Examining Students’ Understanding of Equality
Professional Development Session
Math Teacher Leader Meeting, October 2005
Milwaukee Mathematics Partnership
Developed by
Melissa Hedges, Sharonda Harris, Bernard Rahming, and Beth Schefelker
1. Present Session Goal
To examine the concept of equality through the eyes of mathematics and of students.
2. Opening Activity: Surfacing the Mathematics
(overhead) Solve: 48 + 24 = ___ + 27 and keep track of your strategy.
After a few minutes gather up some of the strategies. Chart out a few strategies and compare
them using relations between the two sides of the equation. Highlight strategies that
calculated the sum of 48 + 24 as 72 and how this was used to find the missing number. Also
highlight strategies that show relationships among the numbers in the two expressions that
made it unnecessary to actually carry out the calculation (e.g., 27 is three more than 24, so
the missing number must be three less than 48).
3. Reflecting on the Meaning of the Equals Sign
Part A: Now that we’ve thought a little about that equal sign, jot down some ideas for the
following questions.
(overhead)
• What does the equal sign mean?
• How would your students answer the same question?
• Why do you think they would answer that way?
Share with a colleague at your table. As you talk through those questions begin to think
about a working definition of the equal sign that might make sense to both you and your
students.
Part B: What were some surprises that surfaced while you shared responses with each other?
(Possible responses: We both think that our students would say the same things. We were
surprised to realize that we never really talked about the meaning of the equal sign before.
We were surprised that we both struggled to say what we thought it meant, students may say
“it tells where the answer is.”)
Well, if we’ve discovered that we never really talk about it, and we are aren’t even quite sure
ourselves what it means, what would we say to our students?
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Milwaukee Mathematics Partnership
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Part C: It is a times like these that we turn to some resources that may help us. The State
Descriptors tells us…. (overhead) State descriptor: It defines the equal sign as ‘the same as.’
So, let’s see how our students do with understanding the meaning of equality.
4. Examining Student Work Samples
These problems (8 + 4 = ___ + 5; 48 + 24 = ___ + 27) were presented to MPS students. At
your tables are some work samples for you to look at. 1st grade work is in blue, 5th and 6th
grade work is in yellow. Feel free to look at both but get fairly familiar with one set.
As you review the work please note the answer that was placed in the box as well as the
reasoning behind the selected answer.
(overhead)
Feel free to review both sets of work but get familiar with one. As a table, work through the
following questions:
• What might account for the variety of answers/responses to the prompt?
• How did the students get the answers that they did?
• What does this work show us about their understanding?
• Do you notice any similarities between the blue set and the yellow set? (both sets
represent similar stages of understanding and confusion)
Debrief using the above questions and the following information. Debrief with the group the
variety of answers displayed and how they may have been achieved.
2nd grade work:
8 + 4 = ___ + 7 with student responses of 5, 12, 19 (How were these answers achieved?
Answer of 5:
Student G: Beginnings of “equality.” The student recognized a relation between the 2 sides,
mainly that both sides of the equal sign had to represent the same number, and filled in the
blank by calculating the answer.
Answer of 12 and 19:
Student J: 8 + 4 and ignored the 12. Then added all there numbers to get 19
Answer of 12:
Student H – added 8 + 4
Answer of 19:
Student I: Added 8 + 4 + 7 (did not subtotal 8 + 4 = 12)
5th and 6th grade work:
48 + 24 = ___ + 27 with student responses of 45, 72, 99 (How were these answers achieved?
(Gather the “wrong” answers from the group first.)
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Answer 72:
Student K: Adds 48 + 24 to get 72 and ignoring the 27. (calculating answers, not looking for
relations)
Answer 72 and 99:
Student E: Adds 48 and 24 to get 72 then adding on 27 to get 99, ignoring the location of the
equal sign. Inserts 99 in blank immediately following equal sign.
Student F: Calculates 48 + 24 = 72 then extends the problem and adds 38 + 24 + 27 to get
99. Notation looks like this: 48 + 24 = 72 + 27 = 99 (calculating answer, not looking for
relations.)
Answer 45:
Student E: Adding 48 and 24 to get 72, subtract 27 from 72 to get 45.
The students in these examples were able to use relations to calculate their answers.
Student D: Finds the difference of 3 between 24 and 27 and then substracts 3 from 48 to get
45. This student used the relations between the two sides of the equation and a saw relation
among the numbers in the two expressions that made it virtually unnecessary to actually
carry out the calculation.
Students B and C: Subtracts 3 away from 48 to get 45 and adding 3 onto 24 to get 27. Each
used the relations between the two sides of the equation and a saw relation among the
numbers in the two expressions that made it virtually unnecessary to actually carry out the
calculation. There is a difference in notation here. That is why they are included The
thinking behind why they did what they did might also be different. It might not be necessary
to include both.
Student A: Recognized a relation between the two sides, mainly that both sides of the equal
sign had to represent the same number and filled in the blank by calculating the answer.
5. Student Results Summary
Part A: Results from Students in MPS
Take a look at the table summarizing how selected grade levels at did with the various
tasks. (Share data from 2nd grade for 8 + 4 = ___ + 5 as well as data from 8th – 5th grade for
48 + 24 = ___ + 27)
Comment on the fact “similar thinking errors” happened with a multiplication prompt of 8 x
___ = 4 x 18 for the upper grade students.
Part B: Results from Other Students
Just so we don’t think only MPS students are in this situation turn to page 9 of the Thinking
Mathematically: Integrating Arithmetic and Algebra in Elementary School book. You will
see similar results. The authors of this book, which we will be using this year to help guide
our learning about algebra, would “reassure” us that:
(overhead)
“Most elementary students, and many older students as well, do not understand that the
equal sign denotes the relation between two equal quantities.”
(Carpenter, Franke, & Levi, 2003, p. 9)
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Part C: Expectations for Our Students
I guess here it would be important for us to ask ourselves: Where would we like our students
to be? We would like them to understand that the equal sign represents a relation. Why?
(overhead)
“A limited conception of what the equal sign means is one of the major stumbling blocks in
learning algebra.”
(Carpenter, Franke, & Levi, 2003, p. 22)
What might we expect from students as they solve these types of problems demonstrating
their understanding of relations among the numbers on both sides of the equal sign?
6. Video Clip of Student Reasoning
We know that some students are calculating the correct answer but as we have already seen,
“not all answers are created equal.” Let’s take a look at a video clip that accompanies your
book. Watch as a kindergarten student works to solve 5 + 4 = ___ + 3.
After the video has been viewed, turn to your partner and talk through what you saw him do.
Then as a table group respond to the following: (overhead) What does he understand about
the equal sign? What strategies does he use to compute his answer?
This student recognized a relation between the two sides, mainly both sides of the equal sign
had to represent the same number and filled in the blank by calculating the answer.
7. Transcript of Student Reasoning
Now let’s take a look at a transcript in your book, page 13 Gina. Read through what Gina
does with that same problem of 8 + 4 = ___ + 5 as well as 57 + 85 = __ + 84.
Turn to your partner and talk through her thinking. (overhead) Compare Gina’s strategy to
the kindergartner’s strategy. Compare her strategy on the two problems.
8. Reading Jigsaw
What information might be helpful to us as we work with students to develop their
understanding of the equal sign? Our book will give us good suggestions as to what to avoid
and what to emphasize. Each person at your table will read one of the 6 sections and be
prepared to share the “big idea” from that section. In the event that you wish to use portions
of this section with your staff we’ve created a study sheet where you can take notes as you
read and listen to each other.
(overhead)
Share these sections among the people at your group:
 Benchmarks p. 19
 What to Avoid p. 20
 The Use of the Equal Sign is a Convention p. 21
 Smoothing the Transition to Algebra p. 22
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 Where Do Children’s Misconceptions About the Equal Sign Come From? p. 22
 Children’s Conceptions p. 23
Read your section and be prepared to share an important idea with your table. Use the
study sheet for notes if you do not want to write in your book.
9. Closure: Pulling the Big Ideas Together
We’ve gotten some very good background information during our time together. We’ve
looked at what the equal sign means, our own understanding and that of elementary and
middle grade students, common conceptions and misconceptions and did some reading to
help us bring some of this together.
What are three big ideas you could take back to your staff about the equal sign? (Chart these
ideas.) For example, teacher content knowledge is a very important factor when we consider
improving our students’ skills and abilities.
Reference
Carpenter, T. P., Franke, M. L., & Levi, L. (2004). Thinking mathematically: Integrating
arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
www.Heinemann.com
Note
This material was developed for the Milwaukee Mathematics Partnership (MMP), an initiative
of the Milwaukee Partnership Academy (MPA). It is based upon work supported by the National
Science Foundation under Grant No. EHR-0314898. Any opinions, findings and conclusions or
recommendations expressed in this material are those of the authors and do not necessarily
reflect the views of the National Science Foundation (NSF).
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Milwaukee Mathematics Partnership
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