Building Engaging Math Tasks

Normally, I’m not big on just sharing a link to another person’s blog, but I think this particular blog entry warrants it.

Last week at the NCTM Annual Meeting and Exposition, I was fortunate to attend a session with Dan Meyer about why word problems are typically the bane of our students’ existence. Generally, they are a bunch of words on paper with no real connection to “real-life” math – or anything else students are interested in.

Dan’s blog contains a large number of engaging tasks here. He also has a post about how to design your own engaging math tasks here.

While you’re there, be sure to check out Dan’s other posts – he is one of my favorite folks to follow. You can also find him on Twitter by following @ddmeyer.

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Moms Talking Numbers

Interesting article in the New York Times about how mothers talk numbers and number concepts with boys more often than girls. Interestingly enough, they said there is no proof that this influences later beliefs that math is “not a girl thing” but given that familiarity breeds liking, wouldn’t that make sense? Moms – please talk math with your daughters!!

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Finding things you never knew you had…

While looking for some information this morning for a teacher regarding our state test, I found something that just might turn out to be useful.  I tend to regard high-stakes tests as a necessary evil, but sometimes good things do come out of them.  For example, what I found in my massive files this morning was a document called “10 Practical Instructional Strategies for Grades K-8.”  It is centered around preparing students to take the CMT (Connecticut Mastery Test), but the strategies could be applied to any mathematics classroom.

10 PRACTICAL INSTRUCTIONAL STRATEGIES FOR GRADES K-8

Preparing students for the Connecticut Mastery Test (CMT) should be an ongoing
process. The process requires sufficient instructional time and appropriate instructional
strategies. While it is certainly appropriate to conduct some form of review, “cramming”
is far less effective than an ongoing set of instructional practices that naturally and
continually prepare students for the test specifically and for higher levels of
understanding generally. A sound K-8 mathematics program embeds these strategies into
all instructional planning.

Strategy 1: Asking “Why?”
Probably the best way to implement a “thinking curriculum” – a curriculum that is
language-rich, focuses on meaning and values alternative approaches – is by regularly
asking students “Why?” A simple, “How do you know that?” or “Can you explain how
you got your answer?” or the basic, “Can you explain to the class why you think that?”
forms the basis of a mathematics curriculum that goes beyond merely correct answers. A
student who can explain his or her answers often has a stronger understanding of
mathematics and can help other students develop understanding. Questions like, “How
did you get 17?” or “Why did you add?” give students powerful opportunities to
communicate their understandings and give teachers powerful tools to assess the degree
of understanding. Classrooms where students are regularly explaining how and why, both
orally and in writing, are classrooms that effectively prepare students for many of the
open-ended items on the CMT.

Strategy 2: Embed In Context, Present As A Problem
Consider the vast difference between “Find the quotient of 20 ÷ 1.79” on the one hand,
and “How many hamburgers, each costing $1.79, can be purchased if you have a $20
bill?” Both problems expect that students can divide. However, the former directs
students to a single long division algorithm with a three-digit divisor that isn’t even tested on the CMT. The latter places the mathematics in a context and expects students to
understand that division is an appropriate operation to use to solve a practical problem. In addition, the latter encourages estimation and raises the issue of sales tax, all of which is assessed on the CMT. Most importantly, the contextualized problem shows students that mathematics is a useful tool.
Strategy 3: Ongoing Cumulative Review
One of the most effective strategies for fostering mastery and retention of critical
mathematical skills is daily, cumulative review at the beginning of every lesson. Rarely
does one master something new after one or two lessons and one or two homework
assignments. Many teachers call this “warm-ups” or daily-math. Five to eight quick
problems to keep skills sharp can be delivered orally or via visual methods. Every day
teachers should present:

  •  a fact of the day (e.g., 7 x 6);
  •  an estimate of the day (e.g., What is a rough estimate of the cost of 55 items at
  • $4.79 each?);
  •  a measure of the day (e.g., About how many meters wide is our classroom?);
  •  a place value problem of the day (e.g., What number is 100 more than
  • 1,584?);
  •  a word problem of the day; and
  •  any other exercise or problem that reinforces weaker, newer or problematic
  • skills and concepts.

This form of review, often patterned after the types of items and item formats used on the CMT, embeds review for the test in what is recognized as sound instructional practice.

Strategy 4: Ensure A Language-Rich Classroom
Like all languages, mathematics must be encountered orally and in writing. Like all
vocabulary, mathematical terms must be used again and again in context until they
become internalized. Just as young children confuse left and right until they develop
strategies and connections to distinguish between the two, older children confuse area
and perimeter until they link area to covering and perimeter to border. A language-rich
classroom, in which mathematical terminology is regularly used in discussions, solving
problems and in writing, can make a big difference in how effectively children learn
mathematics. Posting vocabulary in the room, perhaps on a word wall, is one way to
ensure that mathematical terminology is used on a daily basis. While not exhaustive, the
vocabulary word list found in each grade-level section of this handbook should be used to
ensure that the language used and expected on the CMT is never new to students. 12

Strategy 5: Use Every Number As A Chance To Build Number Sense
The development of number sense is one of the overarching goals of mathematics at the
elementary level. Number sense is a comfort with numbers that includes estimation,
mental math, numerical equivalents, a sense of order and magnitude, and a welldeveloped understanding of place value. The development of number sense must be an ongoing feature of all instruction. A review of CMT 4 reveals how much of the test
focuses on these critical number sense understandings. A simple strategy for
incorporating number sense development into all instruction is to pause regularly and,
regardless of the specific mathematics being taught, ask questions such as the following:

  • ƒ Which is most or greatest? How do you know?
  • ƒ Which is least or smallest? How do you know?
  • ƒ What else can you tell me about those numbers? For example, “they are both odd,” “all are mixed numbers,” “their product is about 18 because you can round.”
  • ƒ How else can we express .2 (2/10, 1/5, 20%, .20)?

Incorporating this strategy into daily instruction creates a mind-set that the numbers in
every problem posed and in every chart or graph used can strengthen and reinforce
number sense. For example, in a simple word problem that asks students to find the sum
of 57 and 67, teachers can first “pluck” the numbers from the problem and ask students to
list four things they can say about the two numbers. Consider how much mathematics is
reviewed when students suggest findings such as the following:

  • ƒ I see two two-digit numbers.
  • ƒ Both numbers are odd.
  • ƒ There is a difference of 10 between the numbers.
  • ƒ The 67 is greater than the 57.
  • ƒ The ones digit is the same and the tens digit is one apart.
  • ƒ One number is prime and the other is composite.
  • ƒ I see 124. 13

Strategy 6: Draw A Picture (Mental or Real)
We say casually that “a picture is worth a thousand words” but we seldom connect
mathematical concepts to their pictorial representations. A significant proportion of the
CMT asks for pictorial equivalents of mathematics ideas. A powerful way to help
students visualize the mathematics they are learning, or to reinforce understanding, is
with mental images or pictures that students actually draw or create. Consider how
infrequently we ask students to, “Show me with your hands about eight inches” or, “Use
your fingers to show me an area of about 10 square inches.” Consider how important it is
that students can draw pictures of fractions and mixed numbers like ¾ or 2 ½ and of
decimals like .3 and 1.2. Consider how powerful a class discussion about the different
pictures for “three-quarters” can be when students show three quarters (25-cent pieces), a shaded pizza slice, a window pane, three stars out of four shapes, a ruler, a measuring cup and simply ¾! Consistently embedding, “Can you draw a picture of…?” and “Can you
show me what that would look like?” into instruction can pay rich benefits in both
student understanding and in CMT scores.

Strategy 7: Build From Graphs, Charts and Tables
Many real-world applications of mathematics arise from the data presented in graphs,
charts and tables. This is why so many of the CMT items are based on data and include
graphs, charts and tables. To best prepare students for these contexts, as well as develop
the essential skills of making sense of data and drawing conclusions from data that is
presented in graphs, charts and tables, teachers are encouraged to make far greater use of
these forms of data presentation. Given a graph or table, students can be asked (similar to Strategy 5) to identify five things they see in the graph or table. In addition, students can be asked to draw two appropriate conclusions from the data and justify those conclusions. So consider “milking” the graphs and charts found in your textbook or data that students find during “data scavenger hunts” by copying the graph, chart or table for students and asking them to create five questions that could be answered by the information in the graph or table. Ask students to share their questions and generate a list of the best questions for future use.

Strategy 8: How Big? How Much? How Far?
No strand of mathematics assessed on the Connecticut Mastery Test produces student
scores as consistently weak as the measurement strand. Rather than leave all
measurement to a single chapter that is often skipped entirely, teachers are encouraged to make measurement an ongoing part of daily instruction. First, questions like, “How
big?,” “How much?,” “How far?,” “How heavy?” all help to develop measurement
understanding. Second, measurements of things such as arm span, book weight, area of
circles, or breath-holding times all provide great sets of data and, therefore, use
measurement to gather data that is analyzed and generalized – integrating many important aspects of mathematics. Finally, more involved projects like determining the number of students that can fit in a classroom or the number of hours students have been alive are wonderful opportunities to keep measurement on the front burner of daily instruction.

Strategy 9: Omit What Is No Longer Important
A significant amount of time and energy is expended by teachers and students on skills
considered less important by national and state standards and not even assessed on the
CMT, the Connecticut Academic Performance Test (CAPT) or the SAT. District
mathematics curriculums must become more focused on what is truly valued and teachers must give themselves and each other permission to skip textbook pages that no longer serve useful purposes. In fact, the proverbial “mile-wide, inch-deep” curriculum that results in far more coverage of topics than mastery of key concepts undermines many efforts to raise student achievement. In addition, time that is no longer spent on
increasingly irrelevant skills – particularly those done most often with a calculator – frees up valuable minutes and hours for increasingly important skills like estimation, algebraic reasoning and problem solving. So carefully review what is NOT assessed on the CMT – particularly complex, multidigit computation – and redirect what is taught to focus on those skills and concepts that have lasting value and that ARE assessed.

Strategy 10: Focus On Sense-Making As Well As Correct Answers
One of the most powerful test-taking skills for multiple-choice items is the artful
elimination of obviously absurd answers. However, identifying such “obviously absurd
answers” – for example, a sales tax of $129 dollars instead of $1.29 on a $20 item –
requires a mind-set that mathematics makes sense. This “minds-on” approach to
instruction is in sharp contrast to the rote regurgitation of rules and procedures to get
correct answers to exercises that all too often comprises mathematics instruction. For
example, when teaching how to convert mixed numbers to improper fractions and vice
versa, it is imperative to teach why these forms are equivalent. Students who only know
how to multiply and add and not why 3 ¾ is equivalent to 15/4 are at a disadvantage in
life and on the CMT. Focusing on the why – that is, focusing on understanding and sensemaking – emerges from consistent use of many of the preceding strategies, particularly 1 and 6.Teachers can improve instruction significantly by adopting the mind-set that good mathematics instruction begins with an answer. That is, when a student responds (for example) “17,” the next question should be something like, “How did you get that?” When the student responds “I added” or “I rounded” or “I took about half,” the next question should be something like, “Why did you do that?”

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2011 in review

The WordPress.com stats helper monkeys prepared a 2011 annual report for this blog.

Here’s an excerpt:

A San Francisco cable car holds 60 people. This blog was viewed about 3,300 times in 2011. If it were a cable car, it would take about 55 trips to carry that many people.

Click here to see the complete report.

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Googleplex?? For Kindergartners??

Last school year, I got an email from a Kindergarten teacher that said:

“My kids are curious about, of all things, googleplex! Is it really a 1 with 100 zeroes?”

Here was my answer:

Googleplex is the headquarters of Google. 🙂

A googolplex is the number. It is built upon the concept that a googol is a 1 with 100 zeroes. A googlplex is a one with a googol zeroes after it! Here are a couple of links that might help, but it’s kind of tricky to me. Interestingly enough, while researching this for you, I found that Carl Sagan (the famous astronomer of Cosmos fame) said that it would be physically impossible to write out a googolplex in numbers because it would take up more space than the universe provides! How crazy is that!  Check out these links for more information…

What is a googol?

What is a googolplex?

Hope this helps!

Your resident math nerd,

Brenda 🙂

What is the most unusual math question you’ve gotten from a student?

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Instructional Coaching vs. Content Coaching…is it really a contest?

Since I became an instructional coach, I’ve heard varying opinions about the importance of content knowledge in being a successful coach. 

Currently, I am a coach for K-5 math.  In my position, I need to know about best practices in teaching, but I also focus on the content.  Is the teacher accurate in what he/she is teaching?  Do they themselves understand the concepts well enough to scaffold learning for students? 

I co-coached (new word??) last week with one of our district’s general instructional coaches.  Her focus is on the indicators of our district’s teacher evaluation plan – things like differentiation, feedback, and questioning.  She asked for my help with a teacher who was looking for some additional ideas for questioning in math.  I gave her a list of my favorite questions, which I have gleaned from many different sources including NCTM. (Click here for that list). 

The other coach and I both observed the beginning part of the lesson, after which students split up into small groups of 3 or 4 to work on their given task.  They were completing an assignment that had to do with finding information in graphs and tables, and answering questions regarding inferences they could make from the data provided.  One of the tables the students were given had to do with a ratio of people with a certain occupation compared to all working people at certain points in time (between 1900 and 2000).  Students were then asked to figure out if the ratio of engineers to the total population had increased or decreased between 1900 and 2000, and why they thought that was the case.

The teacher and the other coach had asked me to model some questioning techniques with the small groups.  I asked to let the students start working without interference at first so we could see the thought process going on, and have some completed work as a starting point.  After about five minutes, I started with one particular group that I saw struggling through the problem.  I could see a few things from my quick observation…so that’s where I started.  Here are the questions I asked…

1) What information are you being asked to find? – The students could tell me that they needed to find out whether the ratio of engineers to the total population increased or decreased over time.

2) Where are you finding that information? – They pointed to a graph on the opposite page from where the actual information was.  This led to a quick discussion about being careful to read the titles of graphs and charts to ensure that you are looking at the correct information.

3) What was the ratio of engineers to the total population in 1900?  In 2000?  – Students were able to answer this question with relative ease.  The struggle came when I asked the students to tell me whether the ratio had increased or decreased over time.  The 1900 number was something like 1 in 1,161 and the 2000 number was something close to 1 in 764.  The students said that the ratio had decreased because the second number was smaller.  I then challenged their thinking a bit by asking them…

3) If you had 1 blue M&M in a package that contained 1,161 M&Ms, would it be very likely that you would pick out a blue M&M if you reached into the package? – The students said it would be very unlikely since it was only 1 blue PER 1,161 M&Ms.   I then pushed them to think about the second ratio of 1 to 764. 

4) Would you be more likely to get a blue M&M if you reached into the bag if there were 1 blue PER 764 M&Ms? – The students responded that it would be more common to find a blue if 1 out of every 764 were blue. 

5) So…what does that tell you about the ratio?  Is it getting bigger or smaller? – Given an example they could relate to, they came to the conclusion that the ratio was getting bigger.

Now came the next task…they needed to explain why they thought the ratio was getting larger over time. 

6)  Why do you think there are more engineers now than in 1900? – Crickets chirping and deer-in-headlights stares.

There was only one problem, but it was a big one.  This group of students weren’t equipped with the prior knowledge of what an engineer was.  After a quick explanation, they were able to make the connection and answer the question.   

The second coach and I quickly debriefed with the teacher about this group before moving on to the next one, and we talked about the importance of prior knowledge in answering questions such as the one about the engineers.  Without knowing what an engineer was, the students were unable to make an inference about why that career had increased in popularity over time.  A thorough reading of the lesson by the teacher beforehand might have allowed her to anticipate roadblocks like the one with the engineers, as well as the one with the increasing and decreasing ratios. 

We moved on through the other groups with me modeling some questions, then asking the teacher to try for herself given some of the ones she had heard me use with other groups.

During our debrief after the lesson, we talked about how the questioning techniques I was using were very similar to those used in reading – asking “skimming” questions first, then probing deeper into the thinking to get at misconceptions and alternative ideas.  The teacher agreed to try one thing during her next few lessons – since we’re at the end of the school year, time is short and the remaining lessons are few!  I offered to come in at the beginning of next school year to continue the work on questioning, which I hope the teacher will take me up on!

The general coach and I had a conversation afterwards about how important it really is to make the connections between the coaching and the content, even though a lot of the techniques and methods we use across the curriculum are similar or even the same.  She said that when it came to the ratios, she wouldn’t have known quite how to explain the concept of increasing and decreasing as clearly as I did.  Given the fact that I had taught that particular lesson at least 10-12 times, I knew exactly where the roadblocks were.  We talked about the importance of doing the work of anticipation to head those kind of problems off at the pass.  We both agreed that it was very helpful to have some time to “co-coach” and learn from each other. 

What do you think?  Is content knowledge important, or even necessary, for good coaching?  Please vote in the poll below and let me know your thoughts in the comments!

Posted in Best Practices, Collaboration, Connections, Math/Literacy Connections, Real-life Mathematics, Teaching, Thinking/Cognition | Tagged , , , | Leave a comment

More Twitter for Teachers

The more I learn about Twitter and the more I use it, the more I am finding it a valuable tool for networking and professional learning.  I currently follow Creative Education, (@CreativeEdu) a UK-based company that provides courses and professional learning experiences for teachers.  As part of their website they maintain a blog that has some great posts about Twitter and its use for educators.  Below are links to some blog posts that have great tips for using Twitter!

Twitter for Teachers: A Guide for Beginners

Top 20 Tweeters for Math Teachers

Top Twitter Hashtags for Teachers

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