In recent years mobile apps such as Angry Birds and Candy Crush Saga have become cult favorites, merging mobile apps into the entertainment sector. Developers from the London-based startup, Brainbow hope to do the same thing, only with educational apps. It seems they could be on their way with over 1 million people playing their app, 6 Numbers only 8 weeks after its launch for iOS reports Stuart Dredge of The Guardian.

The app is a puzzle game designed to help the user practice basic arithmetic and problem solving. Players aim to reach a given target total by adding, subtracting, multiplying and dividing six numbers.

UK gamers are spending 22-31 minutes playing 6 Numbers on a typical given day, according to Brainbow. While currently the game’s numbers don’t compare to is non-education counter parts, Brainbow plans to surpass them one day.

The company’s founding team has experience at Amazon, Google, Playfish, EA and the academic world, and is pitching itself as an “edugaming” startup that “turns knowledge into games”. Brain-training, as Nintendo’s Dr Kawashima character would call it.

“6 Numbers is becoming a phenomenon,” says chief operating officer Sagi Shorrer. “We’ve had 1m downloads, and active users in the hundreds of thousands. And it’s the number one puzzle game in many countries around the world.”

Brainbow’s first two games, 6 Numbers and Brainbow Count, were developed without any investors to prove the start-ups resilience and customer demand. Now the company’s investors include the likes of Initial Capital, DN Capital, London Venture Partners and Lifeline Ventures with $1.2 million in funds raised.

The funding will primarily be used to build a full time team to turn out a bigger game due to release this summer, according to chief product officer Xavier Louis.

The company hopes to go beyond math and develop games in all areas of education, with the belief that “any knowledge can be turned into games”.

Brainbow intends of keeping its apps free to play. They want the initial download to be free, and want the players to expect new puzzles as they go in the hopes consumers will play them for years to come.

Some features may eventually cost extra, but the exact details of what that will be has not be released. Currently players can make in-app purchases such as clues, to help them progress through the levels faster.

In the meantime, Brainbow is continuing to build its team, poaching former colleagues from larger companies to join their cause.

“These people have lived in big companies, but they see we have a mission. There’s a purpose here: we turn knowledge into games,” says Shorrer. “And that mission is combined with mobile, games, and data. They see the combination, and that we’re doing fascinating things.”

 

 

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Kids use the computer for everything from socializing to gaming to math homework – that is, until they reach algebra. In 1988 the Rand Corporation reported that instructional software is great for simple math drills and teaching basic procedures, but it faltered beyond that point. Even with today’s advances in technology the market lacks effective software to help students become proficient in algebra and higher levels of math.

John Barnes from Information Week points out that this is a surprising since a quarter of a century ago the technology was developed that teaches strategy, which is a necessary component in order for students to learn how to effectively solve problems on their own.

A basic algebraic equation can have at least six different strategies that could be used to yield the correct solution. The basis of how proficient a student is at solving these problems is their ability to decide which strategy is best for each equation. This becomes more important as students reach higher levels of math.

Yet most educational software does not teach strategy even though the technology has been available for a long time.

Jane Healy, an educational psychologist,  found that childhood environments also lacked development in another critical area for math: executive function.

It was another, bigger piece of the same problem Rand had found. Executive function is the part of the mind that plans, follows, assesses and re-plans a pathway through a complicated process. It’s the difference between following a recipe and cooking from scratch, painting by the numbers and painting, or running a checklist and fixing a motor. It’s essential for all applied math above the most basic level, as well as for critical thinking and everyday reasoning.

It’s imperative that students develop executive function by choosing and using problem solving strategies. Unfortunately, most software is merely just lecture material with animation, graphics and a self check function.

However, there are some programs that have been developed that teach partial strategy and executive function. Barnes has broken them down into three categories:

 – Hinters: These offer a strategy hint with each problem. They at least make students aware that there are strategies, and that your choice of them matters…

– Executors: These go a step further by asking the student to input a problem from a textbook, handout or other program, and then choose a strategy from a list. The software then follows that strategy to write a perfect show-your-work homework answer…

– Steppers: These not only enable strategy selection but also let students verify each step sequentially, encouraging them to try on their own rather than just copy perfect homework…

While these programs are a step in the right direction, the market still lacks software that properly teaches strategy. The good news is the technology exists to create it. Now someone just needs to have the vision and the initiative to do so.

The post Why is Math Education Software Lacking? appeared first on Education News.

A study shows that reading and math skills at age seven can predict how much money an adult will make, reports Lindsay Abrams at The Atlantic.

This surprising discovery was even unexpected by one of the lead researchers Stuart Ritchie, reports Rebecca Klein at the Huffington Post:

 “A lot of psychologists — including us before we did the study! — would have guessed that, since general intelligence is so important, specific skills like reading and math wouldn’t have any extra effects on SES beyond it,” Ritchie wrote. “But we found that these effects do exist — so no matter how smart people were … being better at reading and math at age seven was still significantly linked to SES aged 42.”

The study was lead by Stuart Ritchie and Timothy Bates of the University of Edinburgh. They wanted to what degree academic skill played into the overall equation for a successful life.

They looked at different measures of success at various points in the lives of over 17,000 residents of England, Scotland and Wales who were followed for 50 years after their birth. The measures focused on five different points in the participants lives. The first was socioeconomic class at birth, and researchers looked at the parent’s occupation and whether they owned or rented a home and what size it was.

The second measure was reading and math skills at age seven and how interested students seemed to be in learning subjects. The third measure was intelligence at age eleven and participants’ IQ scores. The fourth was academic motivation at age sixteen — specifically, they looked at how strongly participants agreed with statements such as “School is a waste of time”.

Finally the researchers looked at the students’ adult socioeconomic status at age 42. This included occupation, income and homeowner status.

The results showed that how much money people made at midlife was predicted by math skills at age seven, with a grade level boost in reading corresponded with a salary $7,750 higher at age 42.

Early reading ability proved to also be an indicator, but only girls. The good news is that according to Bates:

 “Math and reading are two of the most intervention-friendly of topics: Practice improves nearly all children.”

The study, titled “Enduring Links From Childhood Mathematics and Reading Achievement to Adult Socioeconomic Status,” is available in Psychological Science via SAGE Journals.

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This is Part Three of a three part article [Part One, Part Two] which provides the description of each of the Standards for Mathematical Practice as written in the Common Core math standards. It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.

by Barry Garelick

SMP 6: Attend to precision

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Being able to calculate accurately and to judge the degree of precision appropriate for a problem is an important skill as is using correct units of measure and labeling axes correctly.   This SMP also seems to be about providing explanations of one’s work; that is being able to show one’s work on a problem in such a way that others can follow how it was solved.  Showing the mathematical steps is for many if not most math teachers an explanation that “attends” to precision.  Students in early grades do not have the language ability to express such an idea which to them is innately obvious and therefore hard to express. Thus, a sensible way to interpret this SMP for the early grades, say K-6 is to let the math “does the talking”, which was previously known as “showing your work”.

Writing an explanation for one’s reasoning is another matter, however.  Many students asked to provide written explanations of their reasoning are stymied as to how to explain what mathematics does quite economically and efficiently.  They often respond: “But that’s what I just did,” or “It just is.”

Admittedly there is an advantage to learning how to express mathematical ideas in words.  Such skill is an essential part of constructing mathematical proofs, and therefore an asset to have in geometry and other math courses.  Thus, if learning to write a written explanation is a desired goal, students should be instructed in how to do so rather than 1) assuming that students automatically know how to do this if they truly “understand” or 2) that such goal is efficiently achieved by students engaging in group discussions to learn the technique from each other.

For example, consider the following problem: “The length of a rectangle is twice the width.  If the length were increased by 3 units and the width by 2 units, the area would be increased by 34 square units.  Find the length and width of the original rectangle.”  A student may readily solve this by representing the problem as (2w + 3)(w + 2) = 2w² + 34, where w and 2w are the width and length of the original rectangle.  To provide instruction on how to explain reasoning, the teacher could ask a student who has solved the problem to work the problem at the board, and ask the student questions.  “How did you represent the length and width?  What do 2w + 3 and w + 2 represent?  Why did you multiply them?  What does the expression 2w² + 34 represent?”  The teacher can also show how diagrams are part of the explanation as well as words.  Students receiving such instruction and doing this routinely once or twice a week in class, as well as providing such explanation for one or two problems in homework assignments will learn.

Expecting that students will learn such technique by working in groups and having discussions with other students is unrealistic. But the view of many reformers however, is that despite a student getting the right answer to a problem, the moment a student stops doing all the intermediate steps/algorithms and/or fails to explain in words how he or she solved the problem, then he or she is using a “trick” or “rote memorization” to jump to the end result, and/or lacks true “understanding” of the mathematical concepts involved. Such a view is inaccurate and unfair. Setting up the equations to solve complex problems requires a great degree of understanding. It entails understanding what the problem is asking, as well as how to express what’s going on in the problem mathematically.

SMP 7: Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

While observation, awareness and recognition of patterns is necessary in mathematics, it is not sufficient.  Some may interpret the SMP in this way, however and conclude that the habits of mind for pattern and structural recognition can and should be developed outside of the context of the material being learned—that is the vehicle which produces the patterns and structure in the first place.  For example, drawing auxiliary lines in geometry is important, but makes sense when students are given instruction in how that is done, and in the context of conducting proofs or solving problems.

Mathematics demands mastery of foundational steps in order to build upon them. As such, it is relentlessly linear.  The reason a coherent, sequential, efficient, and exercise-rich curriculum works is that the brain requires a great deal of repetition over time to consolidate learning in long term memory.  Without such a foundation, students will not be prepared to solve new and complex problems.  Proficiency is also unlikely to come about in a “problem-based learning” setting, in which a problem is posed that may require certain procedures and skills in order to solve the problem—such as factoring.  Having students learn the procedures on an “as needed” or “just in time” basis is ineffective.  Students need to master the skills in order for such procedures to be applied to problems.  Pattern and structure recognition alone won’t do it.

SMP 8: Look for and express regularity in repeated reasoning

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

It is important to make use of repetition in understanding the derivation of a rule.  While this can be done in a direct and efficient manner of instruction, the write up of this SMP can be interpreted as advocating a discovery type approach.  I.e., “By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3”.  Initial guidance about slopes and how to use them in determining if points are on a line can effectively build a foundation for solving more difficult systems later on.  Students can be given problems such as figuring out the slope, as an introduction and means for understanding the derivation of the point-slope form of a line (y₁-y) = (x₁-x)m.  But expecting all students to discover this is a result of working through checking whether points are on the line through a specific point and slope (e.g., (1,2) with slope 3) is unrealistic, as is the expectation that students will discover what repeating decimals are on their own.  Students can still be mathematically proficient even if he or she is provided an explanation. And in fact, once initial instruction and worked examples are provided, homework problems can be scaffolded in difficulty so that students are required to apply the basic information in situations that vary from the initial problem.

Conclusion

Implementing the SMPs using the straightforward and traditional techniques discussed above are what some math teachers have done for years.  On the other hand, those promoting reform-based practices are fearful that more traditional practices will lead to what they believe is an unsatisfactory outcome that they call “skills-based math”.

Based on articles in newspapers on how the SMPs are being interpreted, it is probably not inaccurate to say that the SMPs and the content standards themselves will continue to be implemented along the lines of the reform agenda.  SMPs will be pointed to as justifying the teaching of math in a “just in time” manner, and will foster bad habits of mind. The result will, in my opinion, leave many students with the task of finding the cat that is producing a confounding and puzzling grin.

Read Part One and Part Two of Standards for Mathematical Practice: The Cheshire Cat’s Grin. 

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.

The post Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Three appeared first on Education News.

This is Part Two of a three part article [Part One] which provides the description of each of the Standards for Mathematical Practice as written in the Common Core math standards. It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.

by Barry Garelick

SMP 3: Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

The skills described in this SMP are a necessary part of learning mathematics, and the standard is an appropriate one for students who have gained the understanding, vocabulary, and mathematical tools by which they can conduct such analyses. The analysis and arguments expected of students, therefore, must be appropriate to the grade level.  In lower grades, students are still developing the analytic tools and vocabulary by which to express mathematical ideas and arguments.   In K-5, therefore it is appropriate to have students observe a problem being worked, identify if the problem is being done correctly, and if not, explain what is being done wrong.

In higher grades such as pre-algebra and above, students now have the tools to express mathematical ideas symbolically and also have a greater mathematical vocabulary.  Analyzing arguments and mathematical reasoning can now be done by being able to express the mathematical ideas symbolically and reason and draw conclusions from their manipulation.  In geometry, analysis of arguments is very important since that subject requires students to prove propositions and theorems.

The danger of this SMP is that in early grades, an emphasis on argumentation and understanding may eclipse the importance of learning basic skills, and problem solving procedures.  Students in early grades would be expected to make arguments beyond just recognizing why an approach to a problem was wrong. The SMP states that “elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.”  But this is requiring arguments to be made with inefficient tools that are better made in the later grades when students have the tools to generalize in a formal manner.  Again, this SMP assumes that making such arguments, albeit inefficiently, creates the habit of mind of logic.  The SMP states that students will “reason inductively with data.”  Thus, as in SMP 2 discussed above, students will carry with them a grade school level of inductive reasoning that will not serve them well in higher level math courses.

SMP 4: Model with mathematics

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Modeling is admittedly a trendy term, but it generally means solving problems by representing a situation mathematically, and then solving it.  Using addition and proportions in early grades to solve problems as stated in the SMP has merit and is the approach taken with traditional mathematics teaching.

The traditional approach generally holds that there is one right answer. Such answer can be a set of numbers, called a “solution set”.   The reform approach to math extends the traditional approach by including open-ended and ill-posed problems in the belief that textbook problems are too “nice”. The fact that the textbook provides the data students need to solve the problems is viewed is an educational detriment which will not prepare students for the “real world” of having to find things out for themselves.  These beliefs lead to providing students with messy problems that are said to duplicate the types of problems that are encountered in the “real world” of problem solving where there is “more than one right answer”.

Thus, students are given problems where there is supposedly “more than one” right answer. For example, a problem may say that some children are given $40 to buy supplies for a party for 10 kids.  The problem lists a number of things that they could buy.  The students are asked to decide what to buy, but not go over the limit of $40. Educators don’t realize that mathematicians would define a merit function that codifies the personal choices. There are then mathematical solution techniques they use to find the one solution that meets their requirements. This is a known class of problems, but the math reform approach holds that by having students come up with multiple solutions, they are teaching students to think like mathematicians.  A mathematician would view the problem as having one optimal solution.

SMP 5: Use appropriate tools strategically

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

While spreadsheets and calculators are useful tools that students should learn how to use, mathematical proficiency goes beyond these tools, whether a student can use such tools “strategically” or not.  The SMP states that “mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator”.  In fact, mathematically proficient high school students should know how to graph functions by hand, by knowing the formulae and graphical representations of  conic sections, rational functions, exponential/logarithmic, and periodic functions.  In addition, proficiency includes the knowledge of how such functions are translated and shaped.

Being able to identify external mathematical resources on the internet is useful, but an emphasis on Googling for information at the expense of solving difficult and challenging problems is misguided at best.  The SMP’s opening statement that  “mathematically proficient students consider the available tools when solving a mathematical problem” should be interpreted to mean that at the high school level, the emphasis should be on applying knowledge of mathematical procedures and deductive reasoning–not which calculator or computer program would be best suited for solving a problem.

Part One is available here, and Part Three will be published on Friday, April 26.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.

The post Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Two appeared first on Education News.

This is Part One of a three part article which provides the description of each of the Standards for Mathematical Practice as written in the Common Core math standards. It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.

by Barry Garelick

The Common Core Standards for math are a set of guidelines written for both math and English language arts under the auspices of National Governors Association and the Council of Chief State School Officers. Where they are adopted, the Common Core standards will replace state standards in these subject areas, establishing more common ground for schools nationwide.

The Standards of Mathematical Practice (SMP) are a part of the Common Core math standards.  On the surface, and to those unaware of underlying concerns and issues, the SMPs appear reasonable.   It’s doubtful that any single mathematician or math teacher would disagree with anything in them, in principle.  They are process standards, which address the “habits of mind” of mathematics that are tied to the content standards.   The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular.

Habits of mind make sense when the habits arise naturally out of the material being learned.

Thus, a habit such as “Say in your head what you are doing whenever you are doing math” will have different forms depending on what is being taught. In elementary math it might be “One third of six is two”; in algebra “Combining like terms 3x and 4x gives me 7x”; in geometry “Linear pairs add to 180, therefore 2x + (x +30) = 180”.  Similarly, in fifth or sixth grade, students can learn to use the distributive property to multiply 57 x 3 as 3 x (50 + 7). In algebra, that habit expresses itself more formally: a(b + c) = ab + ac.

But developing “habits of mind” outside of the context of the material being learned is like the Cheshire Cat of Alice in Wonderland.  Such approach forces students to consider a grin well before they are presented with the cat associated with it.  And yet, this is how the SMP are being interpreted.  Based on statements made by school officials and others in education, it appears that the Common Core math standards in general, and the SMP in particular are following the tenets of the math reform ideology that has gained momentum over the last two decades.  In fact, a glance at the agendas of professional development seminars that are being given to teachers on implementing Common Core spend much if not the majority of time on the SMP rather than the content standards themselves.   In fact, the connection between the SMP and the content standards is made clear in the Common Core standards document itself:

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily.  (See  Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content: http://www.corestandards.org/Math/Practice )

Such explanation plays into an ongoing interpretation of the Common Core standards that downplays the importance of procedures and algorithmic efficiency in the name of “understanding”. The unrelenting search for “understanding” in the teaching of mathematics has often trumps the procedural skills and problem solving techniques that lead to such understanding in the first place.  The tension between “understanding” and procedural fluency is one of several significant tensions between two philosophies in math teaching which for lack of better terminology, I will call the “traditional” mode and the “reform math” mode.  The tensions between the two groups who practice and advocate each type have come to be known as “the math wars”.

Among those in the reform math area, there has been a push to interpret the SMPs along reform math ideologies that push certain mathematical “habits of mind” outside of the context in which such habits are learned, as well as a predominate use of collaborative group work and inquiry-based learning.  This article provides the description of each SMP as written in the Common Core math standards. (http://www.corestandards.org/Math/Practice)   It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.

SMP 1:  Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

The SMP writeup describes a problem solving mind-set as well as a variety of problem solving strategies that students should have. It is important to realize that the goal of this SMP comes about after years of experience and practice.  The ability to solve problems and think mathematically develops over time.  Problem solving cannot be taught directly; rather, it is based on mastery of many basic skills.  (See (http://www.ams.org/notices/201010/rtx101001303p.pdf )

Requisite for learning how to solve problems is an explanation of how specific types of problems are solved using worked examples and practice with routine problems.  A set of problems can then escalate in difficulty through careful scaffolding: i.e., by changing aspects of the problem so that students must apply their knowledge of the basic procedure to new forms of the problem.  In this way homework is not just a set of repetitive “exercises”.  Students progress from simple routine problems to those which increase in complexity and are non-routine.   The non-routine problems can then be extended into even more challenging problems.  Such challenging problems should definitely be given but students must be able to use prior knowledge of skills and procedures in solving them.  The goal of math teaching is to provide sufficient opportunities to apply skills and knowledge so that students know how to turn “problems” into routine exercises.

While the approach described above is a sensible and effective interpretation of this SMP, the reform math ideology that is dominating Common Core implementation is likely to reject it.  That philosophy is to regard math as some sort of magical thinking process.  It holds that “understanding” the problem and seeing the big picture is math, while the mechanics of problem solving are just a rote afterthought.  Worked examples and routine problems are generally disparaged as “non-thinking” and “routine achievement”. The reform approach usually manifests itself as giving students a steady diet of “challenging problems” in an effort to build up a problem solving habit of mind that is sometimes referred to as “sense-making”.  Such approach does not accomplish this, however.  Instead, the constant pursuit of “challenging problems” stands in the way of developing fluency with certain classes of problems and building on what one already knows.

The description of this SMP also states that students “consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.”  Such strategy comes from Polya’s classic advice on problem solving.  Students are told Polya’s rules for problem solving at early ages before such rules even make sense.  Polya intended his approach for upper level high school, and college students.  For lower grade students, Polya’s advice is not self-executing and has about the same effect as providing advice on safe bicycle riding by telling a child to “be careful”. For younger students to find simpler problems, they must receive explicit guidance from a teacher–i.e., the teacher often must provide the simpler problem for the student to then use as a template for solving the more difficult one.

SMP 2:  Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Like most of the SMPs, this one is a habit of mind.  The SMP promotes developing habits of mind used in abstract and quantitative reasoning.  It is not directly teachable, however.  Rather, it arises from the practice and mastery of specific mathematical skills and procedures.  Thus, one way to interpret this SMP is to provide students with sufficient instruction and practice in complex, multi-step problems that are appropriate to the class in which they are given.

While abstract and quantitative reasoning are important goals of algebraic thinking, the SMP opens itself up to the prevalent belief in the reform math camp that students can be taught various algebraic habits of mind outside of an actual algebra course.   An example of this type of thinking can be seen in a certain type of problem presented to students in early grades.  For example, the students are shown pictorial problems like black and white beads in a numbered series of growing sequential patterns.  The problem shows the first three patterns and asks students to predict the number of white beads in pattern 5, say.  Students in fifth grade have not yet learned how to represent equations using algebra.  Also, the problem is more of an IQ test than an exercise in math ability. Furthermore, such problems ignore the deductive nature of mathematics. An unintended habit of mind from such problem is to encourage inductive type reasoning.  Students then learn the habit of jumping to conclusions once they identify a pattern, thinking nothing further needs to be done.

Presenting problems outside of a pre-algebra or algebra course which require algebra to solve will likely result in clumsy attempts at solutions that may or may not lead to algebraic thinking. Algebraic thinking is not inherent at such a stage. But there is a big transition that students of these methods will have to make when moving to high school math which is still mostly taught traditionally. Students who use the inductive grade school understandings for the simple part simply can’t make the leap to complex. They see no need to learn actual “algebra” for easy problems because the old understanding works and they can do the problems in their heads.  They cannot, however, solve 2/3(6x + 24) = -3(x – 1) in their heads.  Many such students give up in frustration.

Parts Two and Three will be published on Wednesday, April 24 and Friday, April 26.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.

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If teachers make a substantial impact in student outcomes, students who are having difficulties with mathematics are victims of some very tough luck. According to recent studies on teacher effectiveness, qualified math teachers at a high school level are very difficult to find – especially in places where students chronically struggle with the subject.

How critical is a good teacher for a high schooler having troubles in mathematics? According to Sarah D. Sparks of EdWeek.org, if students don’t catch up to their classmates most of the way by the end of 9th grade, they’re at a much higher risk of dropping out of school before graduating.

It’s a nasty cycle, according to the study presented by Cara Jackson of the University of Maryland last month. In districts where students typically struggle in math, highly qualified math teachers could make the most difference. Yet these are exactly the districts and high schools where highly-qualified teachers are least likely to be found.

Teachers have been unevenly distributed both within schools, in that students in lower academic tracks have had less well-qualified teachers, and across schools, such that qualifications of teachers tend to be lower in disadvantaged, low-income, and high-minority schools . Not surprisingly, these inequities in students’ educational opportunities have been linked to disparities in educational outcomes. The achievement gap between more and less advantaged students can be attributed in part to the inequitable distribution of teachers across schools.

This discrepancy was one of the chief issues that the passage of No Child Left Behind was supposed to address. Yet 7 years after its passage, when it comes to schools that underperform chronically, students who are high-performers in math are still about 10% more likely to be assigned a highly-qualified teacher than those who underperform. The trend appears to be exactly opposite in schools where children are better math students on average. In such cases, administrators appear to assign the best qualified teachers to students who are struggling most.

In short, if you’re struggling in math, your best odds for getting a teacher most able to help you improve is to attend a high-performing school, even if that means enrolling with students who outpace you in the subject.

Though the research on school context suggests teachers prefer working with higher-income and white students, Hanushek, Rivkin and Kain (2004) acknowledged that student characteristics may be proxies for other factors that shape teachers’ preferences. That is, if lower income and minority students attend schools with less attractive working conditions, the patterns of teacher behavior that suggest a preference for wealthier and whiter students might be at least partially explained by preferences for better working conditions. Ingersoll’s work suggests school staffing problems result from a “revolving door”, where large numbers of qualified teachers depart their jobs out of dissatisfaction with aspects of the school environment, such as student discipline problems.

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Many Seattle seniors preparing for graduation might find out that their plans have to be deferred, The Seattle Times reports. The issue is the state’s graduation requirement that asks seniors to pass either a state-mandated math exam or one of a handful of alternatives to receive their diploma this spring.

This is the first year that the math exam is mandatory for high school students. Although Washington State has had reading and writing graduation requirements in place, the addition of a math exam is an attempt to prove that high school diplomas granted by the state are not meaningless. Yet even as the graduation date gets closer, a growing number of kids have still not taken steps to even take the exam, much less actually pass it.

The state has not yet tallied exactly how many students are in jeopardy. A few weeks ago, about 8,000 students statewide had passed the state’s reading and writing exams, but not math.

Since then, many learned that they passed in the most recent round of state math tests in January, or earned a high-enough score on a collection of math work they submitted to the state. Such portfolios are one of the alternative ways to meet the math requirement.

Still, that leaves potentially several thousand students whose futures continue to be up in the air. In Seattle, about 90 students are in that category – planning for graduation without having yet passed the exam.

For them, this past week has not been kind. Many only found their results on Friday – the day after graduation announcements had been published. According to school staff, a number broke down in tears in front of their parents and counselors when told that they were at risk of not graduating on time.

State reading and writing exams became graduation requirements in 2008, but lawmakers hesitated when it came to math because half the state’s sophomores then were failing the state math test on their first try.

In the past few years, students who failed the math exam could still graduate as long as they kept taking — and passing — math classes.

Not so this year, though some are still trying to get lawmakers to agree to another delay.

Many of Seattle’s high- school principals have asked Superintendent José Banda to ask Randy Dorn, the state’s top education official, for a waiver for students for whom math is the only obstacle to graduation. Banda and his top staff are considering that request, a school-district spokeswoman said.

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Math and reading gender gaps persist even in countries working hard to eliminate them, reports Futurity.org’s Rachel Barson-Leeds. Despite the fact that the UK invests extensively in equality efforts, it continues to have the widest gender gap in the world in math, followed closely by the United States, Germany and the Netherlands.

Gijsbert Stoet of the University of Leeds, one of the authors of a study that looked at 1.5 million students from 75 countries, says that the gender gap in math and science persists and is even wider at the top – between the best performing girls and boys. This is a discouraging finding, especially in light of a number of recent reports suggesting that the performance gap is closing.

Stoet goes on to say that this is likely to continue to have an impact on the efforts to bring more women into the STEM fields because there continues to be a gap in performance prior to students entering college and university.

Published in the journal PLOS ONE, the study comes at a time where there are calls in the UK to increase the number of students studying math after the age 16. Recent findings from the Nuffield Foundation, a charitable trust funding research in education, has also identified performance and attitudes towards math are important factors in determining whether or not young people continue with the subject.

At the same time that experts continue to stress over the gender gap in math, they also need to make room for similar problems when it comes to reading. The study shows that girls continue to substantially outperform boys in reading, but with the gap being widest with low-performers.

Geary points out that at its extreme, the gap between genders when it comes to reading is three times wider than the gap in math.

The researchers say girls’ higher scores in reading could lead to advantages in admissions to certain university programs, such as marketing, journalism or literature, and subsequently careers in those fields. Boys lower reading scores could correlate to problems in any career, since reading is essential in most jobs.

“The gender gap in reading gets less attention in the media than the gender gap in maths,” Stoet says. “Yet there was not a single country in which boys exceed girls in reading. And crucially, amongst boys and girls at the lower end of academic performance scale, the gap is not only greater, it is growing.

The good news is that a country that focuses on closing the gender disparity is likely to be rewarded in at least one way. In countries like the UK and the US where the math gap is large, the reading gap is much smaller. In the countries where the gap in performance between women and men in math is smaller, the reading gap is larger.

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The Presbyterian Day School in Memphis, Tennessee has adopted a new method of learning in their classrooms. Gone are the desks facing the blackboard with the teachers in front of it, teaching the same concept to all the students regardless of their individual skills. Instead, the room – which now resembles an employee lounge in a new tech start-up more than anything else – is filled with boys of different ages moving along at their own pace through their math curriculum using modern technology.

What makes it possible is the rigorous regiment of data collection by the school which allows the learning platform to adapt to the skills of every student from the gifted to the struggling. According to headmaster Lee Burns, even in the small classroom the range of ability is wide. Burns says at any time students could be learning material from grades 2 through 9.

Although it was considered a risky experiment, it appears to be working if the math assessment exams are any indication.

Eight days are allotted for each math unit. Boys who think they understand the concepts may take the unit test on Day 4. If they score 90 percent or higher, they’re off to “guided challenge,” a range of math activities that allow them to work on more sophisticated problems, either virtually or with a teacher. Or they might look at the real-life ways math comes into play in people’s lives.

Those who don’t master the materials in the time allotted go to a different space in the building where more intensive help could be provided. There they will work with a tutor over the internet – using messaging tools like Skype – and use adaptive software to test their knowledge so they won’t progress too rapidly or too slowly. They will rejoin their classmates when they’ve caught up and taken the exam again on Day 8.

“Our scores are stronger than before,” Burns said. “And anecdotally, students are much more engaged. So many boys now say their favorite subject is math. They love the approach. They own the learning, so there is great engagement in it.”
PDS is in its second year of smart math, a concept it adapted from things happening in charter schools and other public schools.

One of the schools that served as a model for PDS is the Rocketship Academy in California. Rocketship Schools, a chain of charters around the country, have been experimenting with technology in learning to an unprecedented degree.

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