Book Summary: Accessible Mathematics

Accessible Mathematics--10 Instructional Shifts That Raise Student Achievement, Steven Leinwand (Heinemann, 2009)

The ten instructional shifts are helpfully printed on the inside front cover.  They are:

1. Incorporate ongoing cumulative review into every day's lesson.
2. Adapt what we know works in our reading programs and apply it to mathematics instruction.
3. Use multiple representations of mathematical entities.
4. Create language-rich classroom routines.
5. Take every available opportunity to support the development of number sense.
6. Build from graphs, charts, and tables.
7. Tie the math to such questions as:  "How big?" "How much?" "How far?" to increase the natural use of measurement throughout the curriculum.
8. Minimize what is no longer important.
9. Embed the mathematics in realistic problems and real-world contexts.
10. Make "Why?" "How do you know?" "Can you explain?" classroom mantras.

Chapter 1 We've Got Most of the Answers

The author points out that there are successful math classrooms, but they are fewer than effective classes in other content areas.  So, he suggests that educators examine these successful classes to pick out the strategies being used.  He includes successful classes internationally as well, contrasting, for example, the U.S. and Japan.  A common approach in a U.S. classroom is:

• The teacher instructs students in a concept or skill.
• The teacher solves example problems with the class.
• The students practice on their own while the teacher assists individual students.

Yup!  That sounds very familiar to me.

In Japan:

• The teacher poses a complex, thought-provoking problem.
• Students struggle with the problem.
• Various students present ideas or solutions to the class.
• The class discusses the various solution methods.
• The teacher summarizes the class' conclusions.
• The students practice similar problems.

Now, right away I'm thinking that just because this works in Japan doesn't mean it will work here.  I can think of several roadblocks:  (1) BHS students aren't conditioned to "struggle" well with academic challenge. How do teachers help them to persevere long enough to come up with any ideas/solutions that can be discussed in class?  (2) BHS students aren't particularly interested in math enough to care about anyone's idea or solution method.  How do teachers help them to be interested in math?  (3) BHS students aren't so great at completing practice.  Whether they don't want to, or can't do it because of poor "basic skills" or whatever, we don't get much practice from them.  How do teachers help them to be competent enough to practice and to desire to practice?

With these challenges and questions in mind, I read on, hoping that the book would have some good insights that we can use at Bridgeport to liven up our math program.

Chapter 2 Ready, Set, Review

Instructional Shift 1:  Incorporate ongoing cumulative review into every day's lesson.

Leinwand describes a typical practice in math class--the warm up, or bellwork, or daily mini-review, whatever you want to call it.  He says that the important part to this review is the formative assessment done while "checking" it.  Each review problem offers opportunities to give some quick follow-up questions that hit important concepts.  Also, he says that there are two essential concepts to make sure that every daily review includes:  measurement and estimation.  He warns that many teachers disregard these two strands in their instruction in the hurry to get to algebra or geometry or whatever the curriculum is, but that they are key to students' success.  When I reflect on this, it make sense to me.  I think of all the times I and other teachers have complained that students don't know how to measure (with a ruler or using a clock, let's say), but then do we address this deficiency?  I also think about Dan Meyer's explanation of climbing the entire ladder of abstraction (math teachers may remember a video that I showed during a department meeting in 2011 - 2012).  Dan says that the time for estimation is before students work a more abstract procedure.  It is unnatural to solve a word problem using algebra, then estimate an answer to check if the worked solution makes sense.  Instead, the estimate should come first to provide a basis for the worked solution.  The chapter also made me think of other instructional strategies I've read about lately, namely that practice should be "distributed."  Students will not be successful if you teach a concept for a few days, hammer them with a ton of homework, and then they never see the stuff again.  Instead, the bulk of practice for this concept needs to occur only after students have mastered the concepts, so that they aren't practicing based on misunderstandings or maybe not even attempting the practice because they have no clue.  Leinwand's discussion of systematic cumulative review reminds me that math teachers should purposefully plan for review (in a sense distributed practice) every day for the school year.

Chapter 3 It's Not Hard to Figure Out Why Reading Works Better Than Math

Instructional Shift 2:  Adapt what we know works in our reading programs and apply it to mathematics instruction.

Leinwand describes a typical reading class in which students have read a sentence and the teacher asks some follow-up questions.

• "Can you read the sentence aloud?"
• "Can you tell me where Jane went?"
• "Can you tell me who went to the store?"
• "Can you tell me why Jane might have gone to the store?"
• "Do you think it made sense for Jane to go the store?"

He makes the point that reading teachers take students through a progression from literal to inferential to evaluative comprehension.  Math class should be like that.  "Good math instruction, like the follow-up questions that extend literal comprehension in reading, begins with an answer.  In effective instruction, you never stop when the same three students call out '19,' so you can move on to the next question.  Instead, we keep 'Does that make sense?'--that is, "Do you comprehend?'--in the foreground at all times" (p. 17).

Chapter 4 Picture It, Draw It

Instructional Shift 3:  Use multiple representations of mathematical entities.

Leinwand reminds us that in a typical class of 25 - 30 students only about half may "process the math being taught, see the math being taught, or feel the math being taught in the same way their teacher is seeing it" (p. 21).  Talk about a kick in the teeth!  What this means is that we need to provide students with many different ways to visualize the math they're learning and become knowledgeable about each student to know who responds to which hints.  Also in this chapter Leinwand explains the bar model that is used in Singapore Math.  His advice seems to be for teachers to use pictorial representations and for students to draw, show, and explain what they've drawn or shown.

Chapter 5 Language-Rich Classes

Instructional Shift 4:  Create language-rich classroom routines.

The point here is that sometimes students' problems aren't primarily caused by lack of math understanding, but by lack of understanding English, the basic terms and vocabulary in a problem.  Most of the chapter is devoted to an instructional activity that I suppose could be for a current topic or for review.  The teacher writes a prompt on the board (e.g., a number, an equation, a graph) and students brainstorm for a few minutes about what they see.  From the list that is generated the teacher composes follow-up questions.  In this way, important vocabulary terms are explored and reviewed with the intent to clarify common  misunderstandings.  The summary box (there's one at the end of each chapter) actually contains a little more information including a suggestion to use word walls, a strategy that's used in other content areas at BHS.

Chapter 6 Building Number Sense

Instructional Shift 5:  Take every available opportunity to support the development of number sense.

First, what's number sense?  Leinwand explains it as "a comfort with numbers that includes estimation, mental math, numerical equivalents, a use of referents like 1/2 and 50%, a sense of order and magnitude, and a well-developed understanding of place value" (p. 35).  He suggests a strategy for developing number sense.  Teachers should pause frequently and ask questions such as:

• Which is most or greatest?  How do you know?
• Which is least or smallest?  How do you know?
• What else can you tell me about those numbers?
• How else can we express that number?  Is there still another way?
• About how much would that be?  How did you get that?

I take this to mean that we should be milking the math problems for a lot more than what the question (in the textbook, for example) is asking.  Once you're beyond chapter 1 of the textbook, where basic things like estimation or place value are sometimes reviewed, these things rarely come up in a math class unless the teacher forcibly adds them back in.  Here's something that I've thought occasionally in the past several years as I'm teaching an algebra or geometry topic:  Maybe I'm teaching graphing linear equations, and the students aren't getting it.  "Why don't they understand?" I ask myself.  "They don't have basic skills; they just aren't fluent with the numbers," I sometimes answer.  But, I don't tend to add in this number sense-generating work to these linear graphing questions.  "Not enough time to teach elementary and middle school content and also teach algebra," I'll tell myself.  And so, many students don't succeed with graphing linear equations.  How are any of us better off for the decision not to take some class time and work on the foundation of number sense?  I've written a few other book summaries that are linked to the Livebinder by now, and this is reminding me of Rick Wormeli's advice regarding differentiated instruction:  give the students what they need!

Chapter 7 Milking the Data

Instructional Shift 6:  Build from graphs, charts, and tables.

The advice in this chapter seems kind of similar to that in Chapter 6.  Get more from a math problem than just what's asked in the textbook or worksheet or test.  In this case, when students are presented with a graph, chart or table, ask them "So?"  For example, you might ask student pairs to identify and justify three possible conclusions that can be drawn from data presented.  Several examples are given in the chapter with some suggestions about how a teacher might get the most mathematics out of them.  I appreciated how some of the data was open to interpretation and how a class discussion about students' conclusions would lead to important realizations about data, representations, and vocabulary.

Chapter 8 How Big, How Far, How Much?

Instructional Shift 7:  Tie the math to such questions as How big?  How much?  How far? to increase the natural use of measurement throughout the curriculum.

Math teachers will remember that Dan Meyer, who I referred to earlier, is well-known for using video imagery to hook students.  Often his question is as basic as "How long will it take for the tank to fill up?"  Or, "How deep is the hole?"  In fact, many "textbook" questions are really of this nature, but they seem disguised for some reason behind a bunch of algebra or geometry notation.  Leinwand wants math teachers to push measurement even more, especially by estimating, measuring with standardized units, and measuring with referential units.  This book is about "shifts" and it really would be a shift for us at BHS to do this because we're so used to sticking close to our topic for the day:  linear functions, factoring trinomials, hyperbolas, surface area of cylinders, etc.  In the past, we've justified this because of time--there wasn't time to do the section and a bunch of other stuff too.  But, under the PLA/Transformation Plan, 9th and 10th grade classes will suddenly have a luxury of greater time in double-dosed classes.  So, how should that time be used?  I think that this book is giving some answers to that:  (1) distribute practice, (2) ask students to conceptualize at a deeper level, (3) use more representations and have students draw, show, explain, (4) more emphasis on vocabulary, (5) milk problems for number sense-building experiences, (6) milk problems for multiple representations, (7) milk problems for measurement and estimation.

Chapter 9 Just Don't Do It!

Instructional Shift 8:  Minimize what is no longer important, and teach what is important when it is appropriate to do so.

This shift isn't so much instructional as it is curricular.  And, I don't know how much flexibility teachers really have here in this age of standards, the MMC and now the CCSS.  The Common Core authors keep talking about how they've pared down the curriculum to more closely align with successful nations, but come on!, there's still too content for teachers and students to deal with when students arrive in a course unprepared.  As I write this summary, the newspaper has just reported the Mackinac Center's rating of SASA in the top 20 high schools in the state.  Well, duh, to get into SASA a student has to have a demonstrated record of success, to be at grade level or above.  I bet lowly C-rated, PLA-listed BHS would do pretty well in math too if all of our students started 9th grade at grade level.  And, I'm not in any way disparaging 8th grade teachers or elementary teachers or parents of our students.  I don't know that I even subscribe to the point of view that it's the students themselves who are to blame when they're not successful. There are just too many things that all factor in to any individual student's performance in school.  My point of view as a teacher is that I can only control some of these factors, starting with my beliefs and then moving on to the instructional decisions that I make.  In that last respect I can choose to try to incorporate the strategies from this book.

Chapter 10 Putting It All in Context

Instructional Shift 9:  Embed the mathematics in realistic problems and real-world contexts.

Chapter 11 Just Ask Them "Why?"

Instructional Shift 10:  Make "Why?" "How do you know?" and "Can you explain?" classroom mantras.

Remember Leinwand's premise?  That many math classrooms aren't as successful as classes in other content areas and that students aren't as engaged in math as in other areas?  Most of the shifts have to do with getting math teachers to not opportunities.  I pretty much take this to mean that many math teachers miss opportunities that teachers in other areas don't miss.  Does the social studies teacher take advantage of opportunities for students to personalize the content?  Does the ELA teacher take advantage of opportunities for students to explain their thinking about literature that was read?  Does the science teacher make the content more real and easier to visualize through the use of experiments?  Math teachers could do these things too, but often we don't, maybe because of how we were taught by our teachers.  It worked for us, right?  It should work for them.  But, for most students, it's not working for them at BHS.  Maybe paying attention to these shifts can help.

Chapter 12 Punting Is Simply No Longer Acceptable

This chapter addresses the increased time and depth of planning necessary to enact the shifts and to teach all students to understand math.  It seems so perfectly in line with the type of lesson planning the Transformation Team is calling for at BHS that people might not believe that I came across this book only after the building-wide lesson plan form was created and the committee had developed the expectations for differentiated planning.  Leinwand says that planning needs to include:

• The math content of the lesson.  What skills or concepts are being developed or mastered as a result of this lesson?
• The math tasks of the lesson.  What specific questions, problems, tasks, investigations, or activities will students be working on during the lesson?  (At BHS, these may be differentiated)
• Evidence that the lesson was successful.  (Formative assessment and data teams, anyone?)
• Launch and closure.  Planning exactly how you will use the first 5 minutes of the lesson and outlining what summary will close the lesson and provide a foreshadowing of tomorrow.  (Primacy/Recency and pre-teaching)
• Notes and nuances.  A set of reminders about vocabulary, connections, common mistakes, and typical misconceptions that need to be considered before the lesson and kept in mind during it.  (All important things, but the mention of vocabulary here reminds me of our building-wide commitment to Marzano's 6-step process for teaching vocabulary.)
• Resources and homework.  (Well, at BHS we've committed to a No Homework policy for any double-dosed math classes in 2012 - 2013, but that's not to say that we don't need to plan how we'll distribute practice.)
• Post-lesson reflections.  (The BHS lesson plan form even has a spot for this.)

Chapter 13 We All Have a Role to Play and Teachers Can't Do It Alone

This chapter explore what students, teachers, and principals should all be doing if teachers are to have a fair shot at implementing the shifts.  Collaboration is also mentioned, and I'm gratified to know that we've already began this at Bridgeport through out 6 - 12th grade meetings last year (it was awesome to get to work with the middle school teachers) and a math data team this year.  The data team will just be a start because there's so much more that could be done:  collaboratively created lessons, videotaped lessons for collegial review (by the way, video recording will be a support I'll offer as an instructional coach--I can "tape" your class and we can examine the tape together for whatever data you'd like to look for, totally confidential, nobody will see the thing but you and me and you can have the only copy or delete it when we're done--see the Livebinder Instructional Coach tab for more information), review of course assessments, common readings/discussions for professional development.

All in all, I thought the book contained some valuable recommendations for math teaching.  It was short, not even 100 pages, so it didn't have as many examples of the practices in action as I would've liked, and those that were there were fictional scenarios.  I would've preferred real scenarios so that we know that the strategies really work and aren't idealized bunk.  But even so, there was just a lot of sense to what was being said, and the biggest obstacle seemed to be time and changing what are deeply ingrained habits.  The double-dosed classes will help us on the issue of time.  Our struggle will be our habits.  Hopefully the depth of planning called for by our lesson plan form, expectation to differentiate to groups of students, and the sharing of strategies during data team meetings will help us change habits.