Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the category “EMU”

Reflections: Why “findingEMU?”

There is a blurb on my “about” page that describes the title of my blog:

Enduring Mathematical Understanding is found within, built upon your own foundation, framed by your current perceptions, and constructed from your experiences. By sharing my thoughts, ideas, and ramblings with others, I hope to encourage the growth of positive dialogue towards this lifelong goal for myself and others.

Wow! I really came up with that on my “first day” blogging, what seems like ages but was just over three weeks ago! (I did say “ramblings,” and that’s sure the truth.)

I have thought about writing a blog for quite awhile now. (Still to come – my “From Lurking to Learning” post.) Over time I toyed with different choices:

“MathMom” because I was a part time teacher / part time stay at home mom. My kids were (ARE) both math fanatics, and I would like to share some stories about watching them grow as mathematicians. (While I don’t exactly have their permission, I have put a few memories down on “paper” over the past few weeks.) However, “Math Mama Writes,” a blog that I have read, appreciated, and enjoyed for many years now, already had a claim on that type of title, so I decided to move on. (In addition, my kids have grown and I don’t find nearly as any opportunities to “be” a Math Mom anymore 😦 .)

My next thought was “Making Math Meaningful” (shortened to M^3….ooh!) While it has a nice ring to it, and it IS a part of what I try to do in my classroom, what does “meaningful” mean? Definition: full of meaning (duh!) significance, purpose, or value; purposeful (you mean, like “full of purpose”?); significant. I DO want to help students find meaning in math, see its value, know that it has purpose. However, my lessons are not “chock full of 3-Acts,” so it didn’t seem to quite capture the “spirit” of my classroom.

Recently, the phrase “Enduring Understandings” seems to have taken its place alongside “Standards,” or “Essential Learnings,” but I really like the use of the term “Enduring.” Teachers complain all the time about lack of retention – students can learn something for a week or a month, but it is no where to be found after that. What is it that makes the learning endure?

Now, “Understanding,” that’s a whole ‘nother can of worms. It seems like last spring, but it was only in June that I read Richard Skemp’s paper on Relational Understanding and Instrumental Understanding. I ran across it by linking from a Math Mama Writes post to The Republic of Math blog. (I can’t imagine why I didn’t read it when it first came out – oh yes, I was still in Elementary School!) It is a powerful piece, written 36 years ago, that, for me, emphasizes the difference between when a student says, “I understand!” and when they truly do. “Conceptual Understanding” vs “Procedural or Algorithmic Understanding” are more commonly used today, but I feel that, in reality only “Conceptual” actually reflects true “Understanding.”

So, that’s what it all boils down to for me. How DO I help my students FIND Enduring Mathematical Understanding? I can’t find it FOR them – sometimes I have a hard enough time finding it for ME. I don’t want the “I can do it!” math, I want the deep comprehension down in their core that builds and branches off from what they DO really know and understand. What I do in the classroom: the questions I ask, the answers I “don’t give,” but the way in which I respond to questions, the student dialogue I promote, the activities I provide that lead without “dragging them along,” and the culture of the classroom that I help establish, are all ways that I can help them on their journey.

I love the crafty stuff, notebook organization, games and activities, and “monster” whiteboarding plans (tooooooo Monday links to post!) that I have been bombarded with over the past few weeks, (I think I have at LEAST tripled my Google Reader) and the discussions about SBG, (or sbar?) how to deal with the homework hassles, and uses for technology on Twitter. However, what I really cherish from the past three weeks are the conversations about teaching fraction division with/for understanding (“I hate “magic math”,”) introducing integer operations with meaning (“I abhor – is that too strong a word? – the “follow the rules” approach”, and re-defining slope (NOT just “rise over run” – KA), running across David Coffey’s ideas about flipping homework that gives “practice problems” an entirely different level of meaning from back in February, and finding Michael Pershan’s fabulous new (in addition to his insightful “regular” one) blog, Math Mistakes where readers are faced with the tasks of identifying not only “what don’t they know,” but what DO they understand and thoughts on remediation – how can I meet them where they’re at?

Deep breath. One sentence?! I’m not editing. (Except to add MORE words, I suppose.)

findingEMU – oh, like “Finding Nemo!” Well, yeah, I went for the “catchy” goofy title that resembles a wonderful little Pixar movie, but with an odd looking flightless bird instead of an adorable clownfish.

Only I guess I really didn’t, because I really AM trying to find ways to reach EMU.

Flashback Friday: “Sharing” in Sums

When my own kids were VERY young, we used to play “finger math.” Hold up some fingers on each hand and ask “How much is 2 and 3?” Of course, in time it changed, bit by bit so that “2 and 3” became “2 + 3,” holding up the fingers became their job, and then there were no fingers at all. The specific memory involves asking my daughter “How much is 3 and 5?” Right away she said “Eight!” Since we had been playing for a little while, I teased her, saying “No, it can’t be eight. You told me 4 and 4 was eight!” She replies with “Look, Mama, you give one from the five over to the three and then it’s four and four!”

Why don’t students automatically think that way more often?

I know, some do, but it seems that addition is so often accomplished one of three ways:
-It is a “known fact” they are able to recall immediately.
-They “stack the numbers” (either on paper or mentally) and add using the algorithm.
-Ummm, I’m too lazy so I’ll use a calculator.

Now, occasionally students will use alternative mental strategies, but often only IF the numbers are “compatible” to begin with, such as 25 + 75. Why don’t we encourage them to MAKE the numbers “more compatible,” if they can? Who wouldn’t rather add 40 + 63 than 39 + 64? What about 50 + 75 instead of 48 + 77? How can we improve our students’ “computational FLEXibility” instead of just focusing on their “computational ABility?”

Enduring Mathematical Understanding doesn’t come from knowing how to apply an algorithm, it comes from looking for alternatives, strategies, and shortcuts that arise from a deeper sense of number!

Lesson: Tricky Tables Part 2

If you haven’t yet read the first post in this series, please do before continuing here.

We left off with the students/interview panel having completed an activity involving “filling in” the missing values in tables that increase or decrease at a constant rate. I neglected to tell you that,on the day prior to this activity in class, students were assigned the following homework:


I had also handed it out to the interview panel to work on briefly before the rest of the “lesson.” So, coming in, most of the students had “tried out” these kinds of problems in advance.

Student Strategies
What were some common strategies (both on the homework and the slide problems?) Guess and check was popular with the eighth graders – especially since most of the rates were whole numbers. The further apart the “known values” drifted, the more challenging it became for them to apply this method. However, some continued to stick with that strategy, even after others shared more efficient methods. Just “hearing about” another way to solve a problem doesn’t automatically change your mindset if you are not ready.

Many focused on how far apart the two y values were, and then students (and interview panel) started counting the “spaces” between the values. If there were three spaces between the y values, they divided the difference in the y values by three. Then they realized that didn’t work and adjusted up to dividing by four (often without really thinking about why.) One member of the interview panel came up with a formula for find the amount of increase or decrease: d/(g+1). We then had to define the variables in the discussion portion: d = difference in the y values and g = “gap” (the number of spaces in the gap.)

Next Steps
In class (and during the interview) we pulled back and added some terminology.


I chose to use “rate of change” at this point – reinforcing the idea that we are interested in how “fast” the values are changing. The “non-example” was an eye-opener for some, as they were entirely focused on the y-values and not really looking at x at all.

The next section deals with activities we did in the classroom. The interview “mini-lesson” wrap-up will resume further down.

Expanding the Concept
Before moving on to finding more challenging rates of change, we moved to how this relates to graphs and story situations. In each of these activities I use a “Think-Pair-Share” routine.


Most (but not all) students were able to “see” the rate of change by looking at y-coordinates for consecutive x-coordinates. (My words, not theirs!) They often (but not always) started at the y-intercept (not a “known vocabulary term yet”.) Those who are not as comfortable with reading coordinates off of a graph struggled more. Marking in the points on the graph and identifying coordinates definitely helped. (I use Absolute Board to draw on screen shots of the slides, or shine the projector on the whiteboard instead of the screen and “write” on the image.) We segued from there to defining another term:


I am sure this is not the “classic” definition of slope people are looking for, but it follows from how we had been approaching the problems.
Next up was connecting to story situations:


Students whipped through these like lightning. For future planning I want to stretch them a bit by addressing situations like the ones given below, but maybe not until we reach the next level.
(1) Stu had $100 in savings. Four weeks later he had $200. On average, how much had he saved each week?
(2) On March 3, Sherry put an iTunes gift card on her account that was worth $100. By March 8 her balance was down to $70. On average, how much did she spend each day?
Disclaimer: These are all very much “pseudo context” but at this point we are developing concepts and not “problem-solving.”

At this point I want them to pause and reflect on what they understand and know how to do. Most are feeling quite confident regarding this concept. The biggest challenges were working though finding a process to “calculate” the rate of change, and some are still not there (yet!) Keeping the values fairly simple has allowed them to determine this process on their own rather than being “told what to do.”
Student writing prompt:


Interview panel writing prompt: (no discussion of slope, graphs, or stories)


Writing about math is always a challenge. Even if the students feel comfortable “knowing what to do,” it is still hard to put it into words.

Students have taken the first steps towards developing Enduring Mathematical Understanding regarding the concept of “rate of change” or “slope.” Even if it is just the idea of “how much it goes up to down by,” the concept will be expanded and refined in the days to come.

Flashback Friday: “It’s Like the Diving Board”

Watching diving during the Olympics always brings back a memory from a number of years ago. I taught high school math in my previous district, and this particular story involves a young lady I was fortunate to have in class for three years during a stretch of time where we were phasing in Contemporary Mathematics in Context. She was in the original “pilot” group at the Junior High and then moved to the high school where I first implemented the next three years of the curriculum with that particular group while the rest of the school phased in each course a year later.

We had been working on analyzing the effects of a, b, and c on the graph of y = ax^2 + bx + c. Nothing as fancy as axis of symmetry, vertex, and specific roots, just “How does changing the value of a affect the graph?” “What about b and c?” She had been sick for a few days so she came in at lunch to spend some time working on what she had missed. We started with looking at y = ax^2 and changing the value of a – no problem. Next we investigated y = ax^2 + c, for different values of c, and again the effects were quickly realized. We also had a successful discussion about “why” changing those values affects the graph the way it does. Then it was time to look at b. In the past students usually “discovered” things like “If b is positive it actually moves the graph to the left (and down) but if b is negative it moves to the right (and down.)” “That’s if a is positive, but if a is negative it does the opposite.” Of course the whole “positive – left” and “negative – right” idea is something they have to “remember.” There’s not a lot to grab onto until you start looking at things more algebraically.

With this student we started by looking at a few graphs where b was not zero, and then she said, quite unexpectedly, “Oh, it’s like the diving board.” “Huh?” “It’s like the diving board problems.” It takes me a bit, but then I see the light!

One of the main contexts in which quadratics were first introduced was using diving platform/diving board scenario. Initially it is a 10m platform. Students are told that, due to gravity, any object will fall such that d = 4.9t^2 where t is time in seconds and d is distance in meters. Using that (and the assumption that divers are basically “falling” with negligible effect from the “jump”) students develop the rule h = 10 – 4.9t^2. The graph and table are analyzed, other “free fall” situations are discussed and then (finally the relevant part) it returns to diving. First students are asked to explore what would happen if there was no gravity, and a diver leaves a 3 meter board with a velocity of 5 meters per second. Of course, it is humorous to imagine a diver continuing to fly though the air at a speed of 5 meters per second. Students are well-versed in linear modeling and come up with h = 3 + 5t, and similar rules for different initial height and velocity situations. Then, we come back to reality where gravity DOES exist, and work towards h = 3 + 5t – 4.9t^2. This student went back and grabbed the connection that b is the “speed the diver is traveling when they leave the board,” which translates to the slope or rate of change of the graph right as it crosses the y-axis!

Since that day I have had a new perspective on the effect of b on the graph of y = ax^2 + bx + c. (Maybe it is more useful to write it as y = c + bx + ax^2.) The graph crosses the y-axis at c. If b is positive, it is increasing as it crosses, but if b is negative it is decreasing. It a is positive, it begins to curve so that the “U shape” is right side up, but if a is negative, it begins to curve so that it is upside down. Based on the magnitude of b, you can draw the steepness of the tangent line at the y-intercept (because, of course, that’s what it is!) Based on the magnitude of a, you can draw a “skinny” or “wide” parabola. Try it! No quadratic formula, no factoring, no axis of symmetry – no “math” – and you can sketch a fairly accurate graph. Sure, you won’t have the exact x-intercepts or vertex, but all of the features of the graph are based on understanding and reasoning, not just “remembering.” How had I never seen this before?

Why do students resist mathematics in “story” situations? When concepts are learned within a context they are more likely to develop Enduring Mathematical Understanding!

Lesson: Tricky Tables

Earlier this summer I applied for a job (it was a part-time position to go with my current part-time position, so now I will be teaching full-time . . .but that’s a whole different story) and was asked to “present” a ten minute “mini-lesson” as a part of the interview. Well, in reality, I am not so sure how much can be accomplished in just ten minutes (and I believe it ended up going significantly longer than that) but I pulled out part of a series of activities that I have used with my eighth-graders for the past two years.

The main focus was working first individually, then in pairs/threes on completing the tables below:


It seems like a fairly simple task, finding a constant increase or constant decrease pattern and filling it in, but nonetheless, there were a few challenges. There were both “math and non-math” people on the interview team, and it was interesting to see how they approached the problems.

-What strategies would YOU use to solve the problems?
-How do you think middle schoolers would approach the task?
-What misconceptions might crop up?

-What questions would you ask to guide discussion?
-What would be some “next steps?”

-What Enduring Mathematical Understanding can be developed by completing these tasks?

Here is a link to the next part in the series.

Flashback Friday: Half and Double

When my son was about six years old we would “play math” when we were in the car, or getting dinner ready, or just “playing.” I remember asking him what 7 x 8 was, and hearing a reply that went something like this: “Well, 7 x 8 is the same as 14 x 4, and 14 x 4 is the same as 28 x 2, so. . . 56!” It was not quite what I expected, but then I recalled a brief conversation we had had some time earlier involving the number 12.

Son: “3 x 4 is 12, and so is 6 x 2.”
Me: “Yes. . .?”
Son: “Does that always work?”
Me: “What?”
Son: “That if you double 3 you get 6, and half of 4 is 2.”
Me: “Well, what do you think?”

I could see the gears turning as he thought about it in his mind – picturing the four groups of three changing into the two groups of six. I don’t recall the resolution of the discussion, but apparently it was enough to satisfy him so that it became a strategy that he began to use on a regular basis. I am certain that I did not encourage it as a “method,” just an interesting occurrence.

A few weeks after the “7 x 8” question, I asked it again, expecting that he might remember “56” from his previous “derivation.” Here’s what I got: “Well, 7 x 8 is the same as 14 x 4, and 14 x 4 is the same as 28 x 2, so. . . 56!” He showed exactly the same enthusiasm for “solving the puzzle” that he had a few weeks earlier, not recollecting that he had, in fact, solved it before.

Contrast that with the child who has memorized 7 x 8 = 56, and would be able to repeat it back in the blink of an eye. Who has more Enduring Mathematical Understanding?

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