be about making connections: using a length to represent a number, asking students to consider simple numerical statements in a visual representation, and then asking the students to transfer these representations into expressions containing variables. For example, in one segment (42:48–44:59) the teacher gives an application problem near the end and launches new problem, which serves as a partial summary of the lesson.
(15:03–16:44) The teacher lets the student show the other way of writing the number 17, and says, “Good, well done,” but could have used this as an example of another way to represent a number. Many of the other students may not have understood what the student said.
(17:07–19:39) One student confuses multiplication and addition. The teacher pushes through to an answer, and then another student points out the error. The teacher may have expected the correction to be made earlier and did not want to point out the error himself. Teachers need to consider what to do in such a situation.
The teacher made sure to connect all examples to the ideas he was presenting. He built up from a simple situation (line segment represents number), to a more complicated one (color strip represents number), to an even more complicated one (line segment represents variable name), followed by a non-visual example (variable represents amount of money).
The teacher presented three different approaches in the one lesson—perhaps too many. After the class used color strips with numbers, the teacher went back to line segments for the section on variables and did not use the color strips again. At the end, the teacher introduced non-geometric quantities and representing them by variables.
The national research coordinator describes the main principles in the national and canton guidelines. These principles include:
Principles 2, 3, and 8 touch on some of the same issues as the Principles and Standards of School Mathematics standards of communication, representation, and connections (see National Council of Teachers of Mathematics, 2000). The national research coordinator identifies specific realizations of principles 2, 3, and 7 in the lesson; see the commentary for details.
Technical detailsFile name: NB10311023.
There are differences between the drawings in the video and the corresponding drawings on the lesson graph. In the second lesson segment in the lesson graph, the representation in the video is drawn so that the number 9 appears on the left rather than the right as shown in the lesson graph. In the third lesson segment, representing 17 as 4 • 3 + 5 was given orally only. The lesson graph is missing an incorrect representation for 26 and shows a representation that was only given orally (17 as 4 yellow strips followed by a blue strip).
Students needed to:
Aspects of the lessons are related to future learning of various topics, including:
Questions that stimulate student thinking
Examples of questions that stimulated students’ thinking, individually or in response to one another’s work, include (11:20–11:50), when the teacher probes the students to explain their thinking, and (33:32–34:08) when the teacher asks students to evaluate one another’s responses.
Engaging students
An overriding theme in this lesson was the teacher engaging the students in collective work of ongoing mathematical evaluation. He seems to be creating a community of learners who listen and respond to each other’s solutions and explanations. He draws out students to talk about and explain their mathematical procedures. He encourages the students to talk about their mathematical reasoning. He also seems to be focused on facilitating the students’ collective learning.
Students are regularly active and moving about the room, coming up to the board to explain to their classmates. Examples include:
Difficulty of the lesson
The emphasis of the lessons may have been as much about the development of classroom norms as on the mathematical ideas. It may have been about developing conceptual understanding of the notion of “variable” instead of doing complex examples or using large numbers. It dealt with some abstraction; the students seemed confused when they were using the colored sheets, even with relatively simple numbers.
Students’ use of materials
Whether and how learning to use tools can take over a lesson is an important issue. The students became very preoccupied with using the compass; the teacher may have considered this a valuable tool of mathematics, and worth spending some time having the students work with it.
Sequence of work
This lesson offers an opportunity to examine a sequenced development of the mathematics. The teacher sequences the students’ work: line segments, colored sheets, and variable expression.
Instructional decisions
(23:07) The teacher gives students practice with what they have been doing. This lesson is a good example of removing support gradually. This lesson could be used to learn to anticipate student error and think ahead about how to use it (interpreting colored paper multiplicatively rather than additively).
The use of language
There are a number of issues about language here (e.g., “calculate with variables”). (This may be due in part to translation and different word usage.) But the lesson offers the opportunity to consider (1) usage that is not standard (to those in the U.S.) and (2) the conventional terms in this context. This lesson can increase teachers’ awareness of mathematical language and the conventions surrounding that language.
As illustrated in the US2 lesson (20:54–24:05), defining a term does not necessarily lead to student understanding. The term variable poses even more of a problem, since mathematicians and mathematics educators do not agree on one definition. Some believe variable encompasses both unknowns and values that change; others support the literal meaning of the word, that it only refers to values that change. For more discussion of this issue, see Schoenfeld and Arcavi (1988) and Usiskin (2005).
The following excerpt comes from Gelfand and Shen (1993), which is written for students:
In algebra we gradually make more and more use of letters (such as a,b,c,...,x,y,z, etc.). Traditionally the use of letters (x's) is considered one of the most difficult topics in the school mathematics curriculum. Many years ago primary school pupils studied "arithmetic" (with no x's) and secondary school pupils started with "algebra' (with x's). Later "arithmetic" was renamed "mathematics" and x's were introduced (and created a mess, some people would say).
”We hope that you, dear reader, never had difficulties understanding "what all these letters mean", but we still wish to give you some advice. If you ever want to explain the meaning of letters to your classmates, brothers and sisters, your parents, or your children (some day), just say that the letters are abbreviations for words. Let us explain what we mean.
In the equality
a + b = b + a
the letters a and b mean "the first term" and "the second term". When we write a + b = b + a we mean that any numbers substituted instead of a and b give a true assertion. Therefore, a + b = b + a can be considered as a unified short version of the equalities 1 + 7 = 7 + 1 or 1028 + 17 = 17 + 1028 as well as infinitely many other equalities of the same type.”
See pages 16-17 for more examples of the use of letters to stand for words or numbers.
Gelfand, I.M. & Shen, A. (1993) Algebra. Boston, MA: Birkhäuser.
National Council of Teachers of Mathematics (2000). Principles and standards of school mathematics. Reston, VA: Author.
Schoenfeld, A. H. & Arcavi, A. (1988). On the meaning of “variable.” Mathematics Teacher 81, 420–427.
Usiskin, Z. (Winter/Spring, 2005). The importance of the transition years, grades 7-10, in school mathematics, in UCSMP Newsletter No. 33, 4-10.
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